diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmahj" "b/data_all_eng_slimpj/shuffled/split2/finalzzmahj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmahj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nE-mail, or electronic mail, is one of the most popular forms of communication in the world, with\nover~3.9 billion active email users~\\cite{campaign_monitor}. As a side effect of this rapid growth, \nthe number of unwanted bulk email messages---i.e., spam messages---sent with commercial \nor malicious intent has also grown. According to~\\cite{campaign_monitor},\n60 billion spam emails will be sent each \nday for the next three years. \n\nWhile text-based spam filtering systems are in use by most, \nif not all, e-mail clients~\\cite{machine_learning_for_email_spam_filtering}, spammers \ncan embed messages in attached images to evade such systems---such messages are known as \nimage spam. Image spam detectors based on optical character recognition (OCR) have been \ndeployed to combat such e-mail. As a countermeasure, spammers can modify images \nso as to disrupt OCR based techniques~\\cite{image_spam_hunter}. \n\nIn recent years, deep learning models, such as multi-layer perceptrons and convolutional neural networks, \nhave been successfully applied to the image spam problem~\\cite{image_spam_hunter, \nimage_spam_analysis_and_detection, statistical_feature_extraction_for_classification, \nsupport_vector_machines_for_image_spam, deepimagespam, image_spam_filtering_using_conv, \nconvolutional_neural_networks_for_image_spam}. Note that these techniques do not rely\non OCR, but instead detect image spam directly, based on characteristics of the images.\n\nWith the recent development of perturbation methods, the possibility exists for spammers \nto utilize adversarial techniques to defeat image-based machine learning \ndetectors~\\cite{intriguing_properties}. To date, we are not aware of perturbation techniques\nhaving been used by image spammers, but it is highly likely that this will occur in the\nnear future.\n\nThe main contributions of our research are the following.\n\\begin{itemize}\n\\item We show that the universal perturbation adversarial attack is best suited for the \ntask of bypassing deep learning-based image spam filters.\n\\item We propose a new image transformation-based attack that utilizes the maximization \nof layer activations to produce spam images containing universal perturbations. This technique\nfocuses perturbations in the most salient regions, as well as concentrating natural features \nin the remaining regions.\n\\item We compare our proposed adversarial technique to existing attacks and find that our \napproach outperforms all others in terms of accuracy reduction, computation time per example, \nand perturbation magnitude.\n\\item We generate a large dataset containing both non-spam and adversarial spam images using \nour proposed attack. The authors will make this dataset available to researchers.\n\\end{itemize}\n\nThe remainder of this chapter is organized as follows. In Section~\\ref{sec:bac}, we provide\nan overview of relevant research and related work.\nIn Section~\\ref{sec:adv}, we evaluate adversarial attacks in the context of image spam, \nand in Section~\\ref{sec:univ}, we present our proposed attack.\nFinally, Section~\\ref{sec:con} concludes this chapter, where we have included\nsuggestions for future work.\n\n\n\\section{Background}\\label{sec:bac}\n\n\\subsection{Image Spam Filtering}\n\nThe initial defenses against image spam relied on\noptical character recognition (OCR). In such OCR-based systems, \ntext is extracted from an image, at which point\na traditional text-based spam filter can be used~\\cite{spam_assassin}. \nAs a reaction to OCR-based techniques,\nspammers introduced images with slight modifications, such as \noverlaying a light background of random artifacts on images, which are\nsufficient to render OCR ineffective. The rise of learning algorithms, however, \nhas enabled the creation of image spame filtering systems based directly on \nimage features. \n\nIn~2008, a filtering system using a global image feature-based probabilistic boosting \ntree was proposed, and achieved an~89.44\\%\\ detection rate with a false positive rate \nof~0.86\\%~\\cite{image_spam_hunter}. Two years later, an artificial neural network for \nimage classification was proposed~\\cite{statistical_feature_extraction_for_classification}. \nThese latter authors used were able to classify image spam\nwith~92.82\\%\\ accuracy based on color histograms, \nand~89.39\\%\\ accuracy based on image composition extraction. \n \nThe two image spam detection methods presented \nin~\\cite{image_spam_analysis_and_detection} rely on principal component analysis (PCA)\nand support vector machines (SVM). In addition, the authors \nof~\\cite{image_spam_analysis_and_detection} introduce a new dataset that their \nmethods cannot reliably detect. \nTwo years later, the authors of~\\cite{support_vector_machines_for_image_spam} improved \non the results in~\\cite{image_spam_analysis_and_detection} by training a linear SVM \non~38 image features, achieving~98\\%, \naccuracy in the best case. The authors also introduce a challenge dataset that is even \nmore challenging than the analogous dataset presented \nin~\\cite{image_spam_analysis_and_detection}. \n \nThe recent rise of deep learning, a subfield of machine learning, coupled with advances \nin computational speed has enabled the creation of filtering systems capable of considering \nnot only image features, but entire images at once. In particular, \nconvolutional neural networks (CNN) \nare well suited to computer vision tasks due to their powerful feature extraction capabilities. \n\nIn recent years, CNNs have been applied to the task of image spam detection.\nFor example, in~\\cite{image_spam_filtering_using_conv} a CNN is trained on \nan augmented dataset of spam images, achieving a~6\\%\\ improvement in accuracy,\nas compared to previous work. Similarly, the authors of~\\cite{deepimagespam} \nconsider a CNN, which achieved~91.7\\%\\ accuracy. \nIn~\\cite{convolutional_neural_networks_for_image_spam},\na CNN-based system is proposed, which achieves an accuracy of~99\\%\\ on \na real-world image spam dataset,~83\\%\\ accuracy \non the challenge dataset in~\\cite{image_spam_analysis_and_detection}\n(an improvement over previous works), \nand~68\\%\\ on the challenge dataset in~\\cite{support_vector_machines_for_image_spam}. \n\nFrom the challenge datasets introduced in~\\cite{image_spam_analysis_and_detection} and~\\cite{support_vector_machines_for_image_spam}, we see that\nthe accuracy of machine learning-based filtering systems can be \nreduced significantly with appropriate modifications to spam images. In this research, \nwe show that the accuracy of \nsuch systems can be reduced far more by using the adversarial learning\napproach that we present below. \n\n\\subsection{Adversarial Learning}\n\nThe authors of~\\cite{intriguing_properties} found that by applying an imperceptible \nfilter to an image, a given neural network's prediction can be arbitrarily changed. \nThis filter can be generated from the optimization problem\n\\begin{equation}\\nonumber\n \\begin{split}\n& \\textbf{minimize } \\|r\\|_2 \\\\\n&\\textbf{subject to } f(x+r) = l \\text{ and } x+r \\in [0, 1]^m\n \\end{split}\n\\end{equation}\nwhere $f$ is the classifier, $r$ is the minimizer, $l$ is the target label, and $m$ is the dimension of the image. \nThe resulting modified images are said to be \\textit{adversarial examples}, and the attack presented in~\\cite{intriguing_properties} is known as the \\textit{L-BFGS Attack}. These adversarial examples generalize well to different network architectures and networks.\n \nMore recently, many advances have been made in both adversarial example generation and \ndetection. For example, in~\\cite{adversarial_examples_attacks_and_defenses} a taxonomy \nis proposed\nfor generation and detection methods, as well as a threat model. Based on this threat model, \nthe task of attacking neural network-based image spam detectors requires an attack that is \nfalse-negative (i.e. generative of positive samples misclassified as negative) and black-box \n(i.e. the attacker does not have access to the trained model). \nAttacks on image spam classifiers must satisfy these two criteria. \n\nAfter the introduction of the L-BFGS Attack, the authors of~\\cite{explaining_and_harnessing} \nbuilt on their work in~\\cite{intriguing_properties} by introducing \nthe \\textit{Fast Gradient Sign Method} (FGSM). This method uses the gradient of the \nloss function with respect to a given input image to efficiently create a new image that \nmaximizes the loss, via backpropagation. This can be summarized with the expression\n$$\n \\mbox{adv}_x = x + \\epsilon\\,\\mbox{sign}\\bigl(\\nabla_x J(\\theta, x, y)\\bigr)\n$$\nwhere $\\theta$ is the parameters of the model, $x$ is the input image, $y$ is the target label, \nand $J$ is the cost function used to train the model.\nThese authors also introduce the notion that adversarial examples result from linear \nbehavior in high-dimensional spaces. \n\nThe authors of~\\cite{towards_evaluating_the_robustness} introduce \\textit{C\\&W's Attack}, \na method designed to combat \\textit{defensive distillation}, which consists of training a \npair of models such that there is a low probability of successively attacking both models. \nC\\&W's Attack is a non-box constrained variant of the L-BFGS Attack that is more easily \noptimized and effective against both distilled and undistilled networks. \nThey formulate adversarial example generation as the optimization problem\n\\begin{equation}\\nonumber\n \\begin{split}\n& \\textbf{minimize } D(x, x + \\delta) + c \\cdot f(x + \\delta) \\\\ \n& \\textbf{such that } x + \\delta \\in [0, 1]^n \n\\end{split} \n\\end{equation}\nwhere $x$ is the image, $D$ is one of the three distance metrics described below, and $c$ is a suitably chosen constraint (the authors choose $c$ with binary search). \nThe authors also utilize three distance metrics for measuring perturbation: $L_0$ (the number of altered pixels), $L_2$ (the Euclidean distance), and $L_{\\infty}$ (the maximum change to any of the coordinates), and introduced three subvariants of their attack that aim to minimize each of these distance metrics. \n \nIt is important to note that the previously mentioned attacks require knowledge of the classifier's gradient and, as such, cannot be directly deployed in a black-box attack. In~\\cite{practical_black_box}, the authors propose using a surrogate model for adversarial example generation to enable the transferability of adversarial examples to attack black-box models. Differing from gradient-based methods, the authors of~\\cite{zoo} introduced a method, \n\\textit{Zeroth Order Optimization} (ZOO), which is inspired by\nthe work in~\\cite{towards_evaluating_the_robustness}.\nThe ZOO technique employs gradient estimation, with the most significant downside \nbeing that it is computationally expensive. \n\nThe paper~\\cite{deepfool} introduces the \\textit{DeepFool} attack, which aims to find the minimum\ndistance from the original input images to the decision boundary for adversarial examples. \nThey found that the minimal perturbation needed for an affine classifier is the distance to the separating affine hyperplane, which is expressed (for differentiable binary classifiers) as \n\\begin{equation}\\nonumber\n\\begin{split}\n& \\textbf{argmin}_{\\eta_i} \\|\\eta_i\\|_2 \\\\\n& \\textbf{such that } f(x_i) + \\nabla f(x_i)^T \\eta_i = 0 \n\\end{split}\n\\end{equation}\nwhere $i$ denotes the iteration, $\\eta$ is the perturbation, and $f$ is the classifier.\nIn comparison to FGSM, DeepFool minimizes the magnitude of the perturbation, instead of the number of selected features. This would appear to be ideal for spammers, since it would tend to minimize the effect on an image.\n \nThe \\textit{universal perturbation} attack presented in~\\cite{universal} is also suited to \nthe task at hand. We believe that universal adversarial examples are most likely \nto be deployed by spammers against black-box models due to their simplicity \nand their transferability across architectures. Generating universal \nperturbations is an iterative process, as the goal is to find a vector $v$ that satisfies \n$$\n \\|v\\|_p \\leq \\xi \\mbox{\\ \\ and\\ \\ }\n \\mathbb{P}_{x\\sim \\mu} (\\hat{k}(x+v) \\neq \\hat{k}(x)) \\geq 1 - \\delta\n$$\nwhere $\\mu$ is a distribution of images, $\\hat{k}$ is a \nclassification function that outputs for each image $x$ and a label $\\hat{k}(x)$.\nThe results in~\\cite{universal} show that universal perturbations are misclassified with high probability, suggesting that the existence of such perturbations are correlated to certain regions of the decision boundary of a deep neural network.\n\nFinally, the authors of~\\cite{restoration_as_a_defense} propose input restoration with a \npreprocessing network to defend against adversarial attacks. \nThe authors' defense improved the classification precision of a CNN \nfrom~10.2\\%\\ to~81.8\\%, on average. These results outperform \nexisting input transformation-based defenses. \n\n\\section{Evaluating Adversarial Attacks}\\label{sec:adv}\n\n\\subsection{Experimental Design}\n\nThe two multi-layer perceptron and convolutional neural network architectures presented in~\\cite{convolutional_neural_networks_for_image_spam} are each trained on both of\nthe dataset presented in~\\cite{image_spam_hunter}, which henceforth will be referred to \nas the \\textit{ISH Dataset}, and the dataset presented \nin~\\cite{support_vector_machines_for_image_spam}, which henceforth will be referred to as \nthe \\textit{MD Dataset} (modified Dredze). We use TensorFlow~\\cite{tensorflow} to train\nour models---both architectures have been trained as they were presented in their \nrespective articles on each of the datasets. NumPy~\\cite{numpy} and \nOpenCV~\\cite{opencv} are used for numerical operations and image processing \ntasks, respectively. All computation are performed on a\nlaptop with~8GB ram, using Google Colaboratory's Pro GPU. \n\nThe ISH Dataset contains~928 spam images and~830 non-spam images, \nwhile the MD Dataset contains~810 spam images and~784 non-spam images;\nall images in both datasets are in \\textit{jpg} format. These datasets are summarized\nin Table~\\ref{tab:data}.\n\n\\begin{table}[!htb]\n\\caption{Image spam datasets}\\label{tab:data}\n\\centering\n\\begin{tabular}{ccc}\n\\midrule\\midrule\n\\textbf{Name} & \\textbf{Spam images} & \\textbf{Non-spam images} \\\\\n\\midrule\nISH dataset & \\zz928 & \\zz830 \\\\\nMD dataset & \\zz810 & \\zz784 \\\\\n\\midrule\nTotal & 1738 & 1613 \\\\\n\\midrule\\midrule\n\\end{tabular}\n\\end{table}\n\nDataset preprocessing for the networks \npresented in~\\cite{convolutional_neural_networks_for_image_spam} consist \nof downsizing each of the images such that their dimensions are~$32\\time 32\\times 3$, \napplying zero-parameter Canny edge detection~\\cite{canny} to a copy of the downsized \nimage, and concatenating the downsized image with the copy that had \nCanny edge detection applied. This process results in~$64\\time 32\\time 3$ images, \nwhich are used to train the two neural networks, one for the ISH dataset, and one for the MD dataset. \nThe four resulting models achieved accuracies within roughly~7\\%\\ of the accuracies \nreported in~\\cite{convolutional_neural_networks_for_image_spam}. \n\nTo enable the generation of adversarial examples, four larger models with an input size of 400x400 are also trained on \nthe original datasets. The first few layers of each of these models are simply used to \ndownscale input images such that the original architectures can be used after downscaling. \nThese four alternative models achieve accuracy roughly equivalent to the \noriginal models. The four adversarial attacks \n(FGSM, C\\&W's Attack, DeepFool, and Universal Perturbation) \nutilize these four alternative models to generate adversarial examples that can then be formatted \nas the original datasets to attack the original four models. This procedure attempts to exploit \nthe transferability of adversarial examples to similar architectures. \n\nThe IBM Adversarial Robustness Toolbox (ART)~\\cite{adversarial_robustness_toolbox} is \nused to implement C\\&W's Attack, DeepFool, \nand Universal Perturbations, while FGSM was implemented independently from scratch.\nAn attempt was made to optimize the parameters of each technique---the resulting\nparameters are summarized in Table~\\ref{tab:parms}. Note that\nfor the Universal Perturbation attack, FGSM was used as the base attack, as the IBM ART \nallows any adversarial attack to be used for computing universal perturbations.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n\\caption{Attack parameters}\\label{tab:parms}\n\\centering\n\\begin{tabular}{c|ll}\n\\midrule\\midrule\n\\textbf{Attack} & \\textbf{Description} & \\textbf{Value} \\\\\n\\midrule\nFGSM \n & perturbation magnitude & 0.1 \\\\ \\midrule\n\\multirow{6}{*}{C\\&W's attack} \n & target confidence & 0 \\\\ \n & learning rate & 0.001 \\\\\n & binary search steps & 20 \\\\ \n & maximum iterations & 250 \\\\\n & initial trade-off & 100 \\\\\n & batch size & 1 \\\\ \\midrule\n\\multirow{4}{*}{DeepFool}\n & max iterations & 500 \\\\ \n & overshoot parameter & $10^{-6}$ \\\\ \n & class gradients & 10 \\\\\n & batch size & 1 \\\\ \\midrule\n\\multirow{4}{*}{Universal Perturbation}\n & target accuracy & 0\\% \\\\ \n & max iterations & 250 \\\\\n & step size & 64 \\\\\n & norm & $\\infty$ \\\\\n\\midrule\\midrule\n\\end{tabular}\n\\end{table}\n\nThe metrics used to evaluate each of the four attacks are the average accuracy, \narea under the curve (AUC) of the receiver operating characteristic (ROC) curve, average~$L_2$ perturbation measurement \n(Euclidean distance), and average computation time per example for each of the four models. \nScikit-learn~\\cite{scikit-learn} was used to generate the ROC curves for each attack. \n\nWe use~251 data points for accuracy \nand $L_2$ distances collected for the FGSM, DeepFool, and Universal Perturbation experiments, \nin accordance with the full size of the test dataset, which contains~251 examples for generating adversarial examples. However, only~$28$ data points were \ncollected from the C\\&W's Attack experiment due to \nthe large amount of time required to generate each data point (roughly five minutes per data point). \nThe technique that will be used as the basis of our proposed attack will be selected based \non the performance of each attack, as presented in the next section.\n\n\\subsection{Analysis}\n\n\nThe mean accuracy, computation time per example, and~$L_2$ distance were \nrecorded for each of the four models attacked by each of the attack methods. \nThis data was compiled into the tables discussed in this section. \n\nFrom Table~\\ref{tab:meanacc1} we see that for FGSM, the accuracy of the attacked models \nis shown to vary inconsistently while Figure~\\ref{fig:fgsml2} shows that the distribution of \nthe~$L_2$ distances of the generated adversarial examples skew right. \nBased on these results \nand corresponding density plots of the accuracy and~$L_2$ distance distributions, \nthe FGSM attack can be ruled out as a candidate due to poor accuracy. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mean accuracy per adversarial example}\n \\label{tab:meanacc1}\n \\begin{tabular}{c|cccc}\n \\midrule\\midrule\n \n \\multirow{2}{*}{\\textbf{Model}}\n & \\multirow{2}{*}{\\textbf{FGSM}}\n & \\textbf{C\\&W's}\n & \\multirow{2}{*}{\\textbf{DeepFool}}\n & \\textbf{Universal}\\\\\n & & \\textbf{Attack} & & \\textbf{Perturbation}\\\\\n \\midrule\n MLP (ISH) & 95.2\\% & 89.2\\% & 98.8\\% & 98.7\\%\\\\\n CNN (ISH) & 36.2\\% & 49.6\\% & 61.5\\% & 49.9\\%\\\\\n MLP (MD) & 69.7\\% & 75.6\\% & 93.5\\% & 94.3\\%\\\\\n CNN (MD) & 82.8\\% & 77.2\\% & 14.5\\% & \\zz8.4\\%\\\\\n \\midrule\\midrule\n \\end{tabular}\n\\end{table}\n\nThe mean~$L_2$ (Euclidean) distances of the adversarial examples\nare given in Table~\\ref{tab:meandist1}. The distribution of distances appears to be roughly equivalent across all attacks.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering \n \\caption{Mean $L_2$ (Euclidean) distance of adversarial examples from original images}\n \\label{tab:meandist1}\n \\begin{tabular}{c|cccc}\n \\midrule\\midrule\n \n \\multirow{2}{*}{\\textbf{Model}}\n & \\multirow{2}{*}{\\textbf{FGSM}}\n & \\textbf{C\\&W's}\n & \\multirow{2}{*}{\\textbf{DeepFool}}\n & \\textbf{Universal}\\\\\n & & \\textbf{Attack} & & \\textbf{Perturbation}\\\\\n \\midrule\n MLP (ISH) & 11537.55 & 10321.77 & 11513.26 & 11483.72\\\\\n CNN (ISH) & 11108.44 & 10924.14 & 11216.19 & 11416.58\\\\\n MLP (MD) & \\zz8998.71 & \\zz9185.04 & \\zz9566.02 & \\zz9490.56\\\\\n CNN (MD) & \\zz9144.49 & \\zz9009.91 & \\zz9128.99 & \\zz9381.15\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=7.8cm]{images\/FGSMl2.png}\n\\caption{Density plot of~$L_2$ (Euclidean) distances (Fast Gradient Sign Method)}\n\\label{fig:fgsml2}\n\\centering\n\\end{figure}\n\nDeepFool can also be ruled as a candidate, as the attack has been seen to be \nonly marginally better than the FGSM attack in terms of performance, while also \nhaving a significantly higher average computation time per adversarial example. This\ncan be observed in Table~\\ref{tab:meancomp1}, where the computation time \nper example varies greatly. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mean computation time per adversarial example}\n \\label{tab:meancomp1}\n \\begin{tabular}{c|cccc}\n \\midrule\\midrule\n \n \\multirow{2}{*}{\\textbf{Model}}\n & \\multirow{2}{*}{\\textbf{FGSM}}\n & \\textbf{C\\&W's}\n & \\multirow{2}{*}{\\textbf{DeepFool}}\n & \\textbf{Universal}\\\\\n & & \\textbf{Attack} & & \\textbf{Perturbation}\\\\\n \\midrule\n MLP (ISH) & 0.180 & 269.65 & 19.90 & 4.37\\\\\n CNN (ISH) & 0.038 & 251.01 & \\zz4.75 & 2.87\\\\\n MLP (MD) & 0.164 & 270.58 & 36.30 & 3.71\\\\\n CNN (MD) & 0.165 & 244.47 & \\zz1.48 & 5.23\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\nIn contrast, C\\&W's Attack shows consistent performance in all three metrics at the cost of \nhigh computation time (roughly five minutes per adversarial example).\nThe consistency of this attack is ideal from a spammer's perspective, \nthough the trade-off is a relatively high computation time. In addition, the \nleft skew of this attack with respect to~$L_2$ distance, as presented in Figure~\\ref{fig:cwl2}, \nindicates that the perturbation made to spam images is much lower in comparison to the other attacks.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=8cm]{images\/CWl2.png}\n\\caption{Density plot of~$L_2$ (Euclidean) distances (C\\&W's Attack)}\n\\label{fig:cwl2}\n\\centering\n\\end{figure}\n\nThe Universal Perturbation attack is inconsistent in terms of accuracy, as shown \nin Table~\\ref{tab:meanacc1} where the mean accuracy across the four models is clearly \nshown to fluctuate wildly, but this \nis simply due to the fact that only one perturbation (albeit with varying success across architectures) is applied to all spam images, which is highly advantageous for spammers.\nThe generation and application of this perturbation to an image takes roughly four seconds,\nwhich would result in greater performance in a real-world spam setting in comparison \nto C\\&W's Attack. \n \nTo further compare C\\&W's Attack and the Universal Perturbation attack, \nthe ROC curves of the two are presented in Figure~\\ref{fig:cwroc_uproc}.\nThese ROC curves can be used to quantify the diagnostic ability of the \nmodels attacked by each method.\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/CWroc.png}\n \\end{minipage}\n \\begin{minipage}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/UProc.png}\n \\end{minipage}\n \\caption{ROC curves of C\\&W's Attack and the Universal Perturbation when used to attack the four classifiers}\n \\label{fig:cwroc_uproc}\n\\end{figure}\n\nThe ROC curve for C\\&W's attack is much noisier due to being generated from only~28 \ndata points. Taking this into consideration, it can be inferred that both C\\&W's Attack \nand the Universal Perturbation attack are able to reduce the areas under the ROC curve (AUC) \nof the attacked models to values close to~0.5. This suggests that both attacks are able to reduce the class separation capacity of attacked image spam classifiers to essentially random. \n\nTo analyze the differences in distribution of the accuracy and~$L_2$ distance data collected from \nthe trials conducted on C\\&W's Attack and the Universal Perturbation attack, the Mann-Whitney U Test was utilized via its implementation \nin SciPy~\\cite{scipy}. The Mann-Whitney U Test compares two populations---in \nthis case, the accuracy and~$L_2$ distance data from both attacks for each attacked model.\nThe null hypothesis (H0) for the test is that the probability is~50\\%\\ that a randomly drawn \nvalue from the first population will exceed a value from the second population. The result of each \ntest is a Mann-Whitney U Statistic (not relevant in our case) and a~$p$-value. We use\nthe~$p$-value to determine whether the difference between the data is statistically \nsignificant, where the standard threshold is~$p = 0.05$. The results of these tests are \ngiven in Table~\\ref{tab:mannwhitneyu}.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mann-Whitney U Test results comparing C\\&W's Attack and the Universal Perturbation attack}\n \\label{tab:mannwhitneyu}\n \\begin{tabular}{l|ll}\n \\midrule\\midrule\n \\multicolumn{1}{c|}{\\textbf{Model}}\n & \\multicolumn{1}{c}{\\textbf{Accuracy $p$-value}} \n & \\multicolumn{1}{c}{\\textbf{$L_2$ distance $p$-value}} \\\\\n \\midrule\n MLP ISH & 0.000 (H0 is rejected) & 0.034 (H0 is rejected)\\\\\n CNN ISH & 0.384 (H0 is not rejected) & 0.098 (H0 is not rejected)\\\\\n MLP MD & 0.000 (H0 is rejected) & 0.057 (H0 is not rejected)\\\\\n CNN MD & 0.000 (H0 is rejected) & 0.016 (H0 is rejected)\\\\\n \\midrule\\midrule\n \\end{tabular}\n\\end{table}\n\nThe results in Table~\\ref{tab:mannwhitneyu} imply that the performance of these two attacks (C\\&W's Attack and the Universal Perturbation attack)\nare nearly identical when attacking a CNN trained on the ISH dataset, \nas evidenced in the second row, where the null hypothesis is not rejected. \nHowever, the~$L_2$ distance measurement for spam images \nthat have had the universal perturbation applied should remain constant relative \nto the original spam image. Therefore, the results of these tests suggest that \nthe Universal Perturbation attack is able to achieve similar performance \nto C\\&W's Attack, in terms of perturbation magnitude, \nwith a much lower computation time per example in comparison to C\\&W's Attack. \n\nGiven the above evidence, the Universal Perturbation attack is the best choice for \nimage spam, as it is unrivaled in terms of potential performance in a real-world \nsetting. The key advantages of the Universal Perturbation attack include\nthat it generates a single perturbation to be applied to all spam images, \nand its relatively fast computation time per adversarial example. Therefore,\nUniversal Perturbation will be used as a basis for our image transformation \ntechnique, as discussed and analyzed in the remainder of this paper. A sample adversarial spam image generated with the Universal Perturbation attack is presented in Figure~\\ref{fig:sampleadv}.\n\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=8cm]{images\/newspamimage.png}\n \\caption{Adversarial spam image generated with the Universal Perturbation attack}\n \\label{fig:sampleadv}\n\\end{figure}\n\n\\section{Inceptionism-Augmented Universal Perturbations}\\label{sec:univ}\n\n\\subsection{Procedure}\n\nBased on the results and discussion above, a transformation that\nis applied to spam images prior to generating adversarial examples, since perturbations cannot be transformed after application, should \nmeet the following conditions.\n\\begin{itemize}\n \\item Lower the misclassification rate\n \\item Preserve adversarial effects after image resizing\n \\item Make non-spam features more prominent, while retaining legibility\n\\end{itemize}\nGiven the above criteria, a reasonable approach would be to maximize the presence \nof \"natural features\" in a given spam image. That is, the features characteristic of non-spam images learned by classifiers should be maximized whilst retaining legibility. To accomplish this, the procedure for maximizing \nthe activation of a given output neuron (in this case, the non-spam output neuron), \nas introduced in~\\cite{deepdream}, dubbed \"DeepDream\", can be used to increase the number of natural features in all images from the non-spam subsets of the ISH and MD datasets.\nThis is accomplished by maximizing the activations of the convolutional layer \nwith the greatest number of parameters and output layer in the corresponding CNNs. \nThe resulting two sets of images that have had DeepDream applied (``dreamified'' images) are then grouped into batches of four images. The weighted average of the four images in each batch can then be taken to produce two \nprocessed non-spam datasets of images with high concentrations of natural features, as batches of greater than four images may result in high noise. Each of the images in the resulting two non-spam datasets \nare henceforth referred to as \\textit{natural perturbations}.\n\nTo preserve the adversarial effect that the universal perturbation introduces, \nthe Gradient-weighted Class Activation Mapping (Grad-CAM) technique \nintroduced in~\\cite{gradcam} \nis used to generate a class activation map for each spam \nimage in each dataset. The inverse of each such map is used with a natural perturbation \ngenerated from the same dataset to remove the regions of the \nnatural perturbation where the class activation map is highest. \nBy superimposing the resulting natural perturbations onto the corresponding spam images, \nthe regions where the universal perturbation is most effective are left intact while the \nregions of the spam images affected by the natural perturbations benefit by being\nmore non-spam-like. The presence of natural features in the resulting spam images \nshould also result in robustness against resizing prior to inference by a deep \nlearning-based image spam detection model, as the natural features should be still be somewhat preserved even after being shrunken. \n\nThe universal perturbation is then applied to each of the resulting spam images. \nThe result is that we potentially reduce a deep learning-based image spam \ndetector's accuracy due to the presence of a natural perturbation and a universal adversarial perturbation and retain some sort of adversarial effect in the case of resizing.\nThis procedure also allows for the retention of legible text within spam images. \n\n\\subsection{Implementation}\n\nTo generate our two sets of ``dreamified'' images, the CNN architecture presented in~\\cite{convolutional_neural_networks_for_image_spam} is trained on both \nthe ISH and MD datasets, with inverted labels to allow for the maximization of the \nactivations of the neurons corresponding to non-spam images, as the activations for spam images would be maximized if the labels weren't inverted. \nThese two models are trained with the TensorFlow Keras API, with the \nhyperparameters given in~\\cite{convolutional_neural_networks_for_image_spam}. \nFor each of the models, the convolutional layer with the highest number of parameters \nand the output layer were chosen as the layers in which the activation should be maximized \nvia gradient ascent, as the aforementioned convolutional layer is responsible for recognizing the most complex natural features. Each of the images from the non-spam subsets of the ISH and MD datasets were used for inference on the two CNN models. The CNN models use \nthe losses of the chosen layers to iteratively update the non-spam images with gradient \nascent so that the number of non-spam features is maximized. \nEach non-spam image \nis updated for~64 iterations with an update size of~$0.001$. The resulting ``dreamified'' \nimages are then grouped into batches of~4 and blended via evenly distributed weighted \naddition to produce a total of~392 grayscale images, each of size~$400\\times 400\\times 1$.\nThese~392 grayscale images are evenly split between the ISH dataset and MD datasets.\n \nTo utilize GradCAM, \nthe CNN architecture presented in~\\cite{convolutional_neural_networks_for_image_spam} \nis trained on both the ISH and MD datasets with normal labels. \nFor each image from the spam subsets of the ISH and MD datasets, \nGradCAM is used to generate a corresponding class activation map based \non the activations of the last convolutional layer in each of the two models.\nThis is accomplished by computing the gradient of the top predicted class with \nrespect to the output feature map of the last convolutional layer, \nusing the mean intensity of the gradient over specific feature map channels. \nOpenCV~\\cite{opencv} is then used to upscale each of the class activation maps \nto~$400\\times 400$, convert them to binary format, and invert the result to allow the class activation maps to be applied to the natural perturbations such that only the areas with highest activation will contain the natural perturbations. \nThe bitwise AND of each processed class activation map and a randomly selected \nnatural perturbation can then be used to generate two sets of processed \nnatural perturbations, which are superimposed on the corresponding spam images \nfrom each of the two spam subsets. This procedure results in two subsets \nof spam images with natural perturbations. \n\nLastly, the universal perturbation is generated and applied to all images within \nthe two spam image subsets that have had natural perturbations applied. For this\noperation, we use the IBM Adversarial Robustness Toolbox~\\cite{adversarial_robustness_toolbox}.\nThe hyperparameters for the Universal Perturbation attack remain the same \nas those given in Table~\\ref{tab:parms}, above.\n\n\\subsection{Performance Evaluation}\n\nThe mean accuracy, computation time per example, and~$L_2$ distance were recorded for \neach of the four models attacked using spam images with modified universal perturbations.\nThis is analogous to what was done during the attack selection process. \nThis data has been compiled into the tables discussed in this section.\n\nAs can be seen from the results in Table~\\ref{tab:meanacc2},\nthe proposed method for generating adversarial \nspam images is capable of lowering a learning-based model's accuracy \nto~23.7\\%. In addition, on average, our proposed technique is\nmuch more effective while being evenly distributed in terms of accuracy \non similar learning-based models. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mean accuracy of each model with spam images created by the proposed method}\n \\label{tab:meanacc2}\n \\resizebox{0.85\\textwidth}{!}{\n \\begin{tabular}{c|cccc}\n \\midrule\n \\midrule\n \\textbf{Images} & \\textbf{MLP (ISH)} & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n \\midrule\n Modified spam images & 80.1\\% & 98.8\\% & 98.4\\% & 75.3\\% \\\\\n Modified spam images with & \\multirow{2}{*}{72.2\\%} \n \t\t\t\t\t\t& \\multirow{2}{*}{50.4\\%} \n\t\t\t\t\t\t& \\multirow{2}{*}{78.7\\%} \n\t\t\t\t\t\t& \\multirow{2}{*}{23.7\\%} \\\\\n Universal Perturbations \\\\\n \\midrule\\midrule\n \\end{tabular}}\n\\end{table}\n\nFrom Table~\\ref{tab:meancomp2}, we see that\nin contrast to C\\&W's Attack, which on average takes 258.93 seconds per example, the time necessary to generate adversarial spam images \nwith natural perturbations is significantly lower and comparable to \nthat of the original Universal Perturbation attack.\nThis is another advantage of our proposed attack.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mean computation time per adversarial spam image (in seconds)}\n \\label{tab:meancomp2}\n \\begin{tabular}{cccc}\n \\midrule\\midrule\n \\textbf{MLP (ISH)} & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n \\midrule\n 5.46 & 5.15 & 5.87 & 4.80 \\\\\n \\midrule\\midrule\n \\end{tabular}\n\\end{table}\n\nThe mean~$L_2$ distances and the distribution of the~$L_2$ distances of the modified adversarial spam \nimages are given in Table~\\ref{tab:meandist2}. From Figure~\\ref{fig:lastl2}, we see that the distributions \nof these distances are, on average, not skewed, indicating that the natural perturbations have \nhad a slight negative effect on the spam image $L_2$ distances, as the distributions for the original Universal Perturbation attack were skewed to the left. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n \\centering\n \\caption{Mean~$L_2$ (Euclidean) distance of modified adversarial spam images from original images}\n \\label{tab:meandist2}\n \\begin{tabular}{cccc}\n \\midrule\\midrule\n \\textbf{MLP (ISH)} & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n \\midrule\n 11392.02 & 11309.40 & 9440.69 & 9628.61 \\\\\n \\midrule\\midrule\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=8cm]{images\/lastl2.png}\n \\caption{Density plot of~$L_2$ (Euclidean) distances of the modified adversarial spam images from the original images}\n \\label{fig:lastl2}\n\\end{figure}\n\nThe ROC curves of the models attacked by the proposed method, \nwhich appear in Figure~\\ref{fig:lastroc}, are slightly worse in comparison to \nthat of the original Universal Perturbation attack, suggesting once more that the \nattack is capable of reducing the class separation capacity of attacked image spam \nclassifiers to essentially random.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=8cm]{images\/lastroc.PNG}\n \\caption{ROC curves of each of the four models attacked by the modified spam images generated with the proposed method}\n \\label{fig:lastroc}\n\\end{figure}\n\n\\subsection{Proposed Dataset Analysis}\n\nFigure~\\ref{fig:spamimage2} contains an example of a modified adversarial spam images.\nFrom this image, we observe that\nthe proposed method was able to effectively utilize class activation maps \ngenerated with GradCAM to selectively apply a random natural perturbation \nto the spam image. As discussed in the previous section, this decreases \nclassification accuracy even prior to the application of a universal perturbation. \n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=6cm]{images\/img213.jpg}\n \\caption{Example of modified adversarial spam image generated with the proposed method}\n \\label{fig:spamimage2}\n\\end{figure}\n\nTo fully evaluate the effect of the modified adversarial spam images from the two \nmodified datasets, two sets of class activation maps are generated from the spam \nsubsets of the two datasets using GradCAM and the corresponding CNN models.\nThese activation maps are then averaged to obtain two heatmaps \nfrom the class activation maps, as shown in Figures~\\ref{fig:origishheatmap1} \nand~\\ref{fig:origsvmheatmap1}. For comparison, the same process was applied to \nthe original datasets to obtain Figures~\\ref{fig:ishheatmap1} and~\\ref{fig:svmheatmap1}.\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[b]{0.425\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/ishheatmap1.png}\n \\caption{ISH spam data}\n \\label{fig:ishheatmap1}\n \\end{minipage}\n \\begin{minipage}[b]{0.425\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/svmheatmap1.png}\n \\caption{MD spam data}\n \\label{fig:svmheatmap1}\n \\end{minipage}\n\\end{figure}\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[b]{0.425\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/origishheatmap1.png}\n \\caption{Modified ISH spam data}\n \\label{fig:origishheatmap1}\n \\end{minipage}\n \\begin{minipage}[b]{0.425\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/origsvmheatmap1.png}\n \\caption{Modified MD spam data}\n \\label{fig:origsvmheatmap1}\n \\end{minipage}\n\\end{figure}\n\nAs can be seen in Figures~\\ref{fig:ishheatmap1} and~\\ref{fig:svmheatmap1}, \nthe activation regions for spam images from the original ISH and MD datasets \nare skewed towards the top and bottom. The narrow shape of these regions represent \nthe regions in spam images that generate the highest activations in \nthe neurons of the deep learning-based classifier. \nThe central region of the average class activation map for spam images from the \nMD dataset is much darker in comparison to that of spam images \nfrom the ISH dataset due to the superimposition of natural images \ndirectly onto spam features, \nas described in~\\cite{support_vector_machines_for_image_spam}. \n\nIn contrast, Figures~\\ref{fig:origishheatmap1} and~\\ref{fig:origsvmheatmap1} indicate \nthat the introduction of natural and universal adversarial perturbations \nare able to more evenly distribute the activation regions. This result shows \nthat the spam images from the modified datasets are much closer---in terms \nof natural features---to non-spam images. This also suggests that the proposed \nmethod outperforms the procedure used to generate the original MD dataset \nas outlined in~\\cite{support_vector_machines_for_image_spam}. \n\n\\section{Conclusion and Future Work}\\label{sec:con}\n\nModern deep learning-based image spam classifiers can accurately classify \nimage spam that has appeared to date in the wild. \nHowever, spammers are constantly creating new countermeasures to\ndefeat anti-spam technology. Consequently, the eventual use of adversarial examples \nto combat deep learning-based image spam filters is inevitable. \n\nIn this chapter, four adversarial attacks were selected based on specific restrictions \nand constraints of the image spam problem. These adversarial attacks \nwere evaluated on the CNN and MLP architectures introduced in~\\cite{convolutional_neural_networks_for_image_spam}. For training data,\nwe used the dataset presented in~\\cite{image_spam_hunter} \nand~\\cite{support_vector_machines_for_image_spam}. \nThe Fast Gradient Sign Method (FGSM) attack, C\\&W's Attack, DeepFool, and the \nUniversal Perturbation attack were all evaluated based on mean accuracy reduction, \nmean computation time per adversarial spam image, mean $L_2$ distance from the original spam \nimages, and ROC curves of the attacked classifiers. Through further statistical analysis, \nthe Universal Perturbation was chosen as a base for our proposed image transformation \nattack, due to its versatility and overall high performance \nin terms of accuracy reduction and computation time.\n\nTo maximize the number and intensity of natural features in an attack, the approach \nintroduced in~\\cite{deepdream} for maximizing activations of certain layers in \na deep neural network was used. This technique serves to generate sets of ``natural perturbations'' \nfrom the non-spam subsets of the image spam datasets. These natural perturbations \nwere then modified via the class activation maps of all spam images in both datasets. \nThe class activations were generated using GradCAM from the two convolutional \nneural networks trained on the ISH and MD datasets. These activation maps allow the regions in spam images recognized to contribute most to the spam classification to benefit from a universal adversarial perturbation. \n\nOur technique resulted in comparable---if not greater---accuracy reduction as\ncompared to C\\&W's Attack. In addition, our approach is \ncomputation much more efficient than C\\&W's Attack. \nFurthermore, the nature of our attack implies that the only potential \ncomputational bottleneck is generating the modified natural perturbations.\nThis aspect of the attack\nwould not be an issue in practice, unless a spammer generates \nvast numbers (i.e., in the millions) of modified \nadversarial spam images. \n\nA dataset of modified adversarial \nspam images has been generated by the authors by applying the proposed attack to the spam subsets of the ISH and MD datasets. This dataset will be made freely available to researchers.\n\nFuture work will include evaluating the ability of adversarial attack defense \nmethods. We will consider defensive distillation against adversarial spam images \ngenerated with our proposed attack. The goal of this research will be to develop defenses specifically designed for natural perturbation-augmented adversarial spam images. For example,\nthe subtraction of predicted adversarial perturbations is one path that we intend to pursue. \n\n\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction Definitions and Results}\nOriginally, the Chuang's inequality is a standard estimate in Nevanlinna theory but later on this inequality is used as a valuable tool in the study\nof value distribution of differential polynomials. For example, Recently, using this inequality, some sufficient conditions are obtained for which two differential polynomials sharing a small function satisfies the conclusions of Br\\\"{u}ck conjecture (\\cite{bb}, \\cite{bc1}, \\cite{bc2}, \\cite{bm}). \\par\nAt this point, we recall some notations and definitions to proceed further.\\par\nIt will be convenient to let $E$ denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence.\\par\nLet $f$ be a non constant meromorphic function in the open complex plane $\\mathbb{C}$. For any non-constant meromorphic function $f$, we denote by $S(r, f)$ any quantity satisfying $$S(r, f) = o(T(r, f))\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ (r\\lra \\infty, r\\not\\in E).$$\nA meromorphic function $a(\\not\\equiv \\infty)$ is called a small function with respect to $f$ provided that $T(r,a)=S(r,f)$ as $r\\lra \\infty, r\\not\\in E$.\\par\nWe use $I$ to denote any set of infinite linear measure of $0&=& \\sum_{n=0}^N \\genfrac{(}{)}{0pt}{}{N}{n}^{1\/2}\\cos^{n}(\\theta\/2)\n \\sin^{N-n}(\\theta\/2)\n \\nonumber\\\\\n &\\times& e^{i(N-n)\\phi}| n, N-n \\rangle\\;.\n\\end{eqnarray}\n\nThe Gross-Pitaevskii dynamics can be mapped to that of a nonrigid\npendulum~\\cite{SmerziEtAl97}. Including the term describing the periodic shaking, the \nHamilton function is given by ($z=\\cos^2(\\theta\/2)-\\sin^2(\\theta\/2)$):\n\\begin{eqnarray}\nH_{\\rm mf}& = &\\frac{N\\kappa}{\\Omega}z^2-\\sqrt{1-z^2}\\cos(\\phi)\\nonumber\\\\\n&-&2z\\left(\\frac{\\mu_0}{\\Omega}+\n\\frac{\\mu_1}{\\Omega}\\sin\\left({\\textstyle\\frac{\\omega}{\\Omega}}\\tau\\right)\\right)\\;,\\quad \\tau =t\\Omega\\;.\n\\label{eq:mean}\n\\end{eqnarray}\nIn our case, $z\/2$ is \nthe experimentally measurable~\\cite{GatiOberthaler07} population imbalance which can\nbe used\nto characterise the mean-field dynamics. The dynamics for this system are known to become\nchaotic~\\cite{GuckenheimerHolmes83}. \n\n\n\\section{Multi-particle-entanglement \\& Quantum Fisher information\\label{sec:multi}}\n\nMulti-particle-entanglement~\\cite{Yukalov2003,Vaucher08,DunninghamEtAl05,PoulsenEtAl05,CiracEtAl98,MicheliEtAl03,MahmudEtAl03,Dounas-frazerEtAl07}\nis a hot topic of current research; to experimentally realise ``Schr\\\"odinger-cat'' like\nsuperpositions of BECs is still a challenge of fundamental research. For the periodically driven\ndouble-well potential, the relation between emergence of ``Schr\\\"odinger-cat'' like mesoscopic superpositions\nin phase-space and mean-field chaos has been discovered in Ref.~\\cite{WeissTeichmann08}. An ideal\nexample of such a mesoscopic superposition is shown\nin Fig.~\\ref{fig:cat}; the fidelity~\\cite{fidelity} of some of the highly entangled states found numerically\nwas well above 50\\%~\\cite{WeissTeichmann08}. In this manuscript, we employ the fact that the\nquantum Fisher information can be used to detect multi-particle entanglement~\\cite{PezzeSmerzi2007}.\n\\begin{figure}\n\\vspace{-1cm}\n\\includegraphics[angle=-90,width=\\linewidth]{cat_perfect.jpg}\n\\caption{\\label{fig:cat}(Colour online) An ideal ``Schr\\\"odinger-cat'' like state which is the superposition of\n two atomic coherent states~(\\ref{eq:atomic}) with hardly any overlap. \nThe figure\nshows the projection \n$|\\langle \\psi_{\\rm cat}| \\theta, \\phi\\rangle|^2$\n of the\n``Schr\\\"odinger-cat'' like state $\\psi_{\\rm cat} = \\frac{1}{\\sqrt{2}} \\left( |z=-0.6, \\phi = 1.2 \\rangle + |z = 0.65, \\phi=-2.20 \\rangle\n\\right)$ onto the atomic\ncoherent states $|\\theta, \\phi \\rangle$ (e.q. \\ref{eq:atomic}) in dependece of\n$z$ and $\\phi$.\nWhile\n Ref.~\\cite{WeissTeichmann08} concentrated on identifying such highly entangled state, the\n present manuscript uses a different approach to search highly entangled states: the sufficient condition~(\\ref{eq:sufficient}) derived in\n Ref.~\\cite{PezzeSmerzi2007} by using the quantum Fisher information~(\\ref{eq:fisher})\n (Ref.~\\cite{PezzeSmerzi2007} and references therein).}\n\\end{figure}\n \nBefore defining the quantum Fisher information, we note that the creation and annihilation\noperators can be used to define\n\\begin{eqnarray}\n\\hat{J}_x =& &\\frac12\\left(\\hat{a}^{\\dag}_1\\hat{a}_2^{\\phantom{\\dag}}+\\hat{a}^{\\dag}_2\\hat{a}^{\\phantom{\\dag}}_1\\right)\\;,\\\\\n\\hat{J}_y =& -&\\frac i2\\left(\\hat{a}_1^{\\dag}\\hat{a}_2^{\\phantom{\\dag}}-\\hat{a}_2^{\\dag}\\hat{a}^{\\phantom{\\dag}}_1\\right)\\;,\\\\\n\\hat{J}_z =& &\\frac 12\\left(\\hat{a}_1^{\\dag}\\hat{a}_1^{\\phantom{\\dag}}-\\hat{a}_2^{\\dag}\\hat{a}_2^{\\phantom{\\dag}}\\right)\\;,\n\\end{eqnarray}\nwhich satisfy angular momentum commutation rules. Except for a factor of $N$, the operator $\\hat{J}_z$ is the operator\nof the (experimentally measurable~\\cite{GatiOberthaler07}) population imbalance.\n\nWhile the quantum Fisher information~$F_{\\rm Q}$ can be defined for statistical mixtures, it is\nparticularly simple for pure states~\\cite{PezzeSmerzi2007}:\n\\begin{equation}\nF_{\\rm Q} = 4(\\Delta \\hat{J}_n)^2\\;;\n\\end{equation}\nwhere $\\Delta \\hat{J}_n$ are the mean-square fluctuations of $\\hat{J}$ in direction $n$. In\nthe following we choose the $z$-direction, thus\n\\begin{equation}\n\\label{eq:fisher}\nF_{\\rm Q} = 4(\\Delta \\hat{J}_z)^2\\;.\n\\end{equation}\nFor $N$ particles, a sufficient condition for multi-particle entanglement is given\nby~\\cite{PezzeSmerzi2007}:\n\\begin{eqnarray}\n\\label{eq:sufficient}\nF_{\\rm ent}&>&1\\;,\\\\\nF_{\\rm ent}&\\equiv&\\frac{F_{\\rm Q}}N\\;.\n\\label{eq:flag}\n\\end{eqnarray}\nFor the ideal ``Schr\\\"odinger-cat'' state in real space,\n\\begin{equation}\n|\\psi_{\\rm NOON}\\rangle=\\frac1{\\sqrt{2}}\\left(|N,0\\rangle+|0,N\\rangle\\right)\\;,\n\\end{equation}\nthis entanglement flag reaches a value of $N$. Thus, while Eq.~(\\ref{eq:sufficient}) already\nindicates multi-particle entanglement, values of \n\\begin{equation}\n\\label{eq:highly}\nF_{\\rm ent}\\gg 1\n\\end{equation}\n demonstrate highly entangled\nstates.\n\n\n\n\\section{Results\\label{sec:results}}\nIn order to characterise whether or not the mean-field dynamics are chaotic, Poincar\\'e\nsurface of sections are particularly suited: for a set of initial conditions, the mean-field\ndynamics of the periodically driven system characterised by the angular frequency $\\omega$ is\nplotted at integer multiples of $2\\pi\/\\omega$. Figure~\\ref{fig:sosa} shows a phenomenon which\nis very characteristic for the non-rigid pendulum on which the BEC in a double well was\nmapped within mean-field: the coexistence between chaotic and regular regions.\n\\begin{figure}[th]\n\\includegraphics[angle=-90,width=\\linewidth]{sos_a.jpg}\n\\caption{\\label{fig:sosa}Poincar{\\'e} surface of section for the mean-field system. Closed\n loops characterise stable orbits whereas chaos is represented by irregular dots. The\n parameters are chosen such that they correspond to a one-photon resonance: a tilt of $2\\mu_0\/\\Omega=3.0$, a driving frequency of~$\\omega=3\\Omega$, an interaction of $N\\kappa\/\\Omega=0.8$ and a driving amplitude of $2\\mu_1\/\\Omega=0.9$ (cf.\\ Ref.~\\cite{EckardtEtAl05}).}\n\\end{figure}\n\n While each\ninitial condition will, in general, lead to many dots in plots like Fig.~\\ref{fig:sosa}, we\nproceed in the spirit of Ref.~\\cite{WeissTeichmann08} to characterise the $N$-particle\ndynamics: In Fig.~\\ref{fig:entfig1a} each initial condition leads to only one point: the\nentanglement flag~(\\ref{eq:flag}) for this initial condition after a fixed time~$t\\Omega$. Highly entangled states\n(Eq.~(\\ref{eq:highly})) can be found on the boundary of stable regions; already for short\ntimes (Fig.~\\ref{fig:entfig1a}~a) such highly entangled states can occur. For larger times\n(Fig.~\\ref{fig:entfig1a}~b) many features of the Poincar\\'e surface of section in\nFig.~\\ref{fig:sosa} are visible in the entanglement generation which essentially spreads over\nthe entire chaotic part of the initial conditions.\n\\begin{figure}\n\\vspace*{-1cm}\n\\includegraphics[width=\\linewidth]{plotfig1a.jpg}\n\\caption{\\label{fig:entfig1a}(Colour online) The entanglement flag~(\\ref{eq:flag}) in a two-dimensional\n projection as a function of the initial condition for $N=100$ particles. All other\n parameter as in Fig.~\\ref{fig:sosa}. The $N$-particle wave-function which corresponds to\n the mean-field initial conditions ($z_0$,$\\phi_0$) can be found in\n Eq.~(\\ref{eq:atomic}). In the upper panel, the (experimentally measurable) entanglement\n flag~(\\ref{eq:flag}) is shown after a time of $t\\Omega=10$, in the lower panel the time is\n $t\\Omega=100$. Highly entangled states (cf.\\ Eq.~(\\ref{eq:highly})) occur in the entire\n chaotic regime.\n }\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=-90,width=\\linewidth]{sos_b.jpg}\n\\caption{\\label{fig:sosb}Poincar{\\'e} surface of section for the mean-field system (cf.\\\n Fig.~\\ref{fig:sosa}). The\n parameters are chosen such that they correspond to a $3\/2$-photon resonance~\\cite{EckardtEtAl05} with\n $N\\kappa\/\\Omega=0.1$, $2\\mu_0\/\\Omega=3.0$, $\\omega\/\\Omega=2.08$ and~$2\\mu_1\/\\Omega=1.8$.}\n\\end{figure}\n\nFor parameters which display no chaotic parts in the Poincar\\'e surface of section\n(Fig.~\\ref{fig:sosb}), on short time-scales hardly any entanglement emerges\n(Fig.~\\ref{fig:entfig1b} a). For larger time-scales, entanglement generation occurs on the\nboundaries of stable regions (Fig.~\\ref{fig:entfig1b} b).\n\\begin{figure}\n\\vspace*{-2cm}\n\\includegraphics[width=\\linewidth]{plotfig1b.jpg}\n\\caption{\\label{fig:entfig1b}(Colour online) The entanglement flag~(\\ref{eq:flag}) in a two-dimensional\n projection as a function of the initial condition (cf.\\ Fig.~\\ref{fig:entfig1a}) for $N=100$ particles; all other\n parameter as in Fig.~\\ref{fig:sosb}. For the regular regime given by the parameters of\n Fig.~\\ref{fig:sosb}, after short time-scales of $t\\Omega=10$ (upper panel)\n entanglement-production is very weak (note that the brightness-entanglement-coding differs by\n a factor of 10 from Fig.~\\ref{fig:entfig1b} and the longer time-scales $t\\Omega=100$\n depicted in the lower panel). For larger time-scales, entanglement production mainly occurs\non the boundary of stable regions (lower panel).}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn Ref.~\\cite{WeissTeichmann08} we discovered the relation between mesoscopic ``Schr\\\"odinger-cat''\nlike superpositions in phase space and mean-field entanglement. In the present paper, we\ndemonstrated that also for more general entangled states, multi-particle entanglement can be\na quantum signature of chaos. For regular systems, the general entangled states also\noccur. However, it is restricted to the boundaries of stable regions and only occurs on\nlonger time-scales. While the focus in\nRef.~\\cite{WeissTeichmann08} was on finding particularly highly entangled states for each\ninitial condition, in the present manuscript the entanglement production was shown for each\ninitial condition at the same point in time, both for short and longer times between the\nonset of the computer experiment and the read-out. \n\nAs an entanglement-flag, we apply the quantum Fisher information for pure states.\nIn this paper we use it in a way which is\nparticularly easy to measure experimentally. However, using Eq.~(\\ref{eq:flag}) assumes a pure state and\nwould thus only be valid in an ideal system without decoherence. For\nmore realistic situations, \nexperimental signatures as in Refs.~\\cite{PiazzaEtAl2008,WeissCastin08} should be\ninvestigated in the future.\n\n\n\n\n\\acknowledgments\n\nWe would like to thank A.~Smerzi for useful discussions and M.~Holthaus for his\ncontinuous support. N.T.\\ acknowledges funding by the Studienstiftung des deutschen Volkes.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWhile within-category demand effects of the marketing mix have been studied extensively, cross-category effects are less well understood \\citep{leeflang:12}. Nevertheless, cross-category effects might be substantial. Some categories are complements, e.g.\\ bacon and eggs studied by \\cite{Niraj08} or cake mix and cake frosting studied by \\cite{Manchanda99}, while others are substitutes, e.g.\\ frozen, refrigerated and shelf-stable juices \\citep{Wedel2004}. But cross-effects also exist among categories that are not complements or substitutes for several reasons. First, as a result of brand extensions, brands are no longer limited to one category \\citep{Erdem98,Kamakura2007,Ma2012}. So advertising and promotion of a brand within one category might spill over to own brand sales in other categories.\nSecond, advertising and promotions generate more store traffic and therefore more sales in other categories \\citep{Bell1998}. \nAnd third, lower expenditures in one category alleviate the budget constraint such that consumers are able to spend more on other, seemingly unrelated, categories \\citep{song:07, Lee2013}.\n\nWhile cross-category effects might be substantial for these reasons, we do not expect that each category's marketing mix variables influence each and every other category. Instead, we expect some cross-category effects to be zero -- or very close to zero -- but we can not a priori exclude them. Therefore, we use an exploratory modeling approach for parsimonious estimation of a product category network. The network allows us to easily identify categories that are influential for or responsive to changes in other categories. Building on a widely used category typology of destination, routine, occasional and convenience categories \\citep{Blattberg95,Briesch2013}, we find that destination categories are most influential, convenience and occasional categories most responsive, and routine categories moderately influential and moderately responsive. \n\nIn order to estimate the cross-category network, this paper presents sparse estimation of the Vector AutoRegressive (VAR) model. The estimation is {\\it sparse} in the sense that some of the within-and cross-category effects in the model can be estimated as exactly zero. Initiated by the work of \\cite{Baghestani91} and \\cite{Dekimpe95}, the VAR Market Response Model has become a standard, flexible tool to measure own- and cross-effects of marketing actions in a competitive environment. The main drawback of the VAR model is the risk of overparametrization because the number of parameters increases quadratically with the number of included categories. \nEarlier studies using the VAR model, like e.g. \\citeauthor{Nijs01} (\\citeyear{Nijs01}; \\citeyear{Nijs2007}); \\cite{Pauwels2002}; \\citeauthor{Srinivasan00} (\\citeyear{Srinivasan00}; \\citeyear{Srinivasan04}); \\cite{Steenkamp05}, were often limited by this overparametrization problem.\nTo overcome this problem, previous research on cross-category effects has limited its attention to a small number of categories by studying substitutes or complements \\citep{Kamakura2007,song:07,Leeflang:08,Bandyopadhyay2009,Ma2012}. We present an estimation technique for cross-category effects in much larger product category networks. The technique allows many parameters to be estimated even with short observation periods. Short observation periods are commonplace in marketing practice since many firms discard data that are older than one year \\citep{Lodish07}. \n\nThis paper contributes to the extant retail literature in a number of important ways. \n(1) Previous cross-category literature largely limits attention to categories that are directly related through substitution, complementarity or brand extensions. We provide evidence that cross-category effects go beyond such directly related categories. \n(2) We introduce the concepts of influence and responsiveness of a product category and position different category types (destination, routine, occasional and convenience) according to these dimensions. \n(3) To identify the cross-category effects, we estimate a large VAR model using an extension of the lasso approach of \\cite{Tibshirani96}. \n\n\nThe remainder of this article is organized as follows. Section 2 positions this paper in the cross-category management literature and describes the conceptual framework that positions category types according to their influence and responsiveness. Section 3 discusses the methodology. We describe the sparse estimator of the VAR model, discuss how to construct impulse response functions ans compare the sparse estimation technique with two Bayesian estimators. In Section 4, a simulation study shows the excellent performance of the proposed methodology in terms of estimation reliability and prediction accuracy. Section 5 presents our data and model, Section 6 our findings on cross-category demand effects. \nWe first identify which categories are most influential and which are most responsive to changes in other categories. Then, we identify the main cross-category effects based on estimated cross-price, promotion and sales elasticities.\n\n\\section{Cross-Category Management}\nThe importance of category management for retailers is widely acknowledged, both as a marketing tool for category performance \\citep{Fader1990, Basuroy2001, Dhar2002} and as an operational tool for planning and logistics \\citep{Rajagopalan2012}. \nSuccessful category management requires retailers to understand cross-category effects of prices, promotions and sales.\nAmong these, the cross-category effects of prices on sales -- which define substitutes and complements -- are the most extensively studied \\citep{Song:06, Bandyopadhyay2009, leeflang:12, Sinistyn2012}. \nCross-category effects of promotions, e.g.\\ feature and display promotions, on sales result from many brands being active in multiple categories \\citep{Erdem02}. Brand associations carry over to products of the same brand in other categories, e.g.\\ through umbrella branding \\citep{Erdem98} or horizontal product line extensions \\citep{Aaker90}. \nLess well understood than the effects of prices and promotions, are the effects of sales in one category on sales in other categories. Such effects might exist because categories are related based on affinity in consumption \\citep{Shankar2014}, because products from various categories are placed close to each other in the shelves \\citep{Bezawada09, Shankar2014}, or because of the budget constraint \\citep{Du2008}. If consumers spend more in a certain category they might, all else equal, spend less in other categories simply because they hit their budget constraint. As a result, cross-category effects might exist between seemingly unrelated categories. \n\nWhen studying these cross-category effects of price, promotion and sales on sales, several asymmetries might arise. \nA first asymmetry concerns within- versus cross-category effects. We expect within-category effects to be more prevalent and larger in size than cross-category effects (e.g. \\citealp{Song:06}; \\citealp{Bezawada09}).\nA second asymmetry concerns category influence versus category responsiveness.\nInfluential categories are important drivers of other category's sales, while sales of responsive categories react to changes in other categories. To identify which categories are more influential or more responsive, we build on a widely used typology of categories described in \\cite{Blattberg95}. \n\n\\cite{Blattberg95} define 4 category types from the consumer perspective: destination, routine, occasional and convenience. \nDestination categories contain goods that consumers plan to buy before they go on a shopping trip, such as soft drinks. \\cite{Briesch2013} show that destination categories are generally categories in which consumers spend a lot of their budget. Retailers typically use a price aggressive promotion strategy and high promotion intensity for these destination categories with the goal of increasing store traffic.\nBecause consumers shop to buy products in the destination categories, destination categories are likely to influence sales in other categories. However, since consumers already plan to buy in the destination categories before entering the store, destination category sales will not be highly responsive \\citep{Shankar2014}. \n\nAbout 55\\% to 60\\% of categories are routine categories \\citep{Pradhan09}. Routine categories are regularly and routinely purchased, such as juices and biscuits. Retailers typically use a consistent pricing strategy and average level of promotion intensity. Because purchases in routine categories can more easily be delayed than purchases in destination categories, we expect routine categories to be more responsive. But, since purchases in routine categories altogether still account for a large portion of the budget, they are also likely to influence sales in other categories. \n\nOccasional categories follow a seasonal pattern or are purchased infrequently. These categories comprise a small proportion of retail expenditures while they contain typically more expensive items, like oatmeal. We therefore expect occasional categories to be less influential and more responsive than destination or routine categories. \n\nFinally, convenience categories are categories that consumers find convenient to pick up during their one-stop shopping trip, like ready-to-eat-meals. These purchase decisions are typically made in the store. Since convenience categories are geared towards consumer convenience and filling impulse needs, we expect them to be highly responsive.\n\n\n\\section{Sparse Vector Auto-Regressive Modeling}\n\n\\subsection{Motivation}\n\nThe aim of this paper is to identify cross-category demand effects in a large product category network.\nTo this end, we use the Vector AutoRegressive (VAR) model. The VAR is ideal for measuring within- and cross-category effects of marketing actions since it accounts for both inertia in marketing spending and performance feedback effects by treating marketing variables as endogenous \\citep{Dekimpe95}. \nOther studies on cross-category effects, like e.g.\\ \\cite{Wedel2004} use a demand model with exogenous prices, or a simultaneous equations model without lagged effects like \\cite{Shankar2014}. However, managers may set marketing instruments strategically in response to market performance and market response expectations. Not accounting for time inertia or feedback effects limits our understanding of how the market functions and misleads managerial insights and prediction.\n\nIdentifying cross-category demand effects using VAR analysis remains challenging because the sheer number of such effects makes them hard to estimate. The number of parameters to be estimated in the VAR rapidly explodes, making standard estimation inaccurate. This undermines the ability to identify important relationships in the data.\nTo overcome an explosion of the number of parameters in the VAR, marketing researchers have used pre-estimation dimension reduction techniques, i.e.\\ they first impose restrictions on the model and then estimate the reduced model. Four such common techniques are (i) treating marketing variables as exogenous (e.g. \\citealp{Nijs01}; \\citealp{Pauwels2002} and \\citealp{Nijs2007}), (ii) estimating submodels rather than a full model (e.g. \\citealp{Srinivasan00}; \\citealp{Srinivasan04}), (iii) aggregating or pooling over, for instance, stores or competitors (e.g. \\citealp{Horvath08}; \\citealp{Slotegraaf2008}), and (iv) applying Least Squares to a restricted model (e.g. \\citealp{Dekimpe95, Dekimpe99_b}; \\citealp{Nijs2007}). Most researchers applying pre-estimation dimension reduction techniques recognize that they do so because of the practical limitations of standard estimation techniques rather than for theoretical reasons (e.g. \\citealp{Srinivasan04} and \\citealp{Bandyopadhyay2009}). \n\nTo address the overparametrization of the VAR, we use sparse estimation. Sparsity means that some of the within- and cross-category effects in the VAR are estimated as exactly zero. \nAs argued in the previous section, from a substantive perspective, we cannot exclude cross-category effects before estimation because cross-category effects might occur between seemingly unrelated categories. \nFrom a methodological perspective, sparse estimation is a powerful solution to handle the overparametrization of the VAR.\nIn our cross-category model, we endogenously model sales, promotion and prices of 17 product categories. Hence, already in a VAR model with one lag, as much as $(3 \\times 17) \\times (3 \\times 17) = 2601$ within- and cross-category effects need to be estimated. Since the sparse estimation procedure puts some of these effects to zero, a more parsimonious model is obtained. Results are easier to interpret and, therefore, the sparse estimation procedure provides actionable insights to managers.\n\n\n\n\\subsection{Extending the Lasso to the VAR model}\nIn situations where the number of parameters to estimate is large relative to the sample size, the Lasso proposed by \\cite{Tibshirani96} provides a solution within the multiple regression model. The Lasso minimizes the least squares criterion penalized for the sum of the absolute values of the regression parameters. This penalization forces some of the estimated regression coefficients to be exactly zero, which results in selection of the pertinent variables in the model. The Lasso method is well established \\citep{Buhlmann2010, Chatterjee2011} and shows good performance in various applied fields \\citep{Wu2009, Fan2011}.\n\nThe Lasso technique can not be directly applied to the VAR model because the VAR model differs from a multiple regression model in two important aspects. First, a VAR model contains several equations, corresponding to a multivariate regression model. Correlations between the error terms of the different equations need to be taken into account.\nSecond, a VAR model is dynamic, containing lagged versions of the same time series as right-hand side variables of the regression equation. Both aspects of VAR models make it necessary to extend the lasso to the VAR context, what the sparse estimator in this paper does.\n\nIt builds further on a sparse estimator of the multivariate regression model \\citep{Rothman10}, and the groupwise lasso for categorical variables \\citep{Yuan06, Meier07}. The estimator is consistent for the unknown model parameters, see \\citet{Meier07} and \\citet{Friedman07}. \n\n\n\\subsection{Model Specification}\nSales, price and promotion are measured for several categories over a certain time period. We collect all these time series in a multivariate time series ${\\bf y}_t$ with $q$ components. In our cross-category demand effects study, ${\\bf y}_t$ contains sales, price and promotion for 17 product categories, hence $q= 3 \\times 17 = 51$. The VAR Market Response Model is given by\n\\begin{equation}\\label{varp}\n{\\bf y}_t = B_1 {\\bf y}_{t-1} + B_2 {\\bf y}_{t-2} + \\ldots + B_p {\\bf y}_{t-p} + {\\bf e}_t \\, ,\n\\end{equation}\nwhere $p$ is the lag length. The autoregressive parameters $B_1$ to $B_p$ are $(q \\times q)$ matrices, which capture both within- and cross-category effects. The elements of these matrices measure the effect of sales, price and promotion in one category on the sales, price and promotion in other categories (including its own). The error term ${\\bf e}_t$ is assumed to follow a $N_q(0,\\Sigma)$ distribution. We assume, without loss of generality, that all time series are mean centered such that no intercept is included.\n\nIf the number of components $q$ in the multivariate time series is large, the number of unknown elements in the sequence of matrices $B_1,\\ldots,B_p$ explodes to $p q^2$, and accurate estimation by standard methods is no longer possible. Sparse estimation, with many elements of the matrices $B_1,\\ldots,B_p$ estimated as zero, brings an outcome: it will not only provide estimates with smaller mean squared error, but also substantially improve model interpretability.\nThe method we propose does not require the researcher to prespecify which entries in the $B_j$ matrices are zero and which are not. Instead, the estimation and variable selection are simultaneously performed. This is particularly of interest in situations where there is no a priori information on which time series is driving which.\n\nThe instantaneous correlations in model \\eqref{varp} are captured in the error covariance matrix $\\Sigma$. If the dimension $q$ is large relative to the number of observations, estimation of $\\Sigma$ becomes problematic. The estimated covariance matrix risks getting singular, i.e.\\ its inverse does not exist. Hence, we also induce sparsity in the estimation of the inverse error covariance matrix $\\Omega=\\Sigma^{-1}$. The elements of $\\Omega$ have a natural interpretation as partial correlations between the error components of the $q$ equations in model \\eqref{varp}. If the $ij$-th element of the inverse covariance matrix is zero this means that, conditional on the other error terms, there is no correlation between the error terms of equations $i$ and $j$.\n\n\n\\subsection{Penalized Likelihood Estimation}\nThis section defines the sparse estimation procedure for the VAR model. The Sparse VAR estimator is defined by minimizing a measure of goodness-of-fit to the data combined with a {\\it penalty} for the magnitude of the model parameters. It is convenient to first recast model \\eqref{varp} in stacked form as\n\\begin{equation}\\label{stacked}\ny = X \\beta + e \\, ,\n\\end{equation}\nwhere $y$ is a vector of length $n q$ containing the stacked values of the time series. If the multivariate time series has length $T$, then $n=T-p$ is the number of time points for which all current and lagged observations are available. The vector $\\beta$ contains the stacked\nvectorized matrices $B_1,\\ldots,B_p$, and $e$ the vector of stacked error terms.\nThe matrix $X=I_q \\otimes X_0$, with $ X_0 = (\\underline{{\\bf Y}}_1, \\ldots, \\underline{{\\bf Y}}_p)$, is of dimension $(n q \\times p q^2)$.\nHere $\\underline{{\\bf Y}}_j$ is an $(n \\times q)$ matrix, containing the values of the $q$ series at lag $j$ in its columns, for $1 \\leq j \\leq p$, with $p$ the maximum lag. The symbol $\\otimes$ stands for the Kronecker product.\n\nThe sparse estimator of the autoregressive parameters $\\beta$ and the inverse covariance matrix $\\Omega=\\Sigma^{-1}$ are obtained by minimizing the negative log likelihood with a groupwise penalization on the $\\beta$ and a penalization on the off-diagonal elements of $\\Omega$:\n\\begin{equation}\\label{mincrit}\n(\\hat{\\beta},\\hat{\\Omega}) = \\underset{(\\beta,\\Omega)}{\\operatorname{argmin}} \\, \\frac{1}{n} (y-X \\beta)^{\\prime} \\tilde{\\Omega} (y-X \\beta) - \\log|\\Omega| + \\lambda_1 \\sum_{g=1}^{G} ||\\beta_g|| + \\lambda_2 \\sum_{k \\neq k'} |\\Omega_{kk'}| \\, ,\n\\end{equation}\nwhere $||u||= (\\sum_{i=1}^{n} u_i^2)^{1\/2}$ is the Euclidean norm and $\\tilde{\\Omega}= \\Omega \\otimes I_n$. \nBy simultaneously estimating $\\beta$ and $\\Omega$, we take the correlation structure between the error terms into account.\nThe vector $\\beta_g$ in \\eqref{mincrit} is a subvector of $\\beta$, containing the regression coefficients for the lagged values of the same time series in one of the $q$ equations in model \\eqref{varp}. The coefficients of the lagged values of the same time series form a group. The total number of groups is $G=q^2$ because there are $q$ groups within each of the $q$ equations.\nThe penalty on the regression coefficients enforces that either \\textit{all} elements of the group $\\hat\\beta_g$ are zero or \\textit{none}. As a result, we take the dynamic nature of the VAR model into account since the estimated $B_j$ matrices, for $j=1,\\ldots,p$, have their zero elements in exactly the same cells.\nThe penalization on the off-diagonal elements of $\\Omega$ induces sparsity in the estimate $\\hat\\Omega$. Finally, the scalars $\\lambda_1$ and $\\lambda_2$ control the degree of sparsity of the regression estimator and the inverse covariance matrix estimator, respectively. The larger these values, the more sparsity is imposed.\nDetails on the algorithm to perform penalized likelihood estimation and the selection of the sparsity parameters $\\lambda_1$ and $\\lambda_2$ can be found in Appendix A.\n\nOur approach is similar to \\cite{Hsu08} who use the Lasso within a VAR context. However, they do not account for the group-structure in the VAR model, nor do they impose sparsity on the error covariance matrix. \n\\cite{Davis12} propose another sparse estimation procedure for the VAR. They infer the sparsity structure of the autoregressive parameters from an estimate of the partial spectral coherence using a two-step procedure. Since variable selection is performed prior to model estimation, the resulting estimator suffers from pre-testing bias. Moreover, the number of parameters might still approach the sample size, leading to unstable estimation or even making estimation infeasible if the number of parameters still exceeds the sample size.\nSparse estimation in economics is a growing field, see \\cite{Fan2011} and references therein for an overview. \n\n\\subsection{Alternative: Bayesian Estimators}\nAn alternative to the sparse estimation technique is to impose prior information in a Bayesian setting. Bayesian regularization techniques have been proposed for the VAR model in \\cite{Litterman80} and are used in various applied fields such as macroeconomics \\citep{Gefang14, Banbura10}, finance \\citep{Carriero12} and marketing \\citep{Lenk09,Fok:12,Bandyopadhyay2009}. They are also applicable to a situation like ours where there are many parameters to be estimated with a limited observation period, and are thus a good benchmark. However, these methods are not sparse, they do not perform variable selection simultaneously with model estimation. The following two paragraphs elaborate on two Bayesian estimators which serve as non-sparse alternatives.\n\\bigskip\n\n{\\it Minnesota Prior.} The original Minnesota prior only specifies a prior distribution for the regression parameters of the VAR model. The error covariance matrix $\\Sigma$ is assumed to be diagonal, and estimated by $\\hat{\\Sigma}_{ii} = \\hat{\\sigma}_{i}^{2}$ with $\\hat{\\sigma}_{i}^{2}$ the standard OLS estimate of the error variance in an AR$(p)$ model for the $i^{th}$ time series \\citep{koop:09}. The prior distribution of the regression parameters is taken to be multivariate normal:\n\\begin{equation}\n\\beta \\sim N(\\underline{\\beta}_{M},\\underline{V}_{M}) \\label{Minnesotaprior}.\n\\end{equation}\nFor the prior mean, the common choice is $\\underline{\\beta}_{M}=0_{Kq}$ for stationary series. The prior covariance matrix $\\underline{V}_{M}$ is diagonal. The posterior distribution is again multivariate normal. Full technical details can be found in \\cite{koop:09}.\n\nThe main advantage of the Minnesota prior is its ease of implementation, since posterior inference only involves the multivariate normal distribution. However, imposing the Minnesota prior only ensures that the parameter estimates are \\textit{shrunken} towards zero, while the Sparse VAR ensures that some parameters will be estimated as \\textit{exactly} zero.\n\\bigskip\n\n{\\it Normal Inverted Wishart Prior.}\nThe Minnesota prior takes the error covariance matrix $\\Sigma$ as fixed and diagonal and, hence, not as an unknown parameter. To overcome this problem, \\cite{Banbura10} impose an inverse Wishart prior on the $\\Sigma$ matrix. More precisely,\n\n\\begin{equation}\n\\beta \\mid \\Sigma \\sim N(\\underline{\\beta}_{NIW},\\Sigma \\otimes \\Omega_{0}) \\text{\\ \\ and \\ \\ } \\Sigma \\sim iW(S_{0},\\nu_{0}), \\label{LBVArRprior}\n\\end{equation}\nwhere $\\underline{\\beta}_{NIW},\\Omega_{0},S_{0}$ and $\\nu_{0}$ are hyperparameters. Under this normal inverted Wishart prior (labeled in the remainder of this paper as ``NIW\"), the posterior for $\\beta$, conditional on $\\Sigma$ is normal, and the posterior for $\\Sigma$ is again inverted Wishart. Full technical details can be found in \\cite{Banbura10}.\n\n\\subsection{Impulse Response Functions}\nImpulse response functions (IRFs) are extensively used to assess the dynamic effect of external shocks to the system such as changes in the marketing mix. An IRF pictures how a change to a certain variable at moment $t$ impacts the value of any other time series at time $t+k$, accounting for interrelations with all other variables. The magnitude of the effect is plotted as a function of $k$. An extensive discussion on the interpretation of the IRF in marketing modeling can be found in \\cite{Dekimpe95}. We use IRFs to gain insight in the dynamics of within and cross-category sales, promotion and price effects on each of the 17 product category sales. The IRFs are easily computed as a function of the Sparse VAR estimator (see \\citealp{Hamilton91}). Since we want to account for correlated error terms, we use generalized IRFs \\citep{Pesaran1998, Dekimpe99a}.\n\nTo obtain confidence bounds for the generalized IRFs estimated by Sparse VAR, we use a residual parametric bootstrap procedure \\citep{Chatterjee2011}. We generate $N_{b} = 1000$ time series of length $ T $ from the VAR model (\\ref{stacked}). The invertible estimate of $ \\Sigma $ delivered by the Sparse VAR estimation procedure is needed to draw random numbers for the $ N_{q}(0,\\Sigma) $ error distribution. For each of these $ N_{b} $ multiple time series, the estimates of the regression parameters are computed. We compute the covariance matrix of the $N_{b}$ bootstrap replicates. For each of the $N_b$ generated series impulse response functions are computed; the 90\\% confidence bounds are then obtained by taking the 5\\% and 95\\% percentiles.\n\n\\section{Estimation and prediction performance}\nWe conduct a simulation study to compare the proposed Sparse VAR with Bayesian methods using the Minnesota and NIW prior. As benchmarks, we include the classical Least Squares (LS) estimator and two restricted versions of LS which are often used in practice. In the 1-step Restricted LS \\citep{Dekimpe95,Dekimpe99_b}, we estimate the model with classical LS, delete all variables with $|$$t$-statistic$|$ $ \\leq 1 $, and re-estimate the model with the remaining variables. We also consider an iterative Restricted LS method described in \\cite{Lutkepohl04} where we fit the full model using LS and sequentially eliminate the variables leading to the largest reduction of BIC until no further improvement is possible, of which a close variant was used by \\cite{Nijs2007}.\n\nWe simulate from a VAR model with $q=10$ dimensions and $p=2$ lags. Each time series has an own auto-regressive structure and we include system dynamics among the different series. The first series leads series two to five, while the sixth series leads time series 7 to 10. Specifically, the data generating processes are given by\n$${\\bf y}_t =\n\\begin{bmatrix}\nB_{1} & 0\\\\\n0 & B_{1}\\\\\n\\end{bmatrix} {\\bf y}_{t-1}\n+\n\\begin{bmatrix}\nB_2 & 0\\\\\n0 & B_2\\\\\n\\end{bmatrix} {\\bf y}_{t-2}\n+ {\\bf e}_t \\, ,\n$$\nwith\n\\footnotesize\n$$\nB_{1} =\n\\begin{bmatrix}\n0.4 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n0.4 & 0.4 & 0.0 & 0.0 & 0.0 \\\\\n0.4 & 0.0 & 0.4 & 0.0 & 0.0 \\\\\n0.4 & 0.0 & 0.0 & 0.4 & 0.0 \\\\\n0.4 & 0.0 & 0.0 & 0.0 & 0.4 \\\\\n\\end{bmatrix} \\text{and}\n\\hspace{0.2cm} \\hspace{0.2cm}\nB_{2} =\n\\begin{bmatrix}\n0.2 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n0.2 & 0.2 & 0.0 & 0.0 & 0.0 \\\\\n0.2 & 0.0 & 0.2 & 0.0 & 0.0 \\\\\n0.2 & 0.0 & 0.0 & 0.2 & 0.0 \\\\\n0.2 & 0.0 & 0.0 & 0.0 & 0.2 \\\\\n\\end{bmatrix}.\n$$\n\\smallskip\n\\normalsize\n\nIn total, there are $p q^2=200$ regression parameters to be estimated with 36 true parameter values different from zero. The 10-dimensional error term ${\\bf e}_t $ is drawn from a multivariate normal with mean zero and covariance matrix $\\Sigma=0.1 I_{10}$. We generate $N_s=1000$ multivariate time series of length 50 according to the above simulation scheme.\n\n\\subsection{Performance measures}\n\nWe evaluate the different estimators in terms of (i) estimation accuracy, (ii) sparsity recognition performance, and (iii) forecast performance.\n\nTo evaluate estimation accuracy, we compute the mean absolute estimation error (MAEE), averaged over the simulation runs and over the 200 parameters\n$$\n\\mbox{MAEE} = \\frac{1}{N_s} \\frac{1}{pq^2} \\sum_{s=1}^{N_s} \\sum_{j=1}^p \\sum_{k,l=1}^q | \\hat{b}^s_{klj} - b_{klj} |,\n$$\nwhere $\\hat{b}^s_{klj}$ is the estimate of $b_{klj}$, the $kl^{th}$\\ element of the matrix $B_j$ corresponding to lag $j$, for the $s^{th}$ simulation run.\n\nConcerning sparsity recognition, we compute the true positive rate and true negative rate\n\\begin{gather}\n\\text{TPR}(\\hat{b},b) = \\dfrac{ \\# \\{ (k,l,j) : \\hat{b}_{klj} \\neq 0 \\ and \\ b_{klj} \\neq 0 \\}}{\\# \\{ (k,l,j) : \\ b_{klj} \\neq 0 \\}} \\nonumber \\\\\n\\text{TNR}(\\hat{b},b) = \\dfrac{ \\# \\{ (k,l,j) : \\hat{b}_{klj} = 0 \\ and \\ b_{klj} = 0 \\}}{\\# \\{ (k,l,j) : \\ b_{klj} = 0 \\}}. \\nonumber\n\\label{sparsityperformance} \n\\end{gather}\nThe true positive rate (TPR) gives an indication on the number of true relevant regression parameters detected by the estimation procedure. The true negative rate (TNR) measures the hit rate of detecting a true zero regression parameter. Both should be as large as possible.\n\nFinally, we conduct an out-of-sample rolling window forecasting exercise. Using the same simulation design as before, we generate multivariate time series of length $T=60$, and use a rolling window of length $S=50$. For all estimation methods, 1-step-ahead forecasts are computed for $t=S,\\ldots,T-1$. Next, we compute the Mean Absolute Forecast Error (MAFE), averaged over all time series and across time\n\\begin{equation}\n\\text{MAFE} = \\frac{1}{T-S} \\frac{1}{q} \\sum_{t=S}^{T-1}\\sum_{i=1}^{q} | \\ \\hat{y}^{(i)}_{t+1} - y^{(i)}_{t+1} \\ | ,\n\\end{equation}\nwhere $y^{(i)}_{t+1}$ is the value of the $i^{th}$ time series at time ${t+1}$.\n\n\\subsection{Results}\nTable \\ref{simulationresults} presents the performance measures of the Sparse VAR, the Bayesian and benchmark methods. The Sparse VAR estimator performs best in terms of estimation accuracy. It attains the lowest value of the MAEE (0.041). A paired $t$-test confirms that the Sparse VAR significantly outperforms the other methods (all $p$-values $<0.001$).\n\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Mean Absolute Estimation Error (MAEE), True Positive Rate (TPR), True Negative Rate (TNR) and Mean Absolute Forecast Error (MAFE), averaged over 1000 simulation runs, are reported for every method. \\label{simulationresults}}\n\\begin{tabular}{lcccccccccccc}\n \\hline\nMethod \t\t\t\t\t&&& MAEE &&& TPR &&& TNR &&& MAFE \\\\ \\hline\nSparse VAR \t\t\t\t\t&&& 0.041 &&& 0.860 &&& 0.848 &&& 0.359 \\\\\nLS \t\t\t\t\t\t\t&&& 0.157 &&& 1 &&& 0 &&& 0.540 \\\\\nRestricted LS: 1-step \t\t&&& 0.121 &&& 0.709 &&& 0.541 &&& 0.520 \\\\\nRestricted LS: Iterative \t&&& 0.116 &&& 0.261 &&& 0.775 &&& 0.516 \\\\\nBayesian: Minnesota \t\t&&& 0.044 &&& 1 &&& 0 &&& 0.355 \\\\\nBayesian: NIW \t\t\t\t&&& 0.077 &&& 1 &&& 0 &&& 0.476 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\nSparsity recognition performance is evaluated using the true positive rate and the true negative rate, reported in Table \\ref{simulationresults}. For the LS and Bayesian estimators, all parameters are estimated as non-zero, resulting in a perfect true positive rate and zero true negative rate. Among the variable selection methods, the Sparse VAR performs best. Sparse VAR achieves a value of the true positive rate of 0.86; 0.85 for the true negative rate.\n\nFinally, we evaluate the forecast performance of the different estimators by the Mean Absolute Forecast Error in Table \\ref{simulationresults}. The Sparse VAR and the Bayesian estimator with Minnesota prior achieve the best forecast performance. A Diebold-Mariano test confirms that these two methods perform significantly better than the others ($p$-values $<0.001$). There is no significant difference in forecast performance between Sparse VAR and the Bayesian estimator with Minnesota prior.\n\n\\subsection{Robustness checks}\n\\textit{Alternative penalty function.} We investigate the robustness of Sparse VAR to the choice of the penalty function. We replace the grouplasso penalty on the regression coefficients with the elastic net penalty \\citep{Zou05}. Elastic net is a regularized regression method that linearly combines the $L_1$ and $L_2$ penalties of respectively lasso and ridge regression. Like the grouplasso, elastic net produces a sparse estimate of the regression coefficients. All other steps of the methodology remain unchanged. We find that the grouplasso penalty performs slightly better than the elastic net penalty in terms of estimation accuracy, sparsity recognition and prediction performance. \n\n\\medskip\n\n\\textit{Sensitivity to the order of the VAR.} We estimate the model with Sparse VAR for different values of $p$ and evaluate the performance. As expected, Sparse VAR attains the best estimation accuracy for the true value $p=2$. The results are, however, very robust to the choice of the order of the VAR. Selecting $p$ too low is slightly worse than selecting $p$ too high. \n\n\\medskip\n\n\\textit{Sensitivity to the sparsity parameters.} The sparsity parameters are selected according to the BIC and this selection is an integral part of the estimation procedure.\nThe results are not sensitive to the value of $\\lambda_2$, which controls the sparsity of $\\widehat{\\Omega}$. The results are more sensitive to the choice of $\\lambda_1$, since it directly influences the sparsity of the autoregressive parameters. It turms out that Sparse VAR still outperforms the other estimators for a large range of $\\lambda_1$ values.\n\n\n\\section{Data and Model}\nWe use the sparse estimation technique for large VARs described in Section 3 to identify cross-category demand effects across 17 categories in the Dominick's Finer Foods database. This database is a well-established source of weekly scanner data from a large Midwestern supermarket chain, Dominick's Finer Foods (e.g. \\citealp{Kamakura2007, Pauwels07}). We first describe the data and model in more detail, and then report on the insights the Sparse VAR generates in the next section.\n\n\nWe use all 17 product categories in the Dominick's Finer Foods database containing food and drink items, a much broader selection of categories than previous studies on cross-category demand effects have considered. A description of each product category can be found in Table \\ref{Categories}. For 15 stores, we obtain weekly sales, pricing and promotional feature and display data for the 17 product categories.\n\n\\linespread{1.2}\n\\begin{table}\n\\small\n\\begin{center}\n\\caption{Description of the 17 categories from Dominick's Finer Foods database that are analyzed in this paper. For each category, we report the proportion of food and drink expenditures. \\label{Categories}}\n\\small\n\\begin{tabular}{lclllc} \\hline\nCategory & Expenditures &&& Category & Expenditures \\\\ \\hline\nSoft Drinks & 22.24\\% \t\t&&& Snack Crackers & 3.04\\%\\\\\nCereals & 13.92\\% \t\t\t&&& Frozen Juices & 2.88\\% \\\\\nCheeses & 10.46\\% \t\t&&& Canned Tuna & 2.80\\% \\\\\nRefrigerated Juices & 7.36\\% \t\t\t&&& Frozen Dinners & 2.00\\% \\\\\nFrozen Entrees & 6.98\\% \t&&& Front-end-candies & 2.00\\% \\\\\nBeer & 6.35\\% \t\t\t\t&&& Cigarettes & 1.49\\% \\\\\nCookies & 6.21\\%\t\t\t\t&&& Oatmeal & 1.43\\%\\\\\nCanned Soup & 4.82\\% \t\t\t&&& Crackers & 1.37\\%\\\\\nBottled Juices& 4.66\\% \t\t&&& & \\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\\normalsize\n\n\n\\noindent\n{\\bf Sales.} Category sales volumes for the 17 categories, measured in dollars per week.\n\n\\noindent\n{\\bf Promotion.} The promotional data include the percentage of SKUs of each category that are promoted (feature and display) in a given week, following \\citet{Srinivasan04}.\n\n\\noindent\n{\\bf Prices.} To aggregate pricing data from the SKU level to the product category level, we follow \\citet{Srinivasan04} and \\citet{Pauwels2002} in using SKU market shares as weights. Prices are not deflated because there is strong evidence that people are sensitive to nominal rather than real price changes \\citep{Shafir1997} over short time periods.\n\n\\medskip\n\nWe use data from January 1993 to July 1994, 77 weeks in total. We neither use \ndata before 1993 since they contain missing observations, nor\nobservations after 1994 since \\citet{Srinivasan04} pointed out that manufacturers made extensive use of `pay-for-performance' price promotions as of 1994, which are not fully reflected in the Dominick's database. This data range is short relative to the dimension of the VAR, which calls for a regularization approach such as the Sparse VAR. For all stores, we collect data on sales, promotion and pricing for all 17 categories. Only for cigarettes, no promotion variable is included in the VAR since none of the SKUs in that category were promoted during the observation period.\n\nWe estimate a separate VAR model for each store, which allows to evaluate the robustness of the findings. The multivariate time series entering the VAR model are the log-differenced sales ($\\mathbf{Y_t}$), differenced promotion ($\\mathbf{M_t}$), and log-differenced prices ($\\mathbf{P_t}$).\\footnote{Following standard practice, we first test for stationarity. A stationarity test of all individual time series using the Augmented Dickey-Fuller test indicates that most time series in levels are integrated of order 1.} The dimensions of the time series are represented in Table \\ref{data}. We use the Vector Autoregressive model, with endogenous promotion and prices,\n\\begin{equation}\\label{eq: application model}\n\\left[ \\begin{matrix} \\mathbf{Y_t} \\\\ \\mathbf{P_t} \\\\ \\mathbf{M_t} \\end{matrix} \\right] =\nB_0 + B_1 \\left[ \\begin{matrix} \\mathbf{Y_{t-1}} \\\\ \\mathbf{P_{t-1}} \\\\ \\mathbf{M_{t-1}} \\end{matrix} \\right]\n+ \\ldots + B_p \\left[ \\begin{matrix} \\mathbf{Y_{t-p}} \\\\ \\mathbf{P_{t-p}} \\\\ \\mathbf{M_{t-p}} \\end{matrix} \\right] + \\mathbf{e_t}.\n\\end{equation}\nAveraged across stores, the selected value of $p$ is two for the Sparse VAR.\nAlso for the Bayesian estimators, the lag order of the VAR is selected using the BIC criterion, which is one for the majority of the stores. \n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Description of the 15 data sets. Each data set contains multivariate time series for sales ($\\textbf{Y}_{t}$), promotion ($\\textbf{M}_{t}$) and prices ($\\textbf{P}_{t}$). \\label{data}}\n\\small\n\\begin{tabular}{cccccc} \\hline\n\\rule{0pt}{3ex} Store & Number of & \\multicolumn{3}{c}{Dimension} & \\\\\n & Time Points & $\\textbf{Y}_{t}$ & $\\textbf{M}_{t}$ & $\\textbf{P}_{t}$ & Total \\\\ \\hline\nStore 1-15 \\rule{0pt}{3ex} & 77 & 17 & 16 & 17 & 50\\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\\section{Empirical Results}\n\nWe focus on the effects of prices, promotions and sales in category A on the sales (or demand) in category B, where A and B belong to the product category network. We first study the direct effects. \nFor instance, there is no direct effect of price of A on sales of B if the corresponding estimated regression coefficients are equal to zero at all lags. \nThen we turn to the complete chain of direct and indirect effects using Impulse Response Functions. \nFor instance, price in category A indirectly influences sales in category B when the price of category A influences the price, promotion or sales in a certain other category C which, in turn, influences the sales of category B. Since we work in a time series setting, both direct and indirect effects are dynamic in the sense that the effect occurs with a certain delay.\n\n\\subsection{A network of product categories}\nWe analyze cross-category demand effects as a network of interlinked product categories of which prices, promotions and sales in one category have an effect on sales in other categories. Recently, network perspectives have been increasingly used by marketing researchers to model, for example, the network value of a product in a product network \\citep{Singer2013} or to investigate the flow of influence in a social network \\citep{Zubcsek11}. In our case, the 17 product categories are the nodes of the network. We estimate the Sparse VAR for 15 stores separately. If the Sparse VAR estimation results indicate, by giving a non-zero estimate, that prices in one category have a direct influence on sales in another category in the majority of the 15 stores, a directed edge is drawn between them. The resulting directed network is plotted in Figure \\ref{crosscatPrice}. Similarly, Figures \\ref{crosscatPromo} and \\ref{crosscatSLS} present cross-category effects of respectively promotion and sales on sales. If promotion or sales in one category directly influence sales in another category, respectively, this is indicated by a directed edge. \n\n\n\\linespread{1.2}\n\\begin{table}\n\\centering\n\\caption{Proportion of nonzero within and cross-category effects of price, promotion and sales on sales, averaged across 15 stores and 17 product categories.} \\label{Within-Cross}\n\\begin{tabular}{lccccccccc}\n \\hline\n &&& Price &&& Promotion &&& Sales \\\\\n \\hline\nWithin-category &&& 34\\% &&& 30\\% &&& 96\\% \\\\\nCross-category &&& 19\\% &&& 21\\% &&& 21\\% \\\\\n \\hline \n\\end{tabular}\n\\end{table}\n\\linespread{1.5}\n\n\nA first important finding is that the cross-category networks are sparse -- not each category influences each and every other category. While the sparse VAR estimation favors zero-effects, it does not enforce them. Here, as many as 78\\% of all estimated effects are zero-effects. Table \\ref{Within-Cross} summarizes the prevalence of within-and cross-category effects. As expected, within-category effects are more common than cross-category effects. For all categories, past values of the own category's sales are selected for almost all stores. Cross-category effects of price on sales (19\\%), promotion on sales (21\\%) and sales on sales (21\\%) are about equally prevalent.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9cm, angle=-90]{PRICE}\n\t\\end{center}\n\t\\caption{Cross-category effect network of prices on sales: a directed edge is drawn from one category to another if its price influences sales in the other category for the majority of stores. \\label{crosscatPrice}} \n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm, angle=-90]{PROMO}\n\t\\end{center}\n\t\\caption{Cross-category effect network of promotions on sales: a directed edge is drawn from one category to another if its promotion influences sales in the other category for the majority of stores. \\label{crosscatPromo}} \n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm, angle=-90]{SLS}\n\t\\end{center}\n\t\\caption{Cross-category effect network of sales on sales: a directed edge is drawn from one category to another if its sales influences sales in the other category for the majority of stores.\\label{crosscatSLS}} \n\\end{figure}\n\n\n\nNext, we focus on category influence and responsiveness in the cross-category network, measured by the number of edges originating from and pointing to a category respectively.\nAs discussed in Section 2, destination categories are expected to be more influential, while convenience categories are expected to be more responsive. \n We discuss which types of categories we find to be most influential and\/or responsive in the cross-category networks of prices on sales, promotion on sales, and sales on sales. \n\nThe most influential categories in the cross-category network of prices on sales are destination categories such as Soft Drinks and Cheeses (cfr.\\ each four outgoing edges in Figure \\ref{crosscatPrice}). This is consistent with our expectations, as Soft Drinks is known to be a destination category \\citep{Briesch2013,Shankar2014,Blattberg95}. Soft Drinks is ranked first and Cheeses third in terms of food and drink expenditures (see Table \\ref{Categories}) and are both heavily promoted by retailers. A price change in either of these categories thus strongly influences the budget constraint, which in turn influences purchase decisions in other categories. In the cross-category network of promotions on sales, Cereals is the most influential category (cfr. five outgoing edges in Figure \\ref{crosscatPromo}). \\cite{Briesch2013} identified Cereals as highly ranked among the destination categories. This is not surprising as cereals are part of daily consumption patterns and are ranked second in terms of food and drink expenditures. In the cross-category effects network of sales on sales in Figure \\ref{crosscatSLS}, we identify again Cheeses as the most influential category. \n\nWe find convenience categories to be highly responsive to changes in other categories.\nThe most prominent price effects are observed for Canned Soup (cfr.\\ five incoming edges in Figure \\ref{crosscatPrice});\nthe most prominent promotion effects for Frozen Dinners, Crackers and Canned Soup (cfr. each three incoming edges in Figure \\ref{crosscatPromo}); \nand the most prominent sales effects for Oatmeal and Crackers (cfr.\\ each four incoming edges in Figure \\ref{crosscatSLS}). \nThese categories are typically bought out of convenience, such as Frozen Dinners and Canned Soup; or bought on occasion, such as Oatmeal and Crackers, counting for a very small percentage of food and drink expenditures (see Table \\ref{Categories}). \n\nRoutine categories such as Bottled Juices, Refrigerated Juices, Frozen Juices and Cookies score moderate-to-high on category influence but are also responsive. \nThis is in line with our expectation of many grocery categories being routine categories that are moderately influential and moderately responsive. Finally, the cigarettes category is least responsive and least influential. This finding is not surprising as cigarettes are addictive, hence, smokers probably have a stable consumption unrelated to food and drinks.\n\nTo confirm the robustness of the results obtained by Sparse VAR, we check whether category responsiveness and influence are consistent across stores. We compute Kendall's coefficient of concordance $W$ for category influence and responsiveness calculated from the graphs in Figures 2-4 at the store level. As $W$ increases from 0 to 1, there is stronger consistency across stores. Table \\ref{KendallW} indicates that all values of Kendall's $W$ are significant.\n\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Kendall's coefficient of concordance across stores of cross-category effects of price, promotion and sales on sales for both category influence and responsiveness. $P$-values are indicated between parentheses. \\label{KendallW}}\n\\begin{tabular}{lccccccccc}\n\\hline\n\n &&& Price &&& Promotion &&& Sales \\\\ \\hline\n Influence \t\t&&& $\\underset{(< 0.001)}{0.40}$ &&& $\\underset{(< 0.001)}{0.56}$ &&& $\\underset{(< 0.001)}{0.30}$ \\\\\n Responsiveness \\rule{0pt}{3ex}\t&&& $\\underset{(<0.001)}{0.30}$ &&& $\\underset{(0.001)}{0.16}$ &&& $\\underset{(< 0.001)}{0.17}$ \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\n\\subsection{Impulse Response Functions}\nFor each store, we estimate the Sparse VAR and compute the corresponding Impulse Response Functions (IRFs). The effect size of an impulse is obtained by summing the absolute values of the responses across the first 10 lags of the IRF, where we take absolute values in order not to average out positive and negative response. We compute effect sizes of impulses in price, promotion or sales in one product category on the sales in the same (within) category or another (cross) category. In Table 6, we report the within and cross-category price, promotion and sales effect sizes, averaged across the 15 stores and the product categories. \n\n\nTable \\ref{Effectsizes} indicates that, for example, a one standard deviation price shock leads to an accumulated absolute change of .004 in own sales growth over a time period of 10 lags. As for the direct effects, we systematically find that within-category effects are larger in magnitude than cross-category effects, especially for sales and prices. For the marketing mix, promotions exert stronger within- as well as cross-category effects than price changes.\n\n\\linespread{1.2}\n\\begin{table}\n\\centering\n\\caption{Size of within and cross-category effects of price, promotion and sales on sales, summed across 10 lags of the IRF, averaged across stores and product categories, and in absolute value.} \\label{Effectsizes}\n\\begin{tabular}{lccccccccc}\n \\hline\n &&& Price &&& Promotion &&& Sales \\\\\n \\hline\nWithin-category &&& 0.004 &&& 0.006 &&& 0.057 \\\\\nCross-category &&& 0.002 &&& 0.005 &&& 0.002 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\linespread{1.5}\n\n\\begin{table}\n\\caption{Cross-category price, promotion and sales effects on sales summed across 10 lags of IRFs and averaged across stores. We present only the five largest positive and negative effects. \\label{Crosscateffects}}\n\\footnotesize\n\\resizebox{0.95\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\\centering\n\\begin{tabular}{lllcc|llllcc}\n\\hline\n\\multicolumn{6}{l}{\\textbf{Cross-category price effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\ \\hline\nPrice && Sales && Effect && Price && Sales && Effect \\\\ \nimpulse && response & & && impulse && response & & \\\\ \\hline\n \\multicolumn{4}{c}{\\underline{Perceived complements}} &&& \\multicolumn{4}{c}{\\underline{Perceived substitutes}} &\\\\\nSoft Drinks && Canned Tuna && -0.0209 && Front-end-candies && Bottled Juices && 0.0120\\\\\nSoft Drinks && Frozen Entrees &&-0.0182&& Soft Drinks && Frozen Juices && 0.0060\\\\\nCanned Tuna && Canned Soup && -0.0173 && Snack Crackers && Beer && 0.0058\\\\\nCereals && Frozen Dinners && -0.0104 && Cookies && Oatmeal && 0.0056\\\\\nBottled Juices && Crackers && -0.0074 && Frozen Juices && Bottled Juices && 0.0023\\\\ \\hline\n\\multicolumn{6}{l}{\\textbf{Cross-category promotion effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\ \\hline\nPromotion && Sales && Effect && Promotion && Sales && Effect \\\\ \nimpulse && response && && impulse && response & & \\\\ \\hline\nBottled Juices && Frozen Entrees && 0.0586 && Oatmeal && Canned Tuna && -0.0214\\\\\nCheeses && Frozen Entrees && 0.0421 && Cheeses && Cookies && -0.0160 \\\\\nCrackers && Frozen Entrees && 0.0246 && Bottles Juices && Canned Tuna && -0.0158 \\\\\nFrozen Dinners && Frozen Entrees && 0.0170&& Refrigerated Juices && Canned Tuna && -0.0128 \\\\\nSnack Crackers && Frozen Entrees && 0.0127 && Cereals && Cheeses && -0.0127 \\\\ \\hline\n\n\\multicolumn{6}{l}{\\textbf{Cross-category sales effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\ \\hline\nSales && Sales && Effect && Sales && Sales && Effect \\\\\nimpulse && response & & && impulse && response && \\\\ \\hline\nFront-end-candies && Soft Drinks && 0.0191 && Snack Crackers && Oatmeal && -0.0154 \\\\\nOatmeal && Frozen Entrees && 0.0123 && Frozen Juices && Frozen Entrees && -0.0120 \\\\\nCanned Tuna && Crackers && 0.0094 && Cereals && Frozen Dinners && -0.0099 \\\\\nFront-end-candies && Beer && 0.0086 && Snack Crackers && Cookies && -0.0087\\\\\nSnack Crackers && Frozen Dinners && 0.0064 && Refrigerated Juices && Canned Tuna &&-0.0084 \\\\\\hline\n\\end{tabular}\n\\end{minipage} }\n\\end{table}\n\nTo get more insight in the sign of the cross-category effects, we summarize each IRF by the sum of the first 10 responses, and average this number over all stores. Table \\ref{Crosscateffects} reports the five largest positive and negative cross-category effects of price, promotion and sales on sales.\n\n\\medskip\n\n\\textit{Cross-category price effects.} We investigate whether consumers perceive categories as complements or as substitutes. Complementary and substitution effects occur between categories because they are consumed together or separately. Following the standard economic definition \\citep{Pashigian98}, complements are defined as goods having a negative cross-price elasticity, whereas substitutes are defined as goods having a positive cross-price elasticity. \n We find evidence of two important drivers of cross-category price effects: consumption relatedness and the budget constraint. \n \nAs an example of consumption relatedness, consider Soft Drink prices and Frozen Juices. An increase in Soft Drink prices makes consumers spend more on other drinks as a compensation, in particular Frozen Juices (see Table \\ref{Crosscateffects}). The joint dynamic effect of a one standard deviation price impulse of Soft Drinks on the sales response growth of Frozen Juices is depicted in Figure \\ref{IRF_PRICE} for the first three stores in the data set. Note that the instantaneous effect is estimated as exactly zero since the Sparse VAR puts the corresponding effect in the $\\widehat\\Sigma$ matrix to zero. We see a sharp increase in Frozen Juices sales growth one week after the soft drink price increase, indicating substitution. However, the next two weeks, sales growth of Frozen Juices slows down, which could indicate stockpiling behavior \\citep{Gangwar13}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14.1cm]{IRFS_SLStoPRICE}\n\\end{center}\n\\caption{Impulse response function: response of frozen juices sales growth to a one standard deviation impulse in the price of soft drinks. \\label{IRF_PRICE}}\n\\end{figure}\n\nAnother example of consumption relatedness is Soft Drinks and Frozen Entrees. As can be seen from Table \\ref{Crosscateffects}, we find a strong negative effect of Soft Drink prices on Frozen Entrees. This might be due to the fact that Soft Drinks and Frozen Entrees are consumed together. We do not find the opposite effect of price changes in Frozen Entrees on the sales of Soft Drinks. This asymmetry arises because Soft Drinks is a destination category (high influence), while Frozen Entrees is a convenience category (highly responsiveness). \n\n\n\n\nConcerning the budget constraint, prominent cross-category price effects are observed for Soft Drinks and Cereals, both destination categories. Soft Drinks and Cereals account for a relatively large proportion of the expenditures of US families (respectively 22\\% and 14\\% of spending on food and drinks, see Table \\ref{Categories}), which indicates that the budget constraint is an important source of cross-category effects. \n\n\\medskip\n\n\\textit{Cross-category promotion effects.}\nThe results in Table \\ref{Crosscateffects} indicate that branding and promotion intensity are important drivers of cross-category promotion effects.\nConcerning branding, cross-category promotion effects are observed for categories that share brands such as Frozen Dinners and Frozen Entrees (e.g. the frozen prepared foods brand ``Stouffer's\").\nConcerning promotion intensity, prominent cross-category promotion effects are observed for categories in which a high percentage of the SKUs is promoted, such as Cheeses and Bottled Juices (respectively 28\\% and 26\\% of SKUs, on average, are promoted in our data.) A promotion impulse in such categories might either trigger join consumption (e.g. Bottled Juices and Frozen Entrees), or deter consumption (e.g. Cheeses and Cookies).\n\n\n\\medskip\n\n\\textit{Cross-category sales effects.} \nIn Table \\ref{Crosscateffects}, we find evidence of two important drivers of cross-category effects of sales on sales: affinity in consumption and the budget constraint. \nProminent cross-category sales effects occur because of affinity in consumption. Some categories are jointly consumed towards a common goal, such as Front-end-candies and Soft Drinks\/Beer (for a light meal); while others such as Snack Crackers and Cookies are purchased as replacements since consumers might perceive them to have a similar functionality.\nConcerning the budget constraint, we find some cross-category sales effects between seemingly unrelated categories such as Refrigerated Juices and Canned Tuna.\n\n\n\\medskip\n\n\nImportantly, the results from Table \\ref{Crosscateffects} are in line with our findings on category influence and responsiveness. Destination categories such as Soft Drinks, Cereals and Cheeses mainly influence sales in other categories through their price, promotion or sales impulses. Convenience categories such as Frozen Entrees and Frozen Dinners are more responsive to changes in other categories. Routine categories, such as Cookies, are moderately influential and moderately responsive, while occasional categories, such as Oatmeal, are highly responsive. \n\n\n\\subsection{Robustness checks}\n\\textit{Alternative penalty function.} We investigate the robustness of the results to the choice of the penalty function. We re-estimate the models using the Sparse VAR with elastic net instead of the grouplasso penalty (a short explanation of the elastic net is given in Section 4). The managerial insights obtained by Sparse VAR with either grouplasso or elastic net are very similar. \nSimilarities are that \n(i) within-category effects are more common and larger in magnitude than cross-category effects, \n(ii) destination categories such as Cheeses and Cereals are very influential,\n(iii) convenience categories such as Frozen Entrees, and occasional categories such as Crackers are very responsive\n(iv) routine categories such as Bottled Juices, Refrigerated Juices and Cookies are both influential and responsive\n(v) the most prominent cross-category effects of price, promotion and sales on sales are highly overlapping.\n\n\\medskip\n\n\\textit{Alternative data period.} We also check the performance of the Sparse VAR on the post-1994 data. Retailers made extensive use of ``pay-for-performance\" price promotions that are not fully reflected in the Dominick's database. The data generating process might have changed in this period. Therefore, we should not assume constant parameter values. We re-estimate the model on the post-1994 data (data from October 1995 until May 1997) and verify its performance. In the post-1994 period, similar conclusions can be drawn with respect to within versus cross-category effects and category influence and responsiveness.\nSome differences are observed in the post-1994 period concerning the impulse response functions. These differences occur due to an altered strategy concerning average pricing and promotion intensity in the 17 product categories in the post-1994 period compared to the 1993-1994 period. Detailed results are available from the authors upon request.\n\n\\medskip\n\n\\textit{Alternative sparsity parameter selection.} Our results are based on the BIC to select the penalty parameters. We also ran the analysis using AIC as a selection criterion for the penalty function. While the model selected by AIC are slightly less sparse, the substantive insights do not change. \n\n\\subsection{Forecast Performance}\nAlthough prediction is not the main goal of the proposed methodology, we deem it important to show that the Sparse VAR can compete with other methods in terms of prediction accuracy. We estimate model \\eqref{eq: application model} for each store and perform a forecast exercise (cfr. Section 4), using a rolling window of length $S=67$. One-step-ahead forecasts of sales for each product category are computed for $t=S,\\ldots,T-1$, with $T=77$. The same estimation methods as in Section 4 are used.\n\nResults on the sales predictions are summarized in Table \\ref{Forecasts} by the Mean Absolute Forecast Error (MAFE), averaged across time and over the 17 product categories and 15 stores. The MAFE should be seen as a measure of forecast accuracy, not as a measure of managerial relevance of the obtained results.\nThe variable selection methods Sparse VAR, 1-step and Iterative Restricted LS perform, on average, better than the methods that don't perform variable selection. This indicates that sparsity improves prediction accuracy. Sparse VAR and Iterative Restricted LS achieve the best forecasting performance. A Diebold-Mariano test \\citep{Diebold95} confirms that latter two methods significantly outperform the other methods. We conclude that the improvement in interpretability of the model obtained by Sparse VAR, as discussed in the previous section, does not come at the cost of lower forecast performance.\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Mean Absolute Forecast Error (MAFE) for category-specific sales, averaged over the 15 stores and the 17 product categories. $P$-values of a Diebold-Mariano test comparing the Sparse VAR to its alternatives are indicated between parentheses. \\label{Forecasts}}\n\\small\n\\begin{tabular}{lcccccc}\n \\hline\n \\rule{0pt}{3ex} & Sparse VAR & & \\multicolumn{2}{c}{Restricted LS} & \\multicolumn{2}{c}{Bayesian Methods} \\\\\n \\rule{0pt}{3ex} & & LS & 1-step & Iterative & Minnesota & NIW \\\\\n \\hline\nMAFE & 736.80 & $\\underset{(<0.01)}{1298.54}$ & $\\underset{(<0.01)}{784.96}$ & $\\underset{(0.38)}{734.82}$ & $\\underset{(<0.01)}{875.47}$ & $\\underset{(<0.01)}{1078.03}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\n\\section{Discussion}\n\nThis paper presents a Sparse VAR methodology to detect the inter-relationships in a large product category network. In the cross-category demand effects application, we detect an important number of cross-category demand effects for a large number of categories. We find that categories have asymmetric roles: While destination categories are more influential, convenience categories are more responsive.\nWe identify main perceived cross-category effects but also detect cross-category effects between categories that are not directly related at first sight. Hence, the need to study -- potentially a large number of -- product categories simultaneously. While cross-category effects are prevalent, many of them are still absent, calling for a sparse estimation procedure that succeeds in highlighting the main inter-relationships in the product category network.\n\nWe identify category influence and responsiveness in our cross-category demand effects application using aggregate store level data. Other cross-category studies, such as \\cite{Russell97,Ainslie98,Russell99,Russell00,Elrod02} use market basket data. Since the availability and use of such market basket data pose difficulties to managers, they rarely use market basket data for category analysis \\citep{Shankar2014}. As managerial decisions are often made at the category level, managers prefer to work with more readily available aggregate store level data. Hence, using aggregate category store level is managerially relevant \\citep{ailawadi:09, leeflang:12}.\n\nA first limitation of our approach is that we use aggregate category data, which might lead to biased estimates when there is heterogeneity on the SKU level \\citep{Dekimpe00}. Second, our model does not allow to estimate cross-category effects on the individual consumer level. Insights into the behavior of consumers are revealed using market basket data, which requires a very different modeling approach. Despite these limitations, aggregate category data are highly relevant from the perspective of category management within the store.\n\nAn important advantage of the Sparse VAR is that it overcomes the dimensionality problem -- it results in a parsimonious model with minimal structural constraints. We show that this leads to more accurate estimation and prediction results as compared to standard Least Squares methods.\nIf the researcher wishes to restrict some of the parameters to zero a priori, using marketing theory, this is of course still possible to implement with the Sparse VAR. The same holds for the reverse, i.e.\\ forcing some variables to be included in the model, which can be done by adjusting the penalty on the regression coefficients in \\eqref{mincrit}.\n\nThe methodology presented in this paper is relevant in a variety of other settings. First, Sparse VAR can be used to study competitive demand effects across many competitors. The VAR is ideal for measuring competitive effects since it is able to capture own- and cross-elasticity of sales to both pricing and marketing spending \\citep{Srinivasan04, Horvath08}.\nTypically only three competitors are included in such studies, while using the Sparse VAR allows for a much larger number to be included. Second, in the field of international marketing research there is an increased interest in studying cross-country spill-over effects, as for example in \\cite{Albuquerque07}, \\cite{VanEverdingen09} and \\cite{Kumar02}. Every country that is added to the data set leads to an increase in the number of cross-country parameters to be estimated. Using the proposed methodology, a large VAR model could be built which allows spill-over effects between many countries. Finally, the Market Response Model could be extended with data on online word of mouth or online search, which are now readily available. Especially in the Big Data era, most companies collect an abundance of variables \\citep{BigData2013}, such that large VAR models will become even larger as more granular data become available. \n\n\\bigskip \\noindent\n{\\bf Acknowledgments.} The authors thank the Editors, Shankar Ganesan and Murali K. Mantrala, and two anonymous referees for their valuable comments that have improved the paper significantly. Financial support from the FWO (Research Foundation Flanders) is also gratefully acknowledged (FWO, contract number 11N9913N).\n\n\n\\begin{appendices}\n\\numberwithin{equation}{section}\n\\section{Penalized Likelihood Estimation}\n\\noindent\nWe iteratively solve the minimization problem \\eqref{mincrit} for $\\beta$ conditional on $\\Omega$ and then for $\\Omega$ conditional on $\\beta$.\n\n\\medskip \\noindent\n{\\it Solving for $\\beta|\\Omega$:} When $\\Omega$ is fixed, the minimization problem in \\eqref{mincrit} is equivalent to minimizing\n\\begin{equation}\\label{mincritbeta}\n\\hat{\\beta}|\\Omega = \\underset{\\beta}{\\operatorname{argmin}} \\frac{1}{n} (\\tilde{y}-\\tilde{X} \\beta)^{\\prime} (\\tilde{y}-\\tilde{X} \\beta) + \\lambda_1 \\sum_{g=1}^{G} ||\\beta_g||_2 \\, ,\n\\end{equation}\nwhere $\\tilde{y}= Py$, $\\tilde{X}=PX$, and $P$ is a matrix such that $P^{\\prime}P=\\tilde{\\Omega}$. \nThe transformation of the data to $\\tilde{y}$ and $\\tilde{X}$ ensures that the resulting model has uncorrelated and homoscedastic error terms. The above minimization problem is convex if $\\Omega$ is nonnegative definite. The minimization problem is equivalent to the groupwise lasso of \\cite{Yuan06}, implemented in the R package \\verb+grplasso+ \\citep{Rgrplasso}.\n\n\n\\noindent\n{\\it Solving for $\\Omega|\\beta$:} When $\\beta$ is fixed, the minimization problem in \\eqref{mincrit} reduces to\n\\begin{equation}\\label{mincritOmega}\n\\hat{\\Omega}|\\beta = \\underset{\\Omega}{\\operatorname{argmin}} \\, \\frac{1}{n} (y-X \\beta)^{\\prime} \\tilde{\\Omega} (y-X \\beta)- \\log|\\Omega| + \\lambda_2 \\sum_{k \\neq k'} |\\Omega_{kk'}| \\, ,\n\\end{equation}\nwhich corresponds to penalized covariance estimation. Using the glasso algorithm\nof \\cite{Friedman07}, available in the R package \\verb+glasso+ \\citep{Rglasso}, the optimization problem in \\eqref{mincritOmega}\nis solved.\n\nWe start the algorithm by taking $\\widehat{\\Omega}=I_q$ and iterate until convergence. We iterate until $max_s |\\hat{\\beta}_{s,i}-\\hat{\\beta}_{s,i-1}|<\\epsilon$, with $\\hat{\\beta}_{s,i}$ the $s^{th}$ parameter estimate in iteration $i$ (same for $\\hat{\\Omega}$) and the tolerance $\\epsilon$ set to $10^{-3}$. \n\n\\paragraph{Selecting the Sparsity Parameters and the order of the VAR}\nWe first determine the optimal values of $\\lambda_1$ and $\\lambda_2$ for a fixed value of $p$, the order of the VAR. The sparsity parameters $\\lambda_1$ and $\\lambda_2$ are selected according to a minimal Bayes Information Criterion (BIC).\nIn the iteration step where $\\beta$ is estimated conditional on $\\Omega$, we solve \\eqref{mincritbeta} over a range of values for $\\lambda_1$ and select the one with lowest value of\n\\begin{equation}\\label{eq: BICbeta}\nBIC_{\\lambda_1} = -2 \\log L_{\\lambda_1} + k_{\\lambda_1} \\log(n),\n\\end{equation}\nwhere $L_{\\lambda_1}$ is the estimated likelihood, corresponding to the first term in \\eqref{mincritbeta}, using sparsity parameter $\\lambda_1$. Furthermore, $k_{\\lambda_1}$ is the number of non-zero estimated regression coefficients and $n$ the number of observations.\nSimilarly, for selecting $\\lambda_2$, we use the BIC given by\n\\begin{equation}\\label{eq: BIComega}\nBIC_{\\lambda_2} = -2 \\log L_{\\lambda_2} + k_{\\lambda_2} \\log(n) \\, .\n\\end{equation}\nFinally, we select the order $p$ of the VAR. We estimate the VAR for different values of $p$. The optimal values of $\\lambda_1$ and $\\lambda_2$ are determined for a each of those values of $p$. We select the order $p$ of the VAR using BIC:\n\\begin{equation}\\label{eq: BICp}\nBIC_{(p, \\lambda_1(p), \\lambda_2(p))} = -2 \\log L_{(p, \\lambda_1(p), \\lambda_2(p))} + k_{(p, \\lambda_1(p), \\lambda_2(p))} \\log(n) \\, , \n\\end{equation}\nwhere $L_{(p, \\lambda_1(p), \\lambda_2(p))}$ and $k_{(p, \\lambda_1(p), \\lambda_2(p))}$ depend on the value $p$ and the optimally chosen values of $\\lambda_1(p)$ and $\\lambda_2(p)$ for that specific value of $p$.\n\\end{appendices}\n\n\n\\linespread{1}\n\\bibliographystyle{asa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Primitive trace map, local injectivity}\n\nLet $(M,g)$ be a smooth closed Riemannian Anosov manifold such as a manifold of negative sectional curvature \\cite{Anosov-67}. Recall that this means that there exists a continuous flow-invariant splitting of the tangent bundle to the unit tangent bundle $\\mathcal{M} := SM$:\n\\[\nT\\mathcal{M} = \\mathbb{R} X \\oplus E_s \\oplus E_u,\n\\]\nsuch that:\n\\begin{equation}\n\\label{equation:anosov}\n\\begin{array}{l}\n\\forall t \\geq 0, \\forall v \\in E_s, ~~ |\\dd\\varphi_t(v)| \\leq Ce^{-t\\theta}|v|, \\\\\n\\forall t \\leq 0, \\forall v \\in E_u, ~~ |\\dd\\varphi_t(v)| \\leq Ce^{-|t|\\theta}|v|,\n\\end{array}\n\\end{equation}\nwhere $(\\varphi_t)_{t \\in \\mathbb{R}}$ is the geodesic flow on $\\mathcal{M}$, and the constants $C,\\theta > 0$ are uniform and the metric $|\\cdot|$ is arbitrary. \n\nLet $\\E \\rightarrow M$ be a smooth Hermitian vector bundle. We denote by $\\mathcal{A}_{\\E}$ the affine space of smooth unitary connections on $\\E$ and $\\mathbb{A}_{\\E}$ the moduli space of connections up to gauge-equivalence, namely a point $\\mathfrak{a} \\in \\mathbb{A}_{\\E}$ is an orbit $\\mathfrak{a} := \\left\\{p^*\\nabla^{\\E} ~|~p \\in C^\\infty(M,\\mathrm{U}(\\E))\\right\\}$ of gauge-equivalent connections, where $\\nabla^{\\E} \\in \\mathfrak{a}$ is arbitrary. We let $\\mathcal{C} = \\left\\{c_1,c_2,...\\right\\}$ be the set of free homotopy classes of loops on $M$ which is known to be in one-to-one correspondence with closed geodesics \\cite{Klingenberg-74}. More precisely, given $c \\in \\mathcal{C}$, there exists a unique closed geodesic $\\gamma_g(c) \\subset M$ in the class $c$. It will be important to make a difference between \\emph{primitive} and \\emph{non-primitive} homotopy classes (resp. closed geodesics): a free loop is said to be primitive if it cannot be homotoped to a certain power (greater or equal than $2$) of another free loop. The set of primitive classes defines a subset $\\mathcal{C}^\\sharp = \\left\\{c_1^\\sharp,c_2^\\sharp,...\\right\\} \\subset \\mathcal{C}$.\n\nGiven a class $\\mathfrak{a} \\in \\mathbb{A}_{\\E}$, a unitary connection $\\nabla^{\\E} \\in \\mathfrak{a}$ and an arbitrary point $x_{c^\\sharp} \\in \\gamma_g(c^\\sharp)$ (for some $c^\\sharp \\in \\mathcal{C}^\\sharp$), the parallel transport $\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp) \\in \\mathrm{U}(\\E_{x_{c^\\sharp}})$, starting at $x_{c^\\sharp}$, with respect to $\\nabla^{\\E}$ and along $\\gamma_g({c^\\sharp})$ depends on the choice of representative $\\nabla^{\\E} \\in \\mathfrak{a}$ since two gauge-equivalent connections have conjugate holonomies. However, the trace does not depend on a choice of $\\nabla^{\\E} \\in \\mathfrak{a}$ and the \\emph{primitive trace map}:\n\\begin{equation}\n\\label{equation:trace}\n\\mathcal{T}^\\sharp : \\mathbb{A}_{\\E} \\ni \\mathfrak{a} \\mapsto \\left(\\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp_1)\\right), \\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp_2)\\right), ...\\right) \\in \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nis therefore well-defined. Observe that the data of the primitive trace map is a rather weak information: in particular, it is \\emph{not} (\\emph{a priori}) equivalent to the data of the conjugacy class of the holonomy along each closed geodesic (and the latter is the same as the non-primitive trace map, where one considers \\emph{all} closed geodesics). One of the main results of this paper is the following:\n\n\\begin{theorem}\n\\label{theorem:injectivity}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$ and let $\\E \\rightarrow M$ be a smooth Hermitian vector bundle. Let $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$ be a \\emph{generic} point. Then, the primitive trace map is \\emph{locally injective} near $\\mathfrak{a}_0$.\n\\end{theorem}\n\nBy \\emph{local injectivity}, we mean the following: there exists $N \\in \\mathbb{N}$ (independent of $\\mathfrak{a}_0$) such that $\\mathcal{T}^\\sharp$ is locally injective in the $C^N$-quotient topology on $\\mathbb{A}_{\\E}$. In other words, for any element $\\nabla^{\\E}_0 \\in \\mathfrak{a}_0$, there exists $\\varepsilon > 0$ such that the following holds; if $\\nabla_{1,2}^{\\E}$ are two smooth unitary connections such that $\\|p_i^*\\nabla_{i}^{\\E} - \\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$ for some $p_i \\in C^\\infty(M,\\mathrm{U}(\\E))$, and $\\mathcal{T}^\\sharp(\\nabla_1^{\\E}) = \\mathcal{T}^\\sharp(\\nabla_2^{\\E})$, then $\\nabla_1^{\\E}$ and $\\nabla_2^{\\E}$ are gauge-equivalent.\n\nWe say that a point $\\mathfrak{a}$ is \\emph{generic} if it enjoys the following two features: \n\n\\begin{itemize}\\label{def:generic}\n\\item[\\textbf{(A)}] $\\mathfrak{a}$ is \\textbf{opaque}. By definition (see \\cite[Section 5]{Cekic-Lefeuvre-20}), this means that for all $\\nabla^{\\E} \\in \\mathfrak{a}$, the parallel transport map along geodesics does not preserve any non-trivial subbundle $\\mathcal{F} \\subset \\E$ (i.e. $\\mathcal{F}$ is preserved by parallel transport along geodesics if and only if $\\mathcal{F} = \\left\\{0\\right\\}$ or $\\mathcal{F} = \\E$). This was proved to be equivalent to the fact that the Pollicott-Ruelle resonance at $z=0$ of the operator $\\mathbf{X} := \\pi^* \\nabla^{\\mathrm{End}}_X$ has multiplicity equal to $1$, with resonant space $\\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$ (here $\\pi : SM \\rightarrow M$ is the projection; $\\nabla^{\\mathrm{End}}$ is the induced connection on the endomorphism bundle, see \\S\\ref{ssection:connections} for further details);\n\n\n\\item[\\textbf{(B)}] $\\mathfrak{a}$ has \\textbf{solenoidally injective generalized X-ray transform} $\\Pi^{\\mathrm{End}(\\E)}_1$ on twisted $1$-forms with values in $\\mathrm{End}(\\E)$. This last assumption is less easy to describe in simple geometric terms: roughly speaking, the X-ray transform is an operator of integration of symmetric $m$-tensors along closed geodesics. For vector-valued symmetric $m$-tensors, this might not be well-defined, and one needs a more general (hence, more abstract) definition involving the residue at $z=0$ of the meromorphic extension of the family $\\mathbb{C} \\ni z \\mapsto (-\\mathbf{X}-z)^{-1}$, see \\S\\ref{section:twisted}. \n\\end{itemize}\nIt was shown in previous articles \\cite{Cekic-Lefeuvre-20,Cekic-Lefeuvre-21-2} that in dimension $n \\geq 3$, properties \\textbf{(A)} and \\textbf{(B)} are satisfied on an open dense subset $\\omega \\subset \\mathbb{A}_{\\E}$ with respect to the $C^N$-quotient topology.\\footnote{More precisely, there exists $N \\in \\mathbb{N}$ and a subset $\\Omega \\subset \\mathcal{A}_{\\E}$ of the (affine) Fr\\'echet space of smooth affine connections on $\\E$ such that $\\omega = \\pi_{\\E}(\\Omega)$ (where $\\pi_{\\E} : \\mathcal{A}_{\\E} \\rightarrow \\mathbb{A}_{\\E}$ is the projection) and\n\\begin{itemize}\n\\item $\\Omega$ is invariant by the action of the gauge-group, namely $p^*\\Omega = \\Omega$ for all $p \\in C^\\infty(M,\\mathrm{U}(\\E))$;\n\\item $\\Omega$ is open, namely for all $\\nabla^{\\E}_0 \\in \\Omega$, there exists $\\varepsilon > 0$ such that if $\\nabla^{\\E} \\in \\mathcal{A}_{\\E}$ and $\\|\\nabla^{\\E}-\\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$, then $\\nabla^{\\E} \\in \\Omega$;\n\\item $\\Omega$ is dense, namely for all $\\nabla^{\\E}_0 \\in \\mathcal{A}_{\\E}$, for all $\\varepsilon > 0$, there exists $\\nabla^{\\E} \\in \\Omega$ such that $\\|\\nabla^{\\E}-\\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$;\n\\item Connections in $\\Omega$ satisfy properties \\textbf{(A)} and \\textbf{(B)}.\n\\end{itemize}} When the reference connection $\\mathfrak{a}$ satisfies only the property \\textbf{(A)} (this is the case for the product connection on the trivial bundle for instance), we are able to show a \\emph{weak local injectivity} result, see Theorem \\ref{thm:weaklocal}. \n\nWe note that the gauge class of a connection is uniquely determined from the holonomies along \\emph{all} closed loops \\cite{Barrett-91, Kobayashi-54} and that within the mathematical physics community our primitive trace map $\\mathcal{T}^\\sharp$ is known as the \\emph{Wilson loop} operator \\cite{Beasley-13, Giles-81, Loll-93, Wilson-74}. In stark contrast, our Theorem \\ref{theorem:injectivity} says that the \\emph{restriction to closed geodesics} of this operator already determines (locally) the gauge class of the connection.\n\n\\subsection{Global injectivity}\n\n\\label{ssection:global-inj}\n\nWe now mention some global injectivity results. We let $\\mathbb{A}_r := \\bigsqcup_{\\E_r \\in \\mathrm{Vect}_r(M)} \\mathbb{A}_{\\E_r}$, where the disjoint union runs over all Hermitian vector bundles $\\E_r \\in \\mathrm{Vect}_r(M)$ of rank $r$ over $M$ up to isomorphisms, and we set:\n\\[\n\\mathbb{A} := \\bigsqcup_{r \\geq 0} \\mathbb{A}_r,\n\\]\nand $\\mathrm{Vect}(M) = \\bigsqcup_{r \\geq 0} \\mathrm{Vect}_r(M)$ be the space of all topological vector bundles up to isomorphisms. A point $\\mathrm{x} \\in \\mathbb{A}$ corresponds to a pair $([\\mathcal{E}],\\mathfrak{a})$, where $[\\mathcal{E}] \\in \\mathrm{Vect}(M)$ is an equivalence class of Hermitian vector bundles and $\\mathfrak{a}$ a class of gauge-equivalent unitary connections.\\footnote{Note that if two smooth Hermitian vector bundles $\\E_1$ and $\\E_2$ are isomorphic as topological vector bundles (i.e. there exists an invertible $p \\in C^\\infty(M,\\mathrm{Hom}(\\E_1,\\E_2))$), then they are also isomorphic as Hermitian vector bundles, that is $p$ can be taken unitary; the choice of Hermitian structure is therefore irrelevant.} \n\nThe space $\\mathbb{A}$ has a natural monoid structure given by the $\\oplus$-operator of taking direct sums (both for the vector bundle part and the connection part). The primitive trace map can then be seen as a \\emph{global} (monoid) homomorphism: \n\\begin{equation}\n\\label{equation:trace-total}\n\\mathcal{T}^\\sharp : \\mathbb{A} \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nwhere $\\ell^\\infty(\\mathcal{C}^\\sharp)$ is endowed with the obvious additive structure. We actually conjecture that the generic assumption of Theorem \\ref{theorem:injectivity} is unnecessary and that the primitive trace map \\eqref{equation:trace-total} should be globally injective if $\\dim(M) \\geq 3$ and $\\dim (M)$ is odd. Let us discuss a few partial results supporting the validity of this conjecture:\n\n\\begin{enumerate}\n\t\\item In \\S\\ref{sssection:line}, we show that the primitive trace map is injective when restricted to \\emph{direct sums of line bundles} when $\\dim(M) \\geq 3$, see Theorem \\ref{theorem:sum}. Note that it was proved by Paternain \\cite{Paternain-09} that the primitive trace map restricted to line bundles $\\mathcal{T}_1^\\sharp : \\mathbb{A}_1 \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp)$ is injective when $\\dim(M) \\geq 3$. \n\n\t\\item In \\S\\ref{sssection:flat}, under the restriction of $\\mathcal{T}^\\sharp$ to \\emph{flat} connections, we show that the primitive trace map $\\mathcal{T}^\\sharp$ is globally injective, see Proposition \\ref{proposition:flat}.\n\t\n\t\\item In \\S\\ref{sssection:negative}, we also obtain a global result in negative curvature under an extra \\emph{spectral condition}, see Proposition \\ref{proposition:negative}. This condition is generic (see Appendix \\ref{appendix:ckts}) and is also satisfied by connections with \\emph{small curvature}, i.e. whose curvature is controlled by a constant depending only on the dimension and an upper bound on the sectional curvature of $(M,g)$ (see Lemma \\ref{lemma:small-curvature}).\n\t\n\t\n\t\\item In \\S\\ref{sssection:topology}, as a consequence of Corollary \\ref{corollary:iso} below, we have that the primitive trace map $\\mathcal{T}^{\\sharp}([\\mathcal{E}],\\mathfrak{a})$ allows to recover the isomorphism class $\\pi^*[\\E]$. In particular if $\\dim M$ is odd, this suffices to recover $[\\E]$, see Proposition \\ref{proposition:topology}.\n\n\\end{enumerate}\n\n\n\n\n\n\n\n\nTheorem \\ref{theorem:injectivity} is inspired by earlier work on the subject, see \\cite{Paternain-09,Paternain-12,Paternain-13,Paternain-lecture-notes,Guillarmou-Paternain-Salo-Uhlmann-16} for instance. Nevertheless, it goes beyond the aforementioned literature thanks to an \\emph{exact Liv\\v{s}ic cocycle Theorem} (see Theorem \\ref{theorem:weak-intro}), explained in the next paragraph \\S\\ref{ssection:exact}. It also belongs to a more general family of \\emph{geometric inverse results} which has become a very active field of research in the past twenty years, both on closed manifolds and on manifolds with boundary, see \\cite{Pestov-Uhlmann-05, Stefanov-Uhlmann-04,Paternain-Salo-Uhlmann-13, Uhlmann-Vasy-16,Stefanov-Uhlmann-Vasy-17,Guillarmou-17-2} among other references.\n\n\n\nTheorem \\ref{theorem:injectivity} can also be compared to a similar problem called the \\emph{marked length spectrum} (MLS) rigidity conjecture, also known as the Burns-Katok \\cite{Burns-Katok-85} conjecture. The latter asserts that if $(M,g)$ is Anosov, then the marked length spectrum\n\\begin{equation}\n\\label{equation:mls}\nL_g : \\mathcal{C} \\rightarrow \\mathbb{R}_+, ~~~L_g(c) := \\ell_g(\\gamma_g(c)),\n\\end{equation}\n(where $\\ell_g(\\gamma)$ denotes the Riemannian length of the curve $\\gamma \\subset M$ computed with respect to the metric $g$), namely the length of all closed geodesics marked by the free homotopy classes of $M$, should determine the metric up to isometry. Despite some partial answers \\cite{Katok-88, Croke-90,Otal-90,Besson-Courtois-Gallot-95,Hamenstadt-99,Guillarmou-Lefeuvre-18}, this conjecture is still widely open. Recently, Guillarmou and the second author proved a local version of the Burns-Katok conjecture \\cite{Guillarmou-Lefeuvre-18} using techniques from microlocal analysis and the theory of Pollicott-Ruelle resonances. \n\n\\subsection{Inverse Spectral problem}\n\n\nThe \\emph{length spectrum} of the Riemannian manifold $(M,g)$ is the collection of lengths of closed geodesics \\emph{counted with multiplicities}. It is said to be \\emph{simple} if all closed geodesics have distinct lengths and this is known to be a generic condition (with respect to the metric), see \\cite{Abraham-70,Anosov-82} (even in the non-Anosov case). Given $\\nabla^{\\E} \\in \\mathfrak{a}$, one can form the \\emph{connection Laplacian} $\\Delta_{\\nabla^{\\E}} := (\\nabla^{\\E})^*\\nabla^{\\E}$ (also known as the Bochner Laplacian) which is a differential operator of order $2$, non-negative, formally self-adjoint and elliptic, acting on $C^\\infty(M,\\E)$. While $\\Delta_{\\nabla^{\\E}}$ depends on a choice of representative $\\nabla^{\\E}$ in the class $\\mathfrak{a}$, its spectrum does not and there is a well-defined \\emph{spectrum map}:\n\\begin{equation}\\label{eq:spectrummap}\n\\mathcal{S} : \\mathbb{A}_{\\E} \\ni \\mathfrak{a} \\mapsto \\mathrm{spec}(\\Delta_{\\mathfrak{a}}),\n\\end{equation}\nwhere $\\mathrm{spec}(\\Delta_{\\mathfrak{a}}) = \\left\\{0 \\leq \\lambda_0(\\mathfrak{a}) \\leq \\lambda_1(\\mathfrak{a}) \\leq ...\\right\\}$ is the spectrum counted with multiplicities. Note that more generally, the spectrum map \\eqref{eq:spectrummap} can be defined on the whole moduli space $\\mathbb{A}$ (just as the primitive trace map \\eqref{equation:trace}). The trace formula of Duistermaat-Guillemin \\cite{Duistermaat-Guillemin-75, Guillemin-73} applied to $\\Delta_{\\mathfrak{a}}$ reads (when the length spectrum is simple):\n\\begin{equation}\n\\label{equation:trace-formula}\n\\lim_{t \\to \\ell(\\gamma_g(c))} \\left(t-\\ell(\\gamma_g(c))\\right) \\sum_{j \\geq 0} e^{-i \\sqrt{\\lambda_j}(\\mathfrak{a})t} = \\dfrac{\\ell(\\gamma_g(c^\\sharp)) \\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c)\\right)}{2\\pi |\\det(\\mathbbm{1}-P_{\\gamma_g(c)})|^{1\/2}} ,\n\\end{equation}\nwhere $\\sharp : \\mathcal{C} \\rightarrow \\mathcal{C}^\\sharp$ is the operator giving the primitive orbit associated to an orbit; $P_\\gamma$ is the Poincar\\'e map associated to the orbit $\\gamma$ and $\\ell(\\gamma)$ its length. Theorem \\ref{theorem:injectivity} therefore has the following straightforward consequence:\n\n\\begin{corollary}\n\\label{corollary:spectral}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$ \\emph{with simple length spectrum}. Then:\n\\begin{itemize}\n\\item Let $\\E \\rightarrow M$ be a smooth Hermitian vector bundle and $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$ be a generic point. Then, the spectrum map $\\mathcal{S}$ is locally injective near $\\mathfrak{a}_0$.\n\\item The spectrum map $\\mathcal{S}$ is also globally injective when restricted to the cases \\emph{(1)-(4)} of the previous paragraph \\S\\ref{ssection:global-inj}.\n\\end{itemize}\n\\end{corollary}\n\nThis corollary simply follows from Theorem \\ref{theorem:injectivity} by observing that under the simple length spectrum assumption, the primitive trace map can be recovered from the equality \\eqref{equation:trace-formula}. Corollary \\ref{corollary:spectral} is analogous to the Guillemin-Kazhdan \\cite{Guillemin-Kazhdan-80,Guillemin-Kazhdan-80-2} rigidity result in which a potential $q \\in C^\\infty(M)$ is recovered from the knowledge of the spectrum of $-\\Delta_g + q$ (see also \\cite{Croke-Sharafutdinov-98,Paternain-Salo-Uhlmann-14-1}). As far as the connection Laplacian is concerned, it seems that Corollary \\ref{corollary:spectral} is the first positive result in this direction. Counter-examples were constructed by Kuwabara \\cite{Kuwabara-90} using the Sunada method \\cite{Sunada-85} but on coverings of a given Riemannian manifolds; hence the simple length spectrum condition is clearly violated. Up to our knowledge, it is also the first positive general result in an inverse spectral problem on a closed manifold of dimension $> 1$ with an \\emph{infinite} gauge-group. \n\nThis gives hope that similar methods could be used in the classical problem of recovering the isometry class of a metric from the spectrum of its Laplace-Beltrami operator \\emph{locally} (similarly to a conjecture of Sarnak for planar domains \\cite{Sarnak-90}). Such a result was already obtained in a neighbourhood of negatively-curved locally symmetric spaces by Sharafutdinov \\cite{Sharafutdinov-09}. See also \\cite{Croke-Sharafutdinov-98} for the weaker deformational spectral rigidity results or \\cite{DeSimoi-Kaloshin-Wei-17, Hezari-Zelditch-19} for recent results in the plane.\n\n\\subsection{Exact Liv\\v{s}ic cocycle theorem}\n\n\\label{ssection:exact}\n\nThe main ingredient in the proof of Theorem \\ref{theorem:injectivity} is the following Liv\\v{s}ic-type result in hyperbolic dynamical systems, which may be of independent interest. It shows that the cohomology class of a unitary cocycle over a transitive Anosov flow is determined by its trace along primitive periodic orbits. We phrase it in a somewhat more general context where we allow non-trivial vector bundles\n\n\\begin{theorem}\n\\label{theorem:weak-intro}\nLet $\\mathcal{M}$ be a smooth manifold endowed with a smooth transitive Anosov flow $(\\varphi_t)_{t \\in \\mathbb{R}}$. For $i\\in \\left\\{1,2\\right\\}$, let $\\E_i \\rightarrow \\mathcal{M}$ be a Hermitian vector bundle over $\\mathcal{M}$ equipped with a unitary connection $\\nabla^{\\mathcal{E}_i}$, and denote by $C_i(x,t) : (\\E_i)_x \\rightarrow (\\E_i)_{\\varphi_t(x)}$ the parallel transport along the flowlines with respect to $\\nabla^{\\E_i}$. If the connections have \\emph{trace-equivalent holonomies} in the sense that for all \\emph{primitive} periodic orbits $\\gamma$, one has\n\\begin{equation}\n\\label{equation:trace-intro}\n\\Tr\\left(C_1(x_\\gamma,\\ell(\\gamma))\\right) = \\Tr\\left(C_2(x_\\gamma,\\ell(\\gamma))\\right),\n\\end{equation}\nwhere $x_\\gamma \\in \\gamma$ is arbitrary and $\\ell(\\gamma)$ is the period of $\\gamma$, then the following holds: there exists $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2,\\E_1))$ such that for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$,\n\\begin{equation}\n\\label{equation:cohomologous}\nC_1(x,t) = p(\\varphi_t x) C_2(x,t) p(x)^{-1}.\n\\end{equation}\n\\end{theorem}\n\nIn the vocabulary of dynamical systems, note that each unitary cocycle is given by parallel transport along some unitary connection and \\eqref{equation:cohomologous} says that the cocycles induced by parallel transport are \\emph{cohomologous}. In particular, in the case of the trivial principal bundle $\\mathrm{U}(r) \\times \\mathcal{M} \\rightarrow \\mathcal{M}$ our theorem can be restated just in terms of $\\mathrm{U}(r)$-cocycles. Note that the bundles $\\E_1$ and $\\E_2$ could be \\emph{a priori} distinct (and have different ranks) but Theorem \\ref{theorem:weak-intro} shows that they are actually isomorphic:\n\n\\begin{corollary}\n\\label{corollary:iso}\nLet $\\E_1,\\E_2 \\rightarrow \\mathcal{M}$ be two Hermitian vector bundles equipped with respective unitary connection $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$. If the traces of the holonomy maps agree as in \\eqref{equation:trace-intro}, then $\\E_1 \\simeq \\E_2$ are isomorphic.\n\n\\end{corollary}\n\nTheorem \\ref{theorem:weak-intro} has other geometric consequences which are further detailed in \\S\\ref{ssection:intro}. Liv\\v{s}ic-type theorems have a long history in hyperbolic dynamical systems going back to the seminal paper of Liv\\v{s}ic \\cite{Livsic-72} and appear in various settings. They were both developed in the Abelian case i.e. for functions (see \\cite{Livsic-72,DeLaLlave-Marco-Moryon-86, Lopes-Thieullen-05,Guillarmou-17-1,Gouezel-Lefeuvre-19} for instance) and in the cocycle case.\n\nSurprisingly, we could not locate any result such as Theorem \\ref{theorem:weak-intro} in the literature. The closest works (in the discrete-time case) are that of Parry \\cite{Parry-99} and Schmidt \\cite{Schmidt-99} which mainly inspired the proof of Theorem \\ref{theorem:weak-intro}. Nevertheless, when considering compact Lie groups, Parry's and Schmidt's results seem to be weaker as they need to assume that the conjugacy classes of the cocycles agree (and not only the traces) and that a certain additional cocycle is transitive in order to derive the same conclusion. The literature is mostly concerned with the discrete-time case, namely hyperbolic diffeomorphisms: in that case, a lot of articles are devoted to studying cocycles with values in a non-compact Lie group (and sometimes satisfying a ``slow-growth\" assumption), see \\cite{DeLaLlave-Windsor-10,Kalinin-11, Sadovskaya-13, Sadvoskaya-17, Avila-Kocsard-Liu-18}. One can also wonder if Theorem \\ref{theorem:weak-intro} could be proved in the non-unitary setting. Other articles such as \\cite{Nitica-Torok-95,Nitica-Torok-98,Nicol-Pollicott-99,Walkden-00,Pollicott-Walkden-01} seem to have been concerned with regularity issues on the map $p$, namely bootstrapping its regularity under some weak \\emph{a priori} assumption (such as measurability only). Let us also point out at this stage that some regularity issues will appear while proving Theorem \\ref{theorem:weak-intro} but this will be bypassed by the use of a recent regularity statement \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20} in hyperbolic dynamics.\n\n\n\\subsection{Organization of the paper}\n\nThe paper is divided in three parts:\n\n\\begin{itemize}\n\n\\item First of all, we prove in Section \\S\\ref{section:livsic} the \\emph{exact Liv\\v{s}ic cocycle} Theorem \\ref{theorem:weak-intro} for general Anosov flows showing that the cohomology class of a unitary cocycle is determined by its trace along closed orbits. The proof is based on the introduction of a new tool which we call \\emph{Parry's free monoid}, denoted by $\\mathbf{G}$, and formally generated by orbits homoclinic to a given closed orbit. We show that any unitary connection induces a unitary representation of the monoid $\\mathbf{G}$ and that trace-equivalent connections have the same character; we can then apply tools from representation theory to conclude. We believe that this notion could be used in other contexts. It is reinvested in a companion paper \\cite{Cekic-Lefeuvre-21-3} to study \\emph{transparent manifolds}, namely Riemannian manifolds with trivial holonomy along closed geodesics.\n\n\\item In subsequent sections, we develop a microlocal framework, based on the theory of Pollicott-Ruelle resonances. We define a notion of \\emph{generalized X-ray transform with values in a vector bundle} which is mainly inspired by \\cite{Guillarmou-17-1,Guillarmou-Lefeuvre-18,Gouezel-Lefeuvre-19}. In \\S\\ref{section:geometry}, we relate the geometry of the moduli space of gauge-equivalent connections with the leading Pollicott-Ruelle resonance of a certain natural operator (called the mixed connection).\n\n\\item Eventually, the main results such as Theorem \\ref{theorem:injectivity} are proved in \\S\\ref{section:proofs}, where we also deduce the global properties of $\\mathcal{T}^\\sharp$ involving line bundles, flat bundles, negatively curved base manifolds, and the topology of bundles.\n\n\n\\end{itemize}\nSome technical preliminaries are provided in Section \\S\\ref{section:tools}. \n\n\\subsection{Perspectives}\n\nWe intend to pursue this work in different directions:\n\n\\begin{itemize}\n\n\\item We expect to be able to show a stability estimate version of Theorem \\ref{theorem:injectivity}, namely that the distance between connections up to gauge-equivalence is controlled (at least locally) by the distance between their images under the primitive trace map $\\mathcal{T}^\\sharp$. We believe that such an estimate should prove that the moduli space of connections up to gauge-equivalence $\\mathbb{A}_{\\E}$ (locally) embeds naturally via the primitive trace map as a closed topological (H\\\"older-continuous) Banach submanifold of $\\ell^\\infty(\\mathcal{C}^\\sharp)$.\n\n\\item We believe that the representation-theoretic approach developed in \\S\\ref{section:livsic} using Parry's free monoid $\\mathbf{G}$ could be pushed further. In a subsequent article \\cite{Cekic-Lefeuvre-21-3}, we will study the particular case of the Levi-Civita connection on the tangent bundle of Anosov manifolds and try to classify the possible images $\\rho(\\mathbf{G}) \\subset \\mathrm{O}(n)$. This problem is intimately connected to the dynamics of the frame flow.\n\n\\item Eventually, the arguments developed in \\S\\ref{section:livsic} mainly rely on the use of homoclinic orbits; to these orbits, we will associate a notion of \\emph{length} which is well-defined as an element of $\\mathbb{R}\/T_\\star\\mathbb{Z}$, for some real number $T_\\star > 0$. We believe that, similarly to the set of periodic orbits where one defines the \\emph{Ruelle zeta function}\n\\[\n\\zeta(s) := \\prod_{\\gamma^\\sharp \\in \\mathcal{G}^\\sharp} \\left(1 - e^{-s \\ell(\\gamma^\\sharp)}\\right), \n\\]\nwhere the product runs over all primitive periodic orbits $\\gamma^\\sharp \\in \\mathcal{G}^\\sharp$ (and $\\ell(\\gamma^\\sharp)$ denotes the orbit period) and shows that this extends meromorphically from $\\left\\{\\Re(s) \\gg 0 \\right\\}$ to $\\mathbb{C}$ (see \\cite{Giulietti-Liverani-Pollicott-13,Dyatlov-Zworski-16}), one could also define a complex function for homoclinic orbits by means of a Poincar\\'e series (rather than a product). It should be a consequence of \\cite[Theorem 4.15]{Dang-Riviere-20} that this function extends meromorphically to $\\mathbb{C}$. It might then be interesting to compute its value at $0$; the latter might be independent of the choice of representatives for the length of homoclinic orbits (two representatives differ by $mT_\\star$, for some $m \\in \\mathbb{Z}$) and could be (at least in some particular cases) an interesting topological invariant as for the Ruelle zeta function on surfaces, see \\cite{Dyatlov-Zworski-17}. \\\\\n\n\\end{itemize} \n\n\\noindent \\textbf{Acknowledgement:} M.C. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 725967). We warmly thank Yannick Guedes Bonthonneau, Yann Chaubet, Colin Guillarmou, Julien March\\'e, Gabriel Paternain, Steve Zelditch for fruitful discussions. We also thank S\\'ebastien Gou\\\"ezel, Boris Kalinin, Mark Pollicott, Klaus Schmidt for answering our questions on Liv\\v{s}ic theory, and Nikhil Savale for providing us with the reference \\cite{Guillemin-73}. Special thanks to Danica Kosanovi\\'c for helping out with the topology part.\n\n\n\\section{Setting up the tools}\n\n\\label{section:tools}\n\n\\subsection{Microlocal calculus and functional analysis} \n\nLet $\\mathcal{M}$ be a smooth closed manifold. Given a smooth vector bundle $\\E \\rightarrow \\mathcal{M}$, we denote by $\\Psi^m(\\mathcal{M}, \\E)$ the space of pseudodifferential operators of order $m$ acting on $\\E$. When $\\E$ is the trivial line bundle, such an operator $P \\in \\Psi(\\mathcal{M})$ can be written (up to a smoothing remainder) in local coordinates as\n\\begin{equation}\n\\label{equation:quantization}\nPf(x) = \\int_{\\mathbb{R}^{n}} \\int_{\\mathbb{R}^{n}} e^{i\\xi \\cdot(x-y)}p(x,\\xi)f(y)dyd\\xi,\n\\end{equation}\nwhere $f$ is compactly supported in the local patch and $p \\in S^m(T^*\\mathbb{R}^{n})$ is a \\emph{symbol} i.e. it satisfies the following estimates in coordinates:\n\\begin{equation}\n\\label{equation:bounds}\n\\sup_{|\\alpha'| \\leq \\alpha, |\\beta'|\\leq \\beta} \\sup_{(x,\\xi) \\in T^*\\mathbb{R}^{n+1}} \\langle \\xi \\rangle^{m - |\\alpha'|} |\\partial_\\xi^{\\alpha'} \\partial_x^{\\beta'} p(x,\\xi)| < \\infty,\n\\end{equation}\nfor all $\\alpha, \\beta \\in \\mathbb{N}^n$, with $\\langle \\xi \\rangle = \\sqrt{1+|\\xi|^2}$. When $\\E$ is not the trivial line bundle, the symbol $p$ is a matrix-valued symbol. Given $p \\in S^m(T^*\\mathcal{M})$, one can define a (non-canonical) \\emph{quantization procedure} $\\Op : S^m(T^*\\mathcal{M}) \\rightarrow \\Psi^m(\\mathcal{M})$ thanks to \\eqref{equation:quantization} in coordinates patches. This also works more generally with a vector bundle $\\E \\rightarrow \\mathcal{M}$ and one has a quantization map $\\Op : S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E)) \\rightarrow \\Psi^m(\\mathcal{M},\\E)$ (note that the symbol is then a section of the pullback bundle $\\mathrm{End}(\\E) \\rightarrow T^*\\mathcal{M}$ satisfying the bounds \\eqref{equation:bounds} in coordinates). There is a well-defined (partial) inverse map $\\sigma_{\\mathrm{princ}} : \\Psi(\\mathcal{M},\\E) \\rightarrow S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E))\/S^{m-1}(T^*\\mathcal{M}, \\mathrm{End}(\\E))$ called the \\emph{principal symbol} and satisfying $\\sigma_{\\mathrm{princ}}(\\Op(p)) = [p]$ (the equivalence class as an element of $S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E))\/S^{m-1}(T^*\\mathcal{M}, \\mathrm{End}(\\E))$).\n\nWe denote by $H^s(\\mathcal{M}, \\E)$ the space of Sobolev sections of order $s \\in \\mathbb{R}$ and by $C^s_*(\\mathcal{M}, \\E)$ the space of H\\\"older-Zygmund sections of order $s \\in \\mathbb{R}$. It is well-known that for $s \\in \\mathbb{R}_+ \\setminus \\mathbb{N}$, $C^s_*$ coincide with the space of H\\\"older-continuous sections $C^s$ of order $s$. Recall that $C^s_*$ is an algebra as long as $s > 0$ and $H^s$ is an algebra for $s > n\/2$.\nIf $P \\in \\Psi^m(\\mathcal{M}, \\E)$ is a pseudodifferential operator of order $m \\in \\mathbb{R}$, then $P : X^{s+m}(\\mathcal{M}, \\E) \\rightarrow X^s(\\mathcal{M}, \\E)$, where $X = H$, $C_*$, is bounded. We refer to \\cite{Shubin-01,Taylor-91} for further details.\n\n\\subsection{Connections on vector bundles}\n\n\\label{ssection:connections}\n\nWe refer the reader to \\cite[Chapter 2]{Donaldson-Kronheimer-90} for the background on connections on vector bundles.\n\n\n\n\n\n\n\\subsubsection{Mixed connection on the homomorphism bundle}\n\n\\label{sssection:connection-induced}\n\nIn this paragraph, we consider two Hermitian vector bundles $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$ equipped with respective unitary connections $\\nabla_1 = \\nabla^{\\E_1}$ and $\\nabla_2 = \\nabla^{\\E_2}$ which can be written in some local patch $U \\subset \\mathbb{R}^n$ of coordinates $\\nabla^{\\E_i} = d +\\Gamma_i$, for some $\\Gamma_i \\in C^\\infty(U,T^*U \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E_i))$. Let $\\Hom(\\E_1,\\E_2)$ be the vector bundle of homomorphisms from $\\E_1$ to $\\E_2$, endowed with the natural Hermitian structure.\n\n\\begin{definition}\\label{definition:mixed}\nWe define the (unitary) \\emph{homomorphism} or \\emph{mixed} connection $\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$ on $\\Hom(\\E_1, \\E_2)$, induced by $\\nabla^{\\E_1}$ and $\\nabla^{\\E_2}$, by the Leibnitz property:\n\\[\n\\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1, \\E_2)), \\forall s \\in C^\\infty(\\mathcal{M}, \\E_1), \\quad \\nabla^{\\E_2}(us) = (\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})}u) \\cdot s + u\\cdot (\\nabla^{\\E_1}s).\n\\]\n\\end{definition}\nEquivalently, it is straightforward to check that this is the canonical tensor product connection induced on $\\Hom(\\E_1, \\E_2) \\cong \\E_2 \\otimes \\E_1^*$ and that in local coordinates we have\n\\begin{equation}\\label{eq:localhom}\n\t\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})} u := du + \\Gamma_2 (\\bullet) u - u \\Gamma_1(\\bullet).\n\\end{equation}\nNote that this definition does not require the bundles to have same rank;\nwe insist on the fact that the mixed connection $\\nabla^{\\Hom(\\nabla_1, \\nabla_2)}$ \\emph{depends on a choice} of connections $\\nabla_1$ and $\\nabla_2$. In the particular case when $\\E_1 = \\E_2 = \\E$ and $\\nabla_1^{\\E} = \\nabla_2^{\\E} = \\nabla^{\\E}$ we will write $\\nabla^{\\mathrm{End}(\\nabla^{\\E})} = \\nabla^{\\Hom(\\nabla^{\\E}, \\nabla^{\\E})}$ for the induced \\emph{endomorphism} connection on $\\mathrm{End}(\\E)$. When clear from the context, we will also write $\\nabla^{\\mathrm{End}(\\E)}$ for the endomorphism connection induced by $\\nabla^{\\E}$.\n\n\n\nGiven a flow $(\\varphi_t)_{t \\in \\mathbb{R}}$, we will denote by $P(x,t) : \\Hom({\\E_1}_x, {\\E_2}_x) \\rightarrow \\Hom({\\E_1}_{\\varphi_t(x)}, {\\E_2}_{\\varphi_t(x)})$ the parallel transport with respect to the mixed connection along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$. Observe that for $u \\in \\Hom({\\E_1}_x, {\\E_2}_x)$, we have:\n\\begin{equation}\n\\label{equation:tp-mixed}\nP(x,t) u = C_2(x,t) u C_1(x,t)^{-1},\n\\end{equation}\nwhere $C_i(x,t) : {\\E_i}_x \\rightarrow {\\E_i}_{\\varphi_t(x)}$ is the parallel transport with respect to $\\nabla^{\\E_i}$ along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$.\n\nRecall that the curvature tensor $F_\\nabla \\in C^\\infty(\\mathcal{M}, \\Lambda^2(T^*M) \\otimes \\mathrm{End}(\\E))$ of $\\nabla = \\nabla^{\\E}$ is defined as, for any vector fields $X, Y$ on $\\mathcal{M}$ and sections $S$ of $\\E$\n\\begin{equation*\n\tF_\\nabla(X, Y)S = \\nabla_X \\nabla_Y S - \\nabla_Y \\nabla_X S - \\nabla_{[X, Y]}S.\n\\end{equation*} \nThen a quick computation using \\eqref{eq:localhom} reveals that:\n\\begin{equation}\n\\label{equation:induced-curvature}\nF_{\\nabla^{\\mathrm{Hom}(\\nabla_1,\\nabla_2)}} u = F_{\\nabla_2} \\cdot u - u \\cdot F_{\\nabla_1}.\n\\end{equation}\nEventually, using the Leibnitz property, we observe that if $p_i \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_i',\\E_i))$ is an isomorphism, then:\n\\begin{align}\n\\label{equation:lien}\n\\begin{split}\n\\nabla^{\\Hom(\\nabla_1, p_2^*\\nabla_2)} u &= (p_2)^{-1} \\nabla^{\\Hom(\\nabla_1, \\nabla_2)} (p_2 u), \\quad \\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1, \\E_2')),\\\\\n\\nabla^{\\Hom(p_1^*\\nabla_1, \\nabla_2)} u &= \\nabla^{\\Hom(\\nabla_1, \\nabla_2)} (u p_1^{-1}) \\cdot p_1,\\quad\\,\\, \\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1', \\E_2)).\n\\end{split}\n\\end{align}\n\n\\subsubsection{Ambrose-Singer formula}\n\nRecall that the celebrated Ambrose-Singer formula (see eg. \\cite[Theorem 8.1]{Kobayashi-Nomizu-69}) determines the tangent space at the identity of the holonomy group with respect to an arbitrary connection, in terms of its curvature tensor. Here we give an integral version of this fact. We start with a Hermitian vector bundle $\\E$ over the Riemannian manifold $(\\mathcal{M}, g)$. Equip $\\E$ with unitary connection $\\nabla = \\nabla^{\\E}$. \n\n\n\nConsider\na smooth homotopy $\\Gamma: [0, 1]^2 \\to \\mathcal{M}$ such that $\\Gamma(0, 0) = p$.\nThe ``vertical'' map $C_{\\uparrow}(s, t): \\E_p \\to \\E_{\\Gamma(s, t)}$ is obtained by parallel transporting with respect to $\\nabla$ from $\\E_p$ to $\\E_{\\Gamma(0, 1)}$, then $\\E_{\\Gamma(0, 1)}$ to $\\E_{\\Gamma(s, 1)}$ and $\\E_{\\Gamma(s, 1)}$ to $\\E_{\\Gamma(s, t)}$, along $\\Gamma(0, \\bullet)$, $\\Gamma(\\bullet, 1)$ and $\\Gamma(s, \\bullet)$, respectively. Next, define the ``horizontal'' map $C_{\\rightarrow}(s, t): \\E_p \\to \\E_{\\Gamma(s, t)}$ by parallel transport with respect to $\\nabla$ from $\\E_p$ to $\\E_{\\Gamma(s, 0)}$ and $\\E_{\\Gamma(s, 0)}$ to $\\E_{\\Gamma(s, t)}$, along $\\Gamma(\\bullet, 0)$ and $\\Gamma(s, \\bullet)$, respectively. For a better understanding, see Figure \\ref{fig:AS1}.\n\t\t\n\t\t\n\t\t\n\nWe are ready to prove the formula:\n\n\n\\begin{lemma}\\label{lemma:ambrosesinger}\n\tThe following formula holds\n\t\\begin{equation}\n\t\tC_{\\uparrow}^{-1}(1, 1) C_{\\rightarrow}(1, 1) - \\mathbbm{1}_{\\E_{p}} = \\int_0^1 \\int_0^1 C_{\\uparrow}(s, t)^{-1} F_\\nabla(\\partial_t, \\partial_s) C_{\\rightarrow}(s, t) \\, dt \\, ds.\n\t\\end{equation}\t\n\\end{lemma}\n\n\n\n\n\n\\begin{proof}\n\tLet $w_1, w_2 \\in \\E_p$; formally, we will identify the connection $\\nabla$ with its pullback $\\Gamma^*\\nabla$ on the pullback bundle $\\Gamma^*\\E$ over $[0, 1]^2$, as well as the curvature $F_\\nabla$ with $\\Gamma^*F_\\nabla$. Then we have the following chain of equalities:\n\t\\begin{align*}\n\t\t&\\langle{w_1, (C_{\\uparrow}(1, 1)^{-1} C_{\\rightarrow}(1, 1) - \\mathbbm{1}_{\\E_p}) w_2}\\rangle = \\langle{C_{\\uparrow}(1, 1)w_1, C_{\\rightarrow}(1, 1)w_2}\\rangle - \\langle{C_{\\uparrow}(0, 1)w_1, C_{\\rightarrow}(0, 1)w_2}\\rangle\\\\\n\t\t &= \\int_0^1 \\partial_s \\langle{C_{\\uparrow}(s, 1) w_1, C_{\\rightarrow}(s, 1) w_2}\\rangle ds\\\\\n\t\t&= \\int_0^1 \\langle{C_{\\uparrow}(s, 1) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, 1) w_2}\\rangle ds\\\\\n\t\t&= \\int_0^1 \\Big[\\int_0^1 \\Big(\\partial_t \\langle{C_{\\uparrow}(s, t) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, t)w_2}\\rangle + \\langle{C_{\\uparrow}(s, 0) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, 0) w_2}\\rangle\\Big)dt\\Big] ds\\\\\n\t\t&= \\int_0^1 \\int_0^1 \\langle{C_{\\uparrow}(s, t) w_1, \\nabla_{\\partial_t} \\nabla_{\\partial_s} C_{\\rightarrow}(s, t) w_2}\\rangle \\,ds\\, dt\\\\\n\t\t&= \\int_0^1 \\int_0^1 \\langle{w_1, C_{\\uparrow}(s, t)^{-1}\\underbrace{(\\nabla_{\\partial_t} \\nabla_{\\partial_s} - \\nabla_{\\partial_s} \\nabla_{\\partial_t})}_{=F_\\nabla(\\partial_t, \\partial_s)} C_{\\rightarrow}(s, t) w_2}\\rangle \\,ds\\, dt,\n\t\\end{align*}\n\tas the Lie bracket $[\\partial_s, \\partial_t] = 0$ and we used the unitarity of $\\nabla$ throughout. This completes the proof, since $w_1$ and $w_2$ were arbitrary.\n\\end{proof}\n\n\\begin{figure}\n \\centering\n\\begin{tikzpicture}[scale = 0.8, everynode\/.style={scale=0.5}]\n\\tikzset{cross\/.style={cross out, draw=black, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt},\ncross\/.default={1pt}}\n\n \t\\draw[thick, ->] (0, 0) -- (6,0) node[right] {\\small $s$};\n\t\t\\draw[thick, ->] (0, 0) -- (0,6) node[left] {\\small $t$};\n\t\t\n\t\n\t\t\\draw[thick] (3, 5) -- (5, 5) -- (5, 0);\n\t\t\n\t\n\t\t\\draw[thick, blue] (0, 0) -- (0, 5) -- (3, 5) -- (3, 3);\n\t\t\n\t\n\t\t\\draw[thick, red] (0, 0) -- (3, 0) -- (3, 3);\n\t\t\n\t\n\t\t\\draw[thick, ->] (3, 1.5)--(3, 1.501);\n\t\t\\draw[thick, ->] (1.5, 0)--(1.501, 0);\n\t\t\n\t\n\t\t\\draw[thick, ->] (0, 2.5)--(0, 2.501);\n\t\t\\draw[thick, ->] (1.5, 5)--(1.501, 5);\n\t\t\\draw[thick, ->] (3, 4)--(3, 3.999);\n\t\t\n\t\n\t\t\\fill (0, 0) node[below left] {\\tiny $p$} circle (1.5pt);\n\t\t\\fill (5, 0) node[below] {\\tiny $\\Gamma(1, 0)$} circle (1.5pt);\n\t\t\\fill (0, 5) node[left] {\\tiny $\\Gamma(0, 1)$} circle (1.5pt);\n\t\t\\fill (5, 5) node[right] {\\tiny $\\Gamma(1, 1)$} circle (1.5pt);\n\t\t\n\t\n\t\t\\fill (3, 3) node[right] {\\small $\\Gamma(s, t)$} circle (1.5pt);\n\t\t\\fill (3, 0) node[below] {\\small $\\Gamma(s, 0)$} circle (1.5pt);\n\t\t\\fill (3, 5) node[above] {\\small $\\Gamma(s, 1)$} circle (1.5pt);\n\\end{tikzpicture}\n \\caption{\\small The homotopy $\\Gamma$ in Lemma \\ref{lemma:ambrosesinger} with the corresponding points in $\\mathcal{M}$: in blue and red are the trajectories along which the parallel transport maps $C_{\\uparrow}$ (vertical) and $C_{\\rightarrow}$ (horizontal) are taken, respectively.}\n \\label{fig:AS1}\n\\end{figure}\n\n\nWe have two applications in mind for this lemma: one if $\\gamma$ is in a neighbourhood of $p$ and we use the radial homotopy via geodesics emanating from $p$, and the second one for the ``thin rectangle'' obtained by shadowing a piece of the flow orbit, see Lemma \\ref{lemma:ASgeometry}.\n\n\n\n\\subsection{Fourier analysis in the fibers} In this paragraph, we recall some elements of Fourier analysis in the fibers and refer to \\cite{Guillemin-Kazhdan-80, Guillemin-Kazhdan-80-2, Paternain-Salo-Uhlmann-14-2, Paternain-Salo-Uhlmann-15, Guillarmou-Paternain-Salo-Uhlmann-16} for further details.\n\n\n\n\n\\subsubsection{Analysis on the trivial line bundle}\n\n\\label{sssection:line-bundle}\n\n\nLet $(M,g)$ be a smooth Riemannian manifold of arbitrary dimension $n \\geq 2$. The unit tangent bundle is endowed with the natural Sasaki metric and we let $\\pi : SM \\rightarrow M$ be the projection on the base. There is a canonical splitting of the tangent bundle to $SM$ as:\n\\[\nT(SM) = \\mathbb{H} \\oplus \\mathbb{V} \\oplus \\mathbb{R} X,\n\\] \nwhere $X$ is the geodesic vector field, $\\mathbb{V} := \\ker \\dd \\pi$ is the vertical space and $\\mathbb{H}$ is the horizontal space: it can be defined as the orthogonal to $\\mathbb{V} \\oplus \\mathbb{R} X$ with respect to the Sasaki metric (see \\cite[Chapter 1]{Paternain-99}). Any vector $Z \\in T(SM)$ can be decomposed according to the splitting\n\\[\nZ = \\alpha(Z) X + Z_{\\mathbb{H}} + Z_{\\mathbb{V}},\n\\]\nwhere $\\alpha$ is the Liouville $1$-form, $Z_{\\mathbb{H}} \\in \\mathbb{H}, Z_{\\mathbb{V}} \\in \\mathbb{V}$. If $f \\in C^\\infty(SM)$, its gradient computed with respect to the Sasaki metric can be written as:\n\\[\n\\nabla_{\\mathrm{Sas}}f = (Xf) X + \\nabla_{\\mathbb{H}} f + \\nabla_{\\mathbb{V}} f,\n\\]\nwhere $\\nabla_{\\mathbb{H}} f \\in \\mathbb{H}$ is the horizontal gradient, $\\nabla_{\\mathbb{V}} f \\in \\mathbb{V}$ is the vertical gradient. We also let $\\mathcal{N} \\to SM$ be the \\emph{normal bundle} whose fiber over each $(x,v) \\in SM$ is given by $(\\mathbb{R} \\cdot v)^\\bot$. The bundles $\\mathbb{H}$ and $\\mathbb{V}$ may be naturally identified with the bundle $\\mathcal{N}$ (see \\cite[Section 1]{Paternain-99}).\n\nFor every $x \\in M$, the sphere $S_xM = \\left\\{ v \\in T_xM ~|~ |v|^2_x = 1\\right\\} \\subset SM$ endowed with the Sasaki metric is isometric to the canonical sphere $(\\mathbb{S}^{n-1},g_{\\mathrm{can}})$. We denote by $\\Delta_{\\mathbb{V}}$ the vertical Laplacian obtained for $f \\in C^\\infty(SM)$ as $\\Delta_{\\mathbb{V}} f(x,v) = \\Delta_{g_{\\mathrm{can}}}(f|_{S_xM})(v)$, where $\\Delta_{g_{\\mathrm{can}}}$ is the spherical Laplacian. For $m \\geq 0$, we denote by $\\Omega_m$ the (finite-dimensional) vector space of spherical harmonics of degree $m$ for the spherical Laplacian $\\Delta_{g_{\\mathrm{can}}}$: they are defined as the elements of $\\ker(\\Delta_{g_{\\mathrm{can}}} + m(m+n-2))$. We will use the convention that $\\Omega_m = \\left\\{0\\right\\}$ if $m< 0$. If $f \\in C^\\infty(SM)$, it can then be decomposed as $f = \\sum_{m \\geq 0} f_m$, where $f_m \\in C^\\infty(M,\\Omega_m)$ is the $L^2$-orthogonal projection of $f$ onto the spherical harmonics of degree $m$.\n\nThere is a one-to-one correspondence between trace-free symmetric tensors of degree $m$ and spherical harmonics of degree $m$. More precisely, the map\n\\[\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr}) \\rightarrow C^\\infty(M,\\Omega_m),\n\\]\ngiven by $\\pi_m^*f(x,v) = f_x(v,...,v)$ is an isomorphism. Here, the index $0-\\Tr$ denotes the space of trace-free symmetric tensors, namely tensors such that, if $(\\e_1,...,\\e_n)$ denotes a local orthonormal frame of $TM$:\n\\[\n\\Tr(f) := \\sum_{i=1}^n f(\\e_i,\\e_i,\\cdot, ...,\\cdot) = 0.\n\\]\nWe will denote by ${\\pi_m}_* : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr})$ the adjoint of this map. More generally, the mapping\n\\begin{equation}\\label{eq:pi_muntwisted}\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M) \\rightarrow \\oplus_{k \\geq 0} C^\\infty(M,\\Omega_{m-2k})\n\\end{equation}\nis an isomorphism. We refer to \\cite[Section 2]{Cekic-Lefeuvre-20} for further details.\n\nThe geodesic vector field acts as $X : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m-1}) \\oplus C^\\infty(M,\\Omega_{m+1})$ (see \\cite{Guillemin-Kazhdan-80-2, Paternain-Salo-Uhlmann-15}). We define $X_+$ as the $L^2$-orthogonal projection of $X$ on the higher modes $\\Omega_{m+1}$, namely if $u \\in C^\\infty(M,\\Omega_m)$, then $X_+u := (Xu)_{m+1}$ and $X_-$ as the $L^2$-orthogonal projection of $X$ on the lower modes $\\Omega_{m-1}$. For $m \\geq 0$, the operator $X_+ : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m+1})$ is elliptic and thus has a finite dimensional kernel (see \\cite{Dairbekov-Sharafutdinov-10}). The operator $X_- : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m-1})$ is of divergence type. The elements in the kernel of $X_+$ are called \\emph{Conformal Killing Tensors (CKTs)}, associated to the trivial line bundle. For $m=0$, the kernel of $X_+$ on $C^\\infty(M,\\Omega_0)$ always contains the constant functions. We call \\emph{non trivial CKTs} elements in $\\ker X_+$ which are not constant functions on $SM$. The kernel of $X_+$ is invariant by changing the metric by a conformal factor (see \\cite[Section 3.6]{Guillarmou-Paternain-Salo-Uhlmann-16}). It is known (see \\cite{Paternain-Salo-Uhlmann-15}) that there are no non trivial CKTs in negative curvature and for Anosov surfaces but the question remains open for general Anosov manifolds. We provide a positive answer to this question \\emph{generically} as a byproduct of our work \\cite{Cekic-Lefeuvre-21-2}.\n\n\n\\subsubsection{Twisted Fourier analysis}\n\n\\label{sssection:twisted-fourier}\n\nWe now consider a Hermitian vector bundle with a unitary connection $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ over $(M,g)$ and define the operator $\\mathbf{X} := (\\pi^*\\nabla^{\\mathcal{E}})_X$ acting on $C^\\infty(SM,\\pi^*\\mathcal{E})$, where $\\pi : SM \\rightarrow M$ is the projection. For the sake of simplicity, we will drop the $\\pi^*$ in the following. If $f \\in C^\\infty(SM,\\mathcal{E})$, then $\\nabla^{\\mathcal{E}}f \\in C^\\infty(SM,T^*(SM) \\otimes \\mathcal{E})$ and we can write\n\\[\n\\nabla^{\\mathcal{E}}f = (\\mathbf{X} f, \\nabla_{\\mathbb{H}} f, \\nabla_{\\mathbb{V}} f),\n\\]\nwhere $\\nabla_{\\mathbb{H}} f \\in C^\\infty(SM, \\mathbb{H}^* \\otimes \\mathcal{E}), \\nabla_{\\mathbb{V}} f \\in C^\\infty(SM, \\mathbb{V}^* \\otimes \\mathcal{E})$. For future reference, we introduce a bundle endomorphism map $R$ on $\\mathcal{N} \\otimes \\E$, derived from the Riemann curvature tensor via the formula $R(x, v)(w \\otimes e) = (R_x(w, v)v) \\otimes e$.\n\nIf $(e_1,...,e_r)$ is a local orthonormal frame of $\\mathcal{E}$, then we define the vertical Laplacian as\n\\[\n\\Delta_{\\mathbb{V}}^{\\mathcal{E}}(\\sum_{k=1}^r u_k e_k) := \\sum_{k=1}^r (\\Delta_{\\mathbb{V}} u_k) e_k.\n\\]\nAny section $f \\in C^\\infty(SM,\\mathcal{E})$ can be decomposed according to $f = \\sum_{m \\geq 0} f_m$, where $f_m \\in \\ker(\\Delta_{\\mathbb{V}}^{\\mathcal{E}} + m(m+n-2))$ and we define $C^\\infty(M,\\Omega_m \\otimes \\mathcal{E}) := \\ker(\\Delta_{\\mathbb{V}}^{\\mathcal{E}} + m(m+n-2)) \\cap C^\\infty(SM,\\mathcal{E})$.\n\nHere again, the operator $\\mathbf{X} : C^\\infty(M,\\Omega_m \\otimes \\E) \\rightarrow C^\\infty(M,\\Omega_{m-1} \\otimes \\E) \\oplus C^\\infty(M,\\Omega_{m+1} \\otimes \\E)$ can be split into the corresponding sum $\\mathbf{X} = \\mathbf{X}_- + \\mathbf{X}_-$. For every $m \\geq 0$, the operator $\\mathbf{X}_+$ is elliptic and has finite dimensional kernel, whereas $\\mathbf{X}_-$ is of divergence type. The kernel of $\\mathbf{X}_+$ is invariant by a conformal change of the metric (see \\cite[Section 3.6]{Guillarmou-Paternain-Salo-Uhlmann-16}) and elements in its kernel are called \\emph{twisted Conformal Killing Tensors} (CKTs). There are examples of vector bundles with CKTs on manifolds of arbitrary dimension. We proved in a companion paper \\cite{Cekic-Lefeuvre-20} that the non existence of CKTs is a generic condition, no matter the curvature of the manifold (generic with respect to the connection, i.e. there is a residual set of the space of all unitary connections with regularity $C^k$, $k \\geq 2$, which has no CKTs)\n\nIt is also known by \\cite{Guillarmou-Paternain-Salo-Uhlmann-16}, that in negative curvature, there is always a \\emph{finite number of degrees} with CKTs (and this number can be estimated thanks to a lower bound on the curvature of the manifold and the curvature of the vector bundle). In other words, $\\ker \\mathbf{X}_+|_{C^\\infty(SM,\\mathcal{E})}$ is finite-dimensional. The proof relies on an energy identity called the Pestov identity. This is also known for Anosov surfaces since any Anosov surface is conformally equivalent to a negatively-curved surface and CKTs are conformally invariant. Nevertheless, and to the best of our knowledge, it is still an open question to show that for Anosov manifolds of dimension $n \\geq 3$, there is at most a finite number of CKTs.\n\n\n\n\n\n\\subsection{Twisted symmetric tensors}\\label{section:twisted}\n\nGiven a section $u \\in C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$, the connection $\\nabla^{\\mathcal{E}}$ produces an element $\\nabla^{\\mathcal{E}}u \\in C^\\infty(M, T^*M \\otimes (\\otimes^m_S T^*M) \\otimes \\mathcal{E})$. In coordinates, if $(e_1, ..., e_r)$ is a local orthonormal frame for $\\mathcal{E}$ and $\\nabla^{\\mathcal{E}} = d + \\Gamma$, for some one-form with values in skew-Hermitian matrices $\\Gamma$, such that $\\nabla^{\\E}e_k = \\sum_{i = 1}^n\\sum_{l = 1}^r \\Gamma_{ik}^{l} dx_i \\otimes e_l$, we have:\n\\begin{equation}\n\\label{equation:nabla-e}\n\\begin{split}\n\\nabla^{\\mathcal{E}}(\\sum_{k=1}^r u_k \\otimes e_k) & = \\sum_{k=1}^r \\big(\\nabla u_k \\otimes e_k + u_k \\otimes \\nabla^{\\mathcal{E}} e_k\\big)\\\\\n& = \\sum_{k=1}^r \\left(\\nabla u_k + \\sum_{l=1}^r \\sum_{i=1}^n \\Gamma_{il}^k u_l \\otimes dx_i \\right) \\otimes e_k,\n\\end{split}\n\\end{equation}\nwhere $u_k \\in C^\\infty(M,\\otimes^m_S T^*M)$ and $\\nabla$ is the Levi-Civita connection. The symmetrization operator $\\mathcal{S}^{\\mathcal{E}} : C^\\infty(M,\\otimes^m T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$ is defined by:\n\\[\n\\mathcal{S}^{\\mathcal{E}}\\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) = \\sum_{k=1}^r \\mathcal{S}(u_k) \\otimes e_k,\n\\]\nwhere $u_k \\in C^\\infty(M,\\otimes^m_S T^*M)$ and in coordinates, writing $u_k = \\sum_{i_1, ..., i_m=1}^n u_{i_1...i_m}^{(k)} dx_{i_1} \\otimes ... \\otimes dx_{i_m}$, we have\n\\[\n\\mathcal{S}(dx_{i_1} \\otimes ... \\otimes dx_{i_m}) = \\dfrac{1}{m!} \\sum_{\\pi \\in \\mathfrak{S}_m} dx_{\\pi(i_1)} \\otimes ... \\otimes dx_{\\pi(i_m)},\n\\]\nwhere $\\mathfrak{S}_m$ denotes the group of permutations of order $m$. For the sake of simplicity, we will write $\\mathcal{S}$ instead of $\\mathcal{S}^{\\mathcal{E}}$. We can symmetrize \\eqref{equation:nabla-e} to produce an element $D^{\\E} := \\mathcal{S} \\nabla^{\\mathcal{E}}u \\in C^\\infty(M, \\otimes^{m+1}_S T^*M \\otimes \\mathcal{E})$ given in coordinates by:\n\\begin{equation}\n\\label{equation:formula-de}\nD^{\\E} \\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) = \\sum_{k=1}^r \\left( Du_k + \\sum_{l=1}^r \\sum_{i=1}^n \\Gamma_{il}^k \\sigma(u_l \\otimes dx_i) \\right) \\otimes e_k,\n\\end{equation}\nwhere $D := \\mathcal{S} \\nabla$ is the usual symmetric derivative of symmetric tensors\\footnote{Beware of the notation: $\\nabla^{\\mathcal{E}}$ is for the connection, $D^{\\E}$ for the symmetric derivative of tensors and $\\nabla^{\\mathrm{End}(\\E)}$ is the connection induced by $\\nabla^{\\mathcal{E}}$ on the endomorphism bundle.}. Elements of the form $Du \\in C^\\infty(M,\\otimes^{m+1}_S T^*M)$ are called \\emph{potential tensors}. By comparison, we will call elements of the form $D^{\\E}f \\in C^\\infty(M,\\otimes^{m+1}_S T^*M \\otimes \\mathcal{E})$ \\emph{twisted potential tensors}. The operator $D^{\\E}$ is a first order differential operator and its expression can be read off from \\eqref{equation:formula-de}, namely:\n\\[\n\\begin{split}\n\\sigma_{\\mathrm{princ}}(D^{\\E})(x,\\xi) \\cdot \\left(\\sum_{k=1}^r u_k(x) \\otimes e_k(x) \\right) & = \\sum_{k=1}^r \\left(\\sigma_{\\mathrm{princ}}(D)(x,\\xi) \\cdot u_k(x)\\right) \\otimes e_k(x) \\\\\n& = i \\sum_{k=1}^r \\sigma(\\xi \\otimes u_k(x)) \\otimes e_k(x),\n\\end{split}\n\\]\nwhere $e_k(x) \\in \\mathcal{E}_x, u_k(x) \\in \\otimes^m_S T^*_xM$ and the basis $(e_1(x),...,e_r(x))$ is assumed to be orthonormal for the metric $h$ on $\\mathcal{E}$. One can check that this is an injective map, which means that $D^{\\E}$ is a left-elliptic operator and can be inverted on the left modulo a smoothing remainder. Its kernel is finite-dimensional and consists of smooth elements\n\nBefore that, we introduce for $m \\in \\mathbb{N}$, the operator\n\\[\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(SM,\\pi^*\\mathcal{E}), \n\\]\ndefined by\n\\[\n\\pi_m^*\\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) (x,v) := \\sum_{k=1}^r ({u_k})_x(v,...,v) e_k(x).\n\\]\nSimilarly to \\eqref{eq:pi_muntwisted}, the following mappings are isomorphisms (see \\cite[Section 2]{Cekic-Lefeuvre-20}):\n\\begin{align*}\n\\pi_m^* &: C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow \\oplus_{k \\geq 0} C^\\infty(M,\\Omega_{m-2k} \\otimes \\E),\\\\\n\\pi_m^* &: C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr} \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\Omega_m \\otimes \\E).\n\\end{align*}\nWe recall the notation $(\\pi^* \\nabla^{\\mathcal{E}})_X := \\mathbf{X}$. The following remarkable commutation property holds (see \\cite[Section 2]{Cekic-Lefeuvre-20}):\n\\begin{equation}\\label{eq:pullback}\n\t\\forall m \\in \\mathbb{Z}_{\\geq 0}, \\quad \\pi_{m+1}^* D^{\\E} = \\mathbf{X} \\pi_m^*.\n\\end{equation}\n\n\nThe vector bundle $\\otimes^m_S T^*M \\otimes \\mathcal{E}$ is naturally endowed with a canonical fiberwise metric induced by the metrics $g$ and $h$ which allows to define a natural $L^2$ scalar product. The $L^2$ formal adjoint $(D^{\\E})^*$ of $D^{\\E}$ is of divergence type (in the sense that its principal symbol is surjective for every $(x,\\xi) \\in T^*M \\setminus \\left\\{ 0 \\right\\}$, see \\cite[Definition 3.1]{Cekic-Lefeuvre-20} for further details). We call \\emph{twisted solenoidal tensors} the elements in its kernel.\n\nBy ellipticity of $D^{\\E}$, for any twisted $m$-tensor $f$ there exists a unique $p \\in (\\ker D^{\\E})^\\perp \\cap C^\\infty(M,\\otimes^{m-1}_S T^*M \\otimes \\mathcal{E}), h \\in C^\\infty(M,\\otimes^{m}_S T^*M \\otimes \\mathcal{E})$ such that:\n\\begin{equation}\\label{eq:decomposition-tt}\nf = D^{\\E}p + h, \\quad (D^{\\E})^*h = 0.\n\\end{equation}\nThe previous decomposition could be extended to other regularities. We define $\\pi_{\\ker (D^{\\E})^*}f := h$ as the $L^2$-orthogonal projection on twisted solenoidal tensors. This can be expressed as:\n\\begin{equation}\n\\label{equation:projection}\n\\pi_{\\ker (D^{\\E})^*} = \\mathbbm{1} - D^{\\E} [(D^{\\E})^*D^{\\E}]^{-1} (D^{\\E})^*,\n\\end{equation}\nwhere $[(D^{\\E})^*D^{\\E}]^{-1}$ is the resolvent of the operator.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Pollicott-Ruelle resonances}\n\nWe explain the link between the widely studied notion of Pollicott-Ruelle resonances (see for instance \\cite{Liverani-04, Gouezel-Liverani-06,Butterley-Liverani-07,Faure-Roy-Sjostrand-08,Faure-Sjostrand-11,Faure-Tsuji-13,Dyatlov-Zworski-16}) and the notion of (twisted) Conformal Killing Tensors introduced in the last paragraph. We also refer to \\cite{Cekic-Lefeuvre-20} for an extensive discussion about this.\n\n\n\n\\subsubsection{Definition of the resolvents}\n\n\n\\label{ssection:resonances}\n\nLet $\\mathcal{M}$ be a smooth manifold endowed with a vector field $X \\in C^\\infty(\\mathcal{M},T\\mathcal{M})$ generating an Anosov flow in the sense of \\eqref{equation:anosov}. Throughout this paragraph, we will always assume that the flow is volume-preserving. It will be important to consider the dual decomposition to \\eqref{equation:anosov}, namely\n\\[\nT^*(\\mathcal{M}) = \\mathbb{R} E_0^* \\oplus E_s^* \\oplus E_u^*,\n\\]\nwhere $E_0^*(E_s \\oplus E_u) = 0, E_s^*(E_s \\oplus \\mathbb{R} X) = 0, E_u^*(E_u \\oplus \\mathbb{R} X) = 0$. As before, we consider a vector bundle $\\mathcal{E} \\rightarrow \\mathcal{M}$ equipped with a unitary connection $\\nabla^{\\E}$ and set $\\mathbf{X} := \\nabla^{\\mathcal{E}}_X$. Since $X$ preserves a smooth measure $\\dd \\mu$ and $\\nabla^{\\mathcal{E}}$ is unitary, the operator $\\mathbf{X}$ is skew-adjoint on $L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu)$, with dense domain\n\\begin{equation}\n\\label{equation:domaine-p}\n\\mathcal{D}_{L^2} := \\left\\{ u \\in L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu) ~|~ \\mathbf{X} u \\in L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu)\\right\\}.\n\\end{equation}\nIts $L^2$-spectrum consists of absolutely continuous spectrum on $i\\mathbb{R}$ and on embedded eigenvalues. We introduce the resolvents\n\\begin{equation}\n\\label{equation:resolvent}\n\\begin{split}\n&\\RR_+(z) := (-\\mathbf{X}-z)^{-1} = - \\int_0^{+\\infty} e^{-t z} e^{-t\\mathbf{X}} \\dd t, \\\\\n&\\RR_-(z) := (\\mathbf{X}-z)^{-1} = - \\int_{-\\infty}^0 e^{z t} e^{-t\\mathbf{X}} \\dd t,\n\\end{split}\n\\end{equation}\ninitially defined for $\\Re(z) > 0$. (Let us stress on the conventions here: $-\\mathbf{X}$ is associated to the positive resolvent $\\RR_+(z)$ whereas $\\mathbf{X}$ is associated to the negative one $\\RR_-(z)$.) Here $e^{-t\\mathbf{X}}$ denotes the propagator of $\\mathbf{X}$, namely the parallel transport by $\\nabla^{\\mathcal{E}}$ along the flowlines of $X$. If $\\mathbf{X} = X$ is simply the vector field acting on functions (i.e. $\\mathcal{E}$ is the trivial line bundle), then $e^{-tX}f(x) = f(\\varphi_{-t}(x))$ is nothing but the composition with the flow.\n\nThere exists a family $\\mathcal{H}^s_\\pm$ of Hilbert spaces called \\emph{anisotropic Sobolev spaces}, indexed by $s > 0$, such that the resolvents can be meromorphically extended to the whole complex plane by making $\\mathbf{X}$ act on $\\mathcal{H}^s_\\pm$. The poles of the resolvents are called the \\emph{Pollicott-Ruelle resonances} and have been widely studied in the aforementioned literature \\cite{Liverani-04, Gouezel-Liverani-06,Butterley-Liverani-07,Faure-Roy-Sjostrand-08,Faure-Sjostrand-11,Faure-Tsuji-13,Dyatlov-Zworski-16}. Note that the resonances and the resonant states associated to them are intrinsic to the flow and do not depend on any choice of construction of the anisotropic Sobolev spaces. More precisely, there exists a constant $c > 0$ such that $\\RR_\\pm(z) \\in \\mathcal{L}(\\mathcal{H}_\\pm^s)$ are meromorphic in $\\left\\{\\Re(z) > -cs\\right\\}$. For $\\RR_+(z)$ (resp. $\\RR_-(z)$), the space $\\mathcal{H}^s_+$ (resp. $\\mathcal{H}^s_-$) consists of distributions which are microlocally $H^s$ in a neighborhood of $E_s^*$ (resp. microlocally $H^{s}$ in a neighborhood of $E_u^*$) and microlocally $H^{-s}$ in a neighborhood of $E_u^*$ (resp. microlocally $H^{-s}$ in a neighborhood of $E_s^*$), see \\cite{Faure-Sjostrand-11,Dyatlov-Zworski-16}. These spaces also satisfy $(\\mathcal{H}^s_+)' = \\mathcal{H}^s_-$ (where one identifies the spaces using the $L^2$-pairing).\n\nFrom now on, we will assume that $s$ is fixed and small enough, and set $\\mathcal{H}_\\pm := \\mathcal{H}_\\pm^s$. We have\n\\begin{equation}\n\\label{equation:betaan}\nH^{s} \\subset \\mathcal{H}_{\\pm} \\subset H^{-s}.\n\\end{equation}\nand there is a certain strip $\\left\\{\\Re(z) > -\\varepsilon_{\\mathrm{strip}}\\right\\}$ (for some $\\varepsilon_{\\mathrm{strip}} > 0$) on which $z \\mapsto \\RR_{\\pm}(z) \\in \\mathcal{L}(\\mathcal{H}_{\\pm})$ is meromorphic (and the same holds for small perturbations of $\\mathbf{X}$).\n\nThese resolvents satisfy the following equalities on $\\mathcal{H}_\\pm$, for $z$ not a resonance:\n\\begin{equation}\n\\label{equation:resolvent-identity}\n\\RR_\\pm(z)(\\mp\\mathbf{X}- z) = (\\mp\\mathbf{X}- z) \\RR_\\pm(z) = \\mathbbm{1}_{\\mathcal{E}}.\n\\end{equation}\nGiven $z \\in \\mathbb{C}$ which not a resonance, we have:\n\\begin{equation}\n\\label{equation:adjoint}\n\\RR_+(z)^* = \\RR_-(\\overline{z}),\n\\end{equation}\nwhere this is understood in the following way: given $f_1, f_2 \\in C^\\infty(\\mathcal{M},\\mathcal{E})$, we have\n\\[\n\\langle \\RR_+(z) f_1, f_2 \\rangle_{L^2} = \\langle f_1, \\RR_-(\\overline{z}) f_2 \\rangle_{L^2}.\n\\]\n(We will always use this convention for the definition of the adjoint.) Since the operators are skew-adjoint on $L^2$, all the resonances (for both the positive and the negative resolvents $\\RR_\\pm$) are contained in $\\left\\{\\Re(z) \\leq 0 \\right\\}$. A point $z_0 \\in \\mathbb{C}$ is a resonance for $-\\mathbf{X}$ (resp. $\\mathbf{X}$) i.e. is a pole of $z \\mapsto \\RR_+(z)$ (resp. $\\RR_-(z)$) if and only if there exists a non-zero $u \\in \\mathcal{H}^s_+$ (resp. $\\mathcal{H}^s_-$) for some $s > 0$ such that $-\\mathbf{X} u = z_0 u$ (resp. $\\mathbf{X} u = z_0 u$). If $\\gamma$ is a small counter clock-wise oriented circle around $z_0$, then the spectral projector onto the resonant states is\n\\[\n\\Pi_{z_0}^{\\pm} = - \\dfrac{1}{2\\pi i} \\int_{\\gamma} \\RR_{\\pm}(z) \\dd z = \\dfrac{1}{2\\pi i} \\int_{\\gamma} (z \\pm \\mathbf{X})^{-1} \\dd z,\n\\]\nwhere we use the abuse of notation that $-(\\mathbf{X}+z)^{-1}$ (resp. $(\\mathbf{X}-z)^{-1}$) to denote the meromorphic extension of $\\RR_+(z)$ (resp. $\\RR_-(z)$). \n\n\n\\subsubsection{Resonances at $z=0$}\n\n\\label{sssection:resonances-zero}\n\nBy the previous paragraph, we can write in a neighborhood of $z=0$ the following Laurent expansion (beware the conventions):\n\\[\n\\RR_+(z) = - \\RR_0^+ - \\dfrac{\\Pi_0^+}{z} + \\mathcal{O}(z).\n\\]\n(Or in other words, using our abuse of notations, $(\\mathbf{X}+z)^{-1} = \\RR_0^+ + \\Pi_0^+\/z + \\mathcal{O}(z)$.) And:\n\\[\n\\RR_-(z) = -\\RR_0^- - \\dfrac{\\Pi_0^-}{z} + \\mathcal{O}(z).\n\\]\n(Or in other words, $(z - \\mathbf{X})^{-1} = \\RR_0^- + \\Pi_0^-\/z + \\mathcal{O}(z)$.) As a consequence, these equalities define the two operators $\\RR_0^{\\pm}$ as the holomorphic part (at $z=0$) of the resolvents $-\\RR_\\pm(z)$. We introduce:\n\\begin{equation}\n\\label{equation:pi}\n\\Pi := \\RR_0^+ + \\RR_0^-.\n\\end{equation}\nWe have:\n\n\\begin{lemma}\n\\label{lemma:relations-resolvent}\nWe have $(\\RR_0^+)^* = \\RR_0^{-}, (\\Pi_0^+)^* = \\Pi_0^- = \\Pi_0^+$. Thus $\\Pi$ is formally self-adjoint. Moreover, it is nonnegative in the sense that for all $f \\in C^\\infty(\\mathcal{M},\\mathcal{E})$, $\\langle \\Pi f, f \\rangle_{L^2} = \\langle f, \\Pi f \\rangle_{L^2} \\geq 0$. Also, $\\langle \\Pi f, f \\rangle = 0$ if and only if $\\Pi f = 0$ if and only if $f = \\mathbf{X} u + v $, for some $u \\in C^\\infty(\\mathcal{M},\\E), v \\in \\ker(\\mathbf{X}|_{\\mathcal{H}_\\pm})$.\n\\end{lemma}\n\n\\begin{proof} See \\cite[Lemma 5.1]{Cekic-Lefeuvre-20}. \\end{proof}\n\n\nWe also record here for the sake of clarity the following identities:\n\\begin{equation}\\label{eq:resolventrelations}\n\\begin{split}\n& \\Pi_0^{\\pm} \\RR_0^+ = \\RR_0^+ \\Pi_0^{\\pm} = 0, \\Pi_0^{\\pm} \\RR_0^- = \\RR_0^- \\Pi_0^{\\pm} = 0,\\\\\n& \\mathbf{X} \\Pi_0^\\pm = \\Pi_0^\\pm \\mathbf{X} = 0, \\mathbf{X} \\RR_0^+ = \\RR_0^+ \\mathbf{X} = \\mathbbm{1} - \\Pi_0^+, -\\mathbf{X} \\RR_0^- = -\\RR_0^- \\mathbf{X} = \\mathbbm{1} - \\Pi_0^-.\n\\end{split}\n\\end{equation}\n\n\n\n\n\\subsection{Generalized X-ray transform}\n\\label{ssection:generalized-x-ray}\nThe discussion is carried out here in the closed case, but could also be generalized to the case of a manifold with boundary.\nWe introduce the operator\n\\[\n\\Pi := \\RR_0^+ + \\RR_0^-,\n\\]\nwhere $\\RR_0^+$ (resp. $\\RR_0^-$) denotes the holomorphic part at $0$ of $-\\RR_+(z)$ (resp. $-\\RR_-(z)$) and $\\Pi^+_0$ is the $L^2$-orthogonal projection on the (smooth) resonant states at $0$. Such an operator was first introduced in the non-twisted case by Guillarmou \\cite{Guillarmou-17-1}. The operator $\\Pi + \\Pi^+_0$ is the derivative of the (total) $L^2$-spectral measure at $0$ of the skew-adjoint operator $\\mathbf{X}$.\n\n\\begin{definition}[Generalized X-ray transform of twisted symmetric tensors]\n\\label{definition:generalized-xray}\nWe define the generalized X-ray transform of twisted symmetric tensors as the operator:\n\\[\n\\Pi_m := {\\pi_m}_* \\left(\\Pi + \\Pi^+_0\\right) \\pi_m^*.\n\\]\n\\end{definition}\n\n\\begin{remark}\\rm\n\tThis allows to define a generalized (twisted) X-ray transform $\\Pi_m$ for an arbitrary unitary connection $\\nabla^{\\mathcal{E}}$ on $\\mathcal{E}$. Indeed, it is not clear a priori if one sticks to the usual definition of the X-ray transform that one can find a ``natural'' candidate for the X-ray transform on twisted tensors. For instance, one could consider the map\n\\[\n\\mathcal{C} \\ni \\gamma \\mapsto I_m^{\\nabla^{\\E}}f (\\gamma) := \\dfrac{1}{\\ell(\\gamma)} \\int_0^{\\ell(\\gamma)} (e^{-t \\mathbf{X}} f) (x_\\gamma, v_\\gamma) \\dd t \\in \\mathcal{E}_{x_\\gamma},\n\\]\nwhere $\\gamma \\in \\mathcal{C}$ is a closed geodesic and $(x_\\gamma,v_\\gamma) \\in \\gamma$. However, this definition does depend on the choice of base point $(x_\\gamma,v_\\gamma) \\in \\gamma$ and it would no longer be true that $I^{\\nabla^{\\mathcal{E}}}_m (D^{\\E}p) (\\gamma) = 0$ unless the connection is transparent.\n\\end{remark}\n\n\nBy \\eqref{eq:pullback} and \\eqref{eq:resolventrelations}, we have the equalities:\n\\begin{equation}\\label{eq:Pi_mproperties}\n(D^{\\E})^* \\Pi_m = 0 = \\Pi_m D^{\\E},\n\\end{equation}\nshowing that $\\Pi_m$ maps the set of twisted solenoidal tensors to itself. We say that the generalised $X$-ray transform is \\emph{solenoidally injective} ($s$-injective) on $m$-tensors, if for all $u \\in C^\\infty(SM, \\E)$ and $f \\in C^\\infty(M, \\otimes_S^mT^*M \\otimes \\E)$\n\\begin{equation}\\label{eq:cohomeqn}\n\t\\mathbf{X} u = \\pi_m^* f \\implies \\exists p \\in C^\\infty(M, \\otimes_S^{m-1}T^*M \\otimes \\E)\\,\\, \\mathrm{such\\,\\, that}\\,\\,f = D^{\\E} p.\n\\end{equation}\nWe have the following:\n\n\\begin{lemma}\\label{lemma:x-ray}\n\tThe generalised $X$-ray transform is $s$-injective of $m$-tensors if and only if $\\Pi_m$ is injective on solenoidal tensors (if this holds, we say $\\Pi_m$ is \\emph{$s$-injective}).\n\\end{lemma}\n\\begin{proof}\n\tAssume that $\\Pi_m f = 0$ and $f$ is a twisted solenoidal $m$-tensor. Then\n\\[\n\\langle \\Pi_m f, f \\rangle_{L^2} = \\langle \\Pi \\pi_m^*f , \\pi_m^*f \\rangle_{L^2} + \\langle \\Pi^+_0 \\pi_m^*f, \\pi_m^* f \\rangle_{L^2} = 0.\n\\]\nBoth terms on the right hand side are non-negative by Lemma \\ref{lemma:relations-resolvent}, hence both of them vanish, and the same Lemma implies that $\\Pi \\pi_m^* f = 0$ and $\\Pi^+_0 \\pi_m^*f = 0$. Thus $\\mathbf{X} u = \\pi_m^*f$ for some smooth $u$, so by the $s$-injectivity of generalised $X$-ray transform we obtain $f$ is potential, which implies $f = 0$.\n\nThe other direction is obvious by \\eqref{eq:resolventrelations}.\n\\end{proof}\n\nNext, we show $\\Pi_m$ enjoys good analytical properties:\n\n\\begin{lemma}\n\\label{lemma:generalized-xray}\nThe operator $\\Pi_m : C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$ is:\n\\begin{enumerate}\n\\item A pseudodifferential operator of order $-1$,\n\\item Formally self-adjoint and elliptic on twisted solenoidal tensors (its Fredholm index is thus equal to $0$ and its kernel\/cokernel are finite-dimensional),\n\\item Under the assumption that $\\Pi_m$ is $s$-injective, the following stability estimates hold:\n\\[\n\\forall s \\in \\mathbb{R},\\,\\, \\forall f \\in H^s(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\cap \\ker (D^{\\E})^*, ~~~ \\|f\\|_{H^s} \\leq C_s \\|\\Pi_m f \\|_{H^{s+1}},\n\\]\nfor some $C_s > 0$ and for some $C > 0$:\n\\[\n\\forall f \\in H^{-1\/2}(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\cap \\ker (D^{\\E})^*, ~~~ \\langle \\Pi_m f, f \\rangle_{L^2} \\geq C \\|\\pi_{\\ker (D^{\\E})^*} f\\|^2_{H^{-1\/2}}.\n\\]\nIn particular, these estimate hold if $(M,g)$ has negative curvature and $\\nabla^{\\E}$ has no twisted CKTs.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nThe proof of the first two points follows from a rather straightforward adaptation of the proof of \\cite[Theorem 2.5.1]{Lefeuvre-thesis} (see also \\cite{Guillarmou-17-1} for the original arguments); we omit it. It remains to prove the third point. \n\nThe first estimate follows from $(2)$, the elliptic estimate and the fact that $\\Pi_m$ is $s$-injective. The last estimate in the non-twisted case follows from \\cite[Lemma 2.1]{Guillarmou-Knieper-Lefeuvre-19} (or \\cite[Theorem 2.5.6]{Lefeuvre-thesis}) and subsequent remarks; the twisted case follows by minor adaptations.\n\n\nIf $(M,g)$ has negative curvature and $\\nabla^{\\E}$ has no twisted CKTs, using Lemma \\ref{lemma:x-ray} and by \\cite[Sections 4, 5]{Guillarmou-Paternain-Salo-Uhlmann-16} we get that $\\Pi_m$ is $s$-injective, proving the claim.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Exact Liv\\v{s}ic cocycle theory}\n\nWe phrase this section in a very general context which is that of a transitive Anosov flow on a smooth manifold. It is of independent interest to the rest of the article. \n\n\\label{section:livsic}\n\n\\subsection{Statement of the results}\n\n\\label{ssection:intro}\n\n\\subsubsection{A weak exact Liv\\v{s}ic cocycle theorem}\n\nLet $\\mathcal{M}$ be a smooth closed manifold endowed with a flow $(\\varphi_t)_{t \\in \\mathbb{R}}$ with infinitesimal generator $X \\in C^\\infty(\\mathcal{M},T\\mathcal{M})$. We assume that the flow is Anosov in the sense of \\eqref{equation:anosov} and that it is \\emph{transitive}, namely it admits a dense orbit\\footnote{Note that there are examples of non-transitive Anosov flows, see \\cite{Franks-Williamas-80}.}. We denote by $\\mathcal{G}$ the set of all periodic orbits for the flow and by $\\mathcal{G}^\\sharp$ the set of all \\emph{primitive} orbits, namely orbits which cannot be written as a shorter orbit to some positive power greater or equal than $2$. \n\n\n\n\nLet $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ be a smooth Hermitian vector bundle of rank $r$ equipped with a unitary connection $\\nabla^{\\mathcal{E}}$. We will denote by \n\\[\nC(x,t) : \\mathcal{E}_x \\rightarrow \\mathcal{E}_{\\varphi_t(x)},\n\\]\n the parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to the connection $\\nabla^{\\mathcal{E}}$. In the more general setting, we may consider $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$, two Hermitian vector bundles, equipped with two respective unitary connections $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$.\nRecall that if $\\nabla^{\\E_2} = p^*\\nabla^{\\E_1}$, for some unitary map $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2, \\E_1))$\\footnote{Here, we denote by $\\mathrm{U}(\\E_2, \\E_1) \\rightarrow \\mathcal{M}$ the bundle of unitary maps from $\\E_2 \\rightarrow \\E_1$. Of course, it may be empty if the bundles are not isomorphic.}, i.e. the connections are gauge-equivalent, then\nparallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ satisfies the commutation relation:\n\\[\nC_1(x,t)= p(\\varphi_t x) C_2(x,t) p(x)^{-1}.\n\\]\nWe say that such cocycles are \\emph{cohomologous}. In particular, given a closed orbit $\\gamma = (\\varphi_t x_0)_{t \\in [0,T]}$ of the flow, one has\n\\[\nC_1(x_0,T) = p(x_0) C_2(x_0,T) p(x_0)^{-1},\n\\]\ni.e. the parallel transport map are conjugate.\n\n\\begin{definition}\n\\label{definition:equivalence}\nWe say that the connections $\\nabla^{\\E_{1,2}}$ have \n\\emph{trace-equivalent holonomies} if for all primitive closed orbits $\\gamma \\in \\mathcal{G}^\\sharp$, we have:\n\\begin{equation}\n\\label{equation:trace-equivalence}\n\\Tr(C_1(x_\\gamma,\\ell(\\gamma))) = \\Tr(C_2(x_\\gamma,\\ell(\\gamma))),\n\\end{equation}\nwhere $x_\\gamma \\in \\gamma$ is arbitrary and $\\ell(\\gamma)$ is the period of $\\gamma$.\n\\end{definition}\n\nThis condition could be \\emph{a priori} obtained with $\\rk(\\E_1) \\neq \\rk(\\E_2)$. We shall see that this cannot be the case.\nThe following result is one of the main theorems of this paper. It seems to improve known results on Liv\\v{s}ic cocycle theory (in particular \\cite{Parry-99,Schmidt-99}), see \\S\\ref{ssection:exact} for a more extensive discussion on the literature.\n\n\\begin{theorem}\n\\label{theorem:weak}\nAssume $\\mathcal{M}$ is endowed with a smooth transitive Anosov flow. Let $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$ be two Hermitian vector bundles over $\\mathcal{M}$ equipped with respective unitary connections $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$. If the connections have \\emph{trace-equivalent holonomies} in the sense of Definition \\ref{definition:equivalence}, then there exists $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2,\\E_1))$ such that: for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$,\n\\begin{equation*}\nC_1(x,t) = p(\\varphi_t x) C_2(x,t) p(x)^{-1},\n\\end{equation*}\ni.e. the cocycles induced by parallel transport are cohomologous. Moreover, $\\E_2 \\simeq \\E_1$ are isomorphic.\n\\end{theorem}\n\nAs we shall see in the proof, for any given $L > 0$, it suffices to assume that the trace-equivalent holonomy condition \\eqref{equation:trace} holds for all primitive periodic orbits of length $\\geq L$ in order to get the conclusion of the theorem. Surprisingly, the rather weak condition \\eqref{equation:trace} implies in particular that the bundles are isomorphic as stated in Corollary \\ref{corollary:iso} and the trace of the holonomy of unitary connections along closed orbits should allow one in practice to classify vector bundles over manifolds carrying Anosov flows. Even more suprisingly, the rank of $\\E_1$ and $\\E_2$ might be \\emph{a priori} different and Theorem \\ref{theorem:weak} actually shows that the ranks have to coincide.\n\nThe idea relies on a key notion which we call \\emph{Parry's free monoid}, whose introduction goes back to Parry \\cite{Parry-99}. This free monoid $\\mathbf{G}$ corresponds (at least formally) to the free monoid generated by the set of homoclinic orbits to a given periodic orbit of a point $x_{\\star}$ (see \\S\\ref{sssection:homoclinic} for a definition) and we shall see that a connection induces a unitary representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ (it is not canonical but we shall see that its important properties are). Geometric properties of the connection can be read off this representation, see Theorem \\ref{theorem:iso} below. Moreover, tools from representation theory can be applied and this is eventually how we will prove Theorem \\ref{theorem:weak}. \n\n\n\\subsubsection{Opaque and transparent connections}\n\nTheorem \\ref{theorem:weak} has an interesting straightforward corollary. Recall that a unitary connection is said to be \\emph{transparent} if the holonomy along all closed orbits is trivial.\n\n\\begin{corollary}\nAssume $\\mathcal{M}$ is endowed with a smooth transitive Anosov flow. Let $\\E \\rightarrow \\mathcal{M}$ be a Hermitian vector bundle over $\\mathcal{M}$ of rank $r$ equipped with a unitary connection $\\nabla^{\\mathcal{E}}$. If the connection is transparent, then $\\E$ is trivial and trivialized by a smooth orthonormal family $e_1,...,e_r \\in C^\\infty(\\mathcal{M},\\E)$ such that $\\nabla^{\\E}_X e_i = 0$.\n\\end{corollary}\n\nThis result will be used in \\cite{Cekic-Lefeuvre-21-3} to study the Levi-Civita connection on Anosov manifolds and the notion of \\emph{transparent manifolds}, namely manifolds with trivial holonomy along closed geodesics. In order to prove the previous corollary, it suffices to apply Theorem \\ref{theorem:weak} with $\\E_1 = \\E$ equipped with $\\nabla^{\\E}$ and $\\E_2 = \\mathbb{C}^r \\times M$ equipped with the trivial flat connection. Then $C_2(x,t) = \\mathbf{1}$ and $(e_1,...,e_n)$ is obtained as the image by $p$ of the canonical basis of $\\mathbb{C}^n$. This corollary seems to be known in the folklore but nowhere written. It is stated in \\cite[Proposition 9.2]{Paternain-12} under the extra-assumption that $\\E \\oplus \\E^*$ is trivial.\n\nThe ``opposite\" notion of transparent connections is that of \\emph{opaque} connections which are connections that do not preserve any non-trivial subbundle $\\mathcal{F} \\subset \\E$ by parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$. It was shown in \\cite[Section 5]{Cekic-Lefeuvre-20} that the opacity of a connection is equivalent to the fact that\n\\[\n\\ker (\\nabla^{\\mathrm{End}}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}) = \\mathbb{C} \\cdot \\mathbf{1}_{\\E}.\n\\]\nAlso note that when $X$ is volume-preserving, this corresponds to the Pollicott-Ruelle (co)resonant states at $0$ associated to the operator $\\nabla^{\\mathrm{End}}_X$. We shall also connect this notion with the representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ of the free monoid:\n\n\\begin{prop}\n\\label{proposition:opaque}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item The connection $\\nabla^{\\E}$ is opaque;\n\\item $\\ker (\\nabla^{\\mathrm{End}}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}) = \\mathbb{C} \\cdot \\mathbf{1}_{\\E}$;\n\\item The representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ is irreducible.\n\\end{enumerate}\n\\end{prop}\n\n\n\n\\subsubsection{Kernel of the endomorphism connection}\n\nThe previous proposition actually follows from a more general statement which we now describe. The representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ gives rise to an orthogonal splitting \n\\[\n\\E_{x_{\\star}} = \\oplus_{i=1}^K \\E_i^{\\oplus n_i},\n\\]\nwhere $\\E_i \\subset \\E_{x_\\star}$ and $n_i \\geq 1$; each factor $\\E_i$ is $\\mathbf{G}$-invariant and the induced representation on each factor is irreducible; furthermore, for $i \\neq j$, the induced representations on $\\E_i$ and $\\E_j$ are not isomorphic. Let $\\mathbb{C}[\\mathbf{G}]$ be the formal algebra generated by $\\mathbf{G}$ over $\\mathbb{C}$ and let $\\mathbf{R} := \\rho\\left( \\mathbb{C}[\\mathbf{G}] \\right)$. By Burnside's Theorem (see \\cite[Corollary 3.3]{Lang-02} for instance), one has that:\n\\[\n\\mathbf{R} = \\oplus_{i=1}^K \\Delta_{n_i} \\mathrm{End}(\\E_i),\n\\]\nwhere $\\Delta_{n_i} u = u \\oplus ... \\oplus u$ for $u \\in \\mathrm{End}(\\E_i)$, the sum being repeated $n_i$-times. We introduce the \\emph{commutant} $\\mathbf{R}'$ of $\\mathbf{R}$, defined as:\n\\[\n\\mathbf{R}' := \\left\\{ u \\in \\mathrm{End}(\\E_{x_\\star}) ~|~ \\forall v \\in \\mathbf{R}, uv = vu\\right\\}.\n\\]\nWe then have:\n\n\\begin{theorem}\n\\label{theorem:iso}\nThere exists a natural isomorphism:\n\\[\n\\Phi : \\mathbf{R}' \\rightarrow \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}.\n\\]\nIn particular these spaces have same dimension, that is\n\\[\n\\dim\\left( \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))} \\right) = \\dim(\\mathbf{R}') = \\sum_{i=1}^K n_i^2.\n\\]\n\\end{theorem}\n\n\n\n\\subsubsection{Invariant sections} To conclude this paragraph, we now investigate the existence of smooth \\emph{invariant} sections of the bundle $\\E \\to \\mathcal{M}$, namely elements of $\\ker \\nabla^{\\E}_X|_{C^\\infty(\\mathcal{M},\\E)}$. First of all, observe that if $u \\in C^\\infty(\\mathcal{M},\\E)$ is an invariant section, then $u_\\star:= u(x_\\star)$ is invariant by the $\\mathbf{G}$-action. The converse is also true:\n\n\n\n\\begin{lemma}\n\\label{lemma:invariant-section}\nAssume that there exists $u_\\star \\in \\E_{x_\\star}$ such that $\\rho(g)u_\\star = u_\\star$ for all $g \\in \\mathbf{G}$. Then, there exists (a unique) $u \\in C^\\infty(\\mathcal{M},\\E)$ such that $u(x_\\star)=u_\\star$ and $\\nabla^{\\E}_X u = 0$.\n\\end{lemma}\n\nSuch an approach turns out to be useful when trying to understand a sort of \\emph{weak version} of Liv\\v{s}ic theory, such as the following: if $\\E \\rightarrow \\mathcal{M}$ is a vector bundle equipped with the unitary connection $\\nabla^{\\E}$ and for each periodic orbit $\\gamma \\in \\mathcal{G}$, there exists a section $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$, then one can wonder if this implies the existence of a global invariant smooth section $u \\in C^\\infty(\\mathcal{M},\\E)$? It turns out that the answer depends on the rank of $\\E$:\n\n\\begin{lemma}\n\\label{lemma:rank2}\nAssume that $\\rk(\\E) \\leq 2$ and that for all periodic orbits $\\gamma \\in \\mathcal{G}$, there exists $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$. Then, there exists $u \\in C^\\infty(\\mathcal{M},\\E)$ such that $\\nabla^{\\E}_X u = 0$.\n\\end{lemma}\n\nWe shall see that the proof of the previous Lemma is purely representation-theoretic and completely avoids the need to understand dynamics and the distribution of periodic orbits. We leave as an exercise for the reader the fact that Lemma \\ref{lemma:rank2} does not hold when $\\rk(\\E) \\geq 3$. A simple counter-example can be built using the following argument: any matrix in $\\mathrm{SO}(3)$ preserves an axis; hence, taking any $\\mathrm{SO}(3)$-connection on a \\emph{real} vector bundle of rank $3$ and then complexifying the bundle, one gets a vector bundle and a connection satisfying the assumptions of Lemma \\ref{lemma:rank2}; it then suffices to produce an $\\mathrm{SO}(3)$-connection without any invariant sections.\n\nWe believe that other links between properties of the representation $\\rho$ and the geometry and\/or dynamics of the parallel transport along the flowlines could be discovered. To conclude, let us also mention that all the results are presented here for complex vector bundles; most of them could be naturally restated for real vector bundles modulo the obvious modifications in the statements.\n\n\n\\subsection{Dynamical preliminaries on Anosov flows}\n\n\\label{section:anosov}\n\n\\subsubsection{Shadowing lemma and homoclinic orbits}\n\n\n\nFix an arbitrary Riemannian metric $g$ on $\\mathcal{M}$. As usual, we define the \\emph{local strong (un)stable manifolds} as:\n\\[\n\\begin{array}{l}\nW^{s}_{\\delta}(x) :=\\left\\{y \\in \\mathcal{M}, ~ \\forall t \\geq 0, d(\\varphi_t y, \\varphi_t x) < \\delta, d(\\varphi_t x, \\varphi_t y) \\rightarrow_{t \\rightarrow +\\infty} 0 \\right\\}, \\\\\nW^{u}_{\\delta}(x) :=\\left\\{y \\in \\mathcal{M}, ~ \\forall t \\leq 0, d(\\varphi_t y, \\varphi_t x) < \\delta, d(\\varphi_t x, \\varphi_t y) \\rightarrow_{t \\rightarrow -\\infty} 0 \\right\\},\n\\end{array}\n\\]\nwhere $\\delta > 0$ is chosen small enough. For $\\delta = \\infty$, we obtain the sets $W^{s,u}(x)$ which are the strong stable\/unstable manifolds of $x$. We also set $W^{s,u}_{\\mathrm{loc}}(x) =: W^{s,u}_{\\delta_0}(x)$ for some fixed $\\delta_0 > 0$ small enough. The \\emph{local weak (un)stable manifolds} $W^{ws,wu}_{\\delta}(x)$ are the set of points $y \\in B(x,\\delta)$ such that there exists $t \\in \\mathbb{R}$ with $|t| < \\delta$ and $\\varphi_t y \\in W^{s,u}_{\\mathrm{loc}}(x)$.\nThe following lemma is known as the \\emph{local product structure} (see \\cite[Theorem 5.1.1]{Fisher-Hasselblatt-19} for more details):\n\n\n\\begin{lemma}\n\\label{lemma:product-structure}\nThere exists $\\varepsilon_0, \\delta_0 > 0$ small enough such that for all $x,y \\in \\mathcal{M}$ such that $d(x,y) < \\varepsilon_0$, the intersection $W^{wu}_{\\delta_0}(x) \\cap W^{s}_{\\delta_0}(y)$ is precisely equal to a unique point $\\left\\{z\\right\\}$. We write $z := [x,y]$.\n\\end{lemma}\n\n\nThe main tool we will use to construct suitable \\emph{homoclinic orbits} is the following classical shadowing property of Anosov flows for which we refer to \\cite[Corollary 18.1.8]{Hasselblatt-Katok-95}, \\cite[Theorem\n5.3.2]{Fisher-Hasselblatt-19} and \\cite[Proposition 6.2.4]{Fisher-Hasselblatt-19}. For the sake of simplicity, we now write $\\gamma=[xy]$ if $\\gamma$ is an orbit segment of the flow with endpoints $x$ and $y$.\n\n\\begin{theorem}\n\\label{theorem:shadowing} There exist $\\varepsilon_0>0$, $\\theta>0$ and $C>0$ with the\nfollowing property. Consider $\\varepsilon<\\varepsilon_0$, and a finite or infinite sequence\nof orbit segments $\\gamma_i = [x_iy_i]$ of length $T_i$ greater than $1$ such\nthat for any $n$, $d(y_n,x_{n+1}) \\leq \\varepsilon$. Then there exists a genuine\norbit $\\gamma$ and times $\\tau_i$ such that $\\gamma$ restricted to $[\\tau_i,\n\\tau_i+T_i]$ \\emph{shadows} $\\gamma_i$ up to $C\\varepsilon$. More precisely, for all $t\\in\n[0, T_i]$, one has\n\\begin{equation}\n\\label{eq:d_hyperbolic}\n d(\\gamma(\\tau_i+t), \\gamma_i(t)) \\leq C \\varepsilon e^{-\\theta \\min(t, T_i-t)}.\n\\end{equation}\nMoreover, $|\\tau_{i+1} - (\\tau_i + T_i)| \\leq C \\varepsilon$. Finally, if the\nsequence of orbit segments $\\gamma_i$ is periodic, then the orbit $\\gamma$ is\nperiodic.\n\\end{theorem}\n\nLet us also make the following important comment. In the previous theorem, one can also allow the first orbit segment $\\gamma_i$ to be\ninfinite on the left, and the last orbit segment $\\gamma_j$ to be infinite on\nthe right. In this case,~\\eqref{eq:d_hyperbolic} should be replaced by: assuming that $\\gamma_i$ is defined on $(-\\infty, 0]$\nand $\\gamma_j$ on $[0,+\\infty)$, we would get for some $\\tilde\\tau_{i+1}$\nwithin $C\\varepsilon$ of $\\tau_{i+1}$, and all $t\\geq 0$\n\\begin{equation*}\n d(\\gamma(\\tilde\\tau_{i+1}-t), \\gamma_i(-t)) \\leq C \\varepsilon e^{-\\theta t}, \\quad d(\\gamma(\\tau_{j}+t), \\gamma_j(t)) \\leq C \\varepsilon e^{-\\theta t}.\n\\end{equation*}\n\n\n\n\n\n\n\\label{sssection:homoclinic}\n\nFix an arbitrary periodic point $x_{\\star} \\in \\mathcal{M}$ of period $T_{\\star}$ and denote by $\\gamma_{\\star}$ its primitive orbit.\n\n\\begin{definition}[Homoclinic orbits]\n\\label{definition:homoclinic}\nA point $p \\in \\mathcal{M}$ is said to be \\emph{homoclinic} to $x_{\\star}$ if $p \\in W^{ws}(x_{\\star}) \\cap W^{wu}(x_{\\star})$ (in other words, $d(\\varphi_{t+t^\\pm_0} p, \\varphi_t x_{\\star}) \\rightarrow_{t \\rightarrow \\pm \\infty} 0$ for some $t^\\pm_0 \\in \\mathbb{R}$). We say that an orbit $\\gamma$ is homoclinic to $x_{\\star}$ if it contains a point $p \\in \\gamma$ that is homoclinic to $x_{\\star}$ and we denote by $\\mathcal{H} \\subset \\mathcal{M}$ the set of homoclinic orbits to $x_{\\star}$.\n\\end{definition}\n\n\nNote that due to the hyperbolicity, the convergence of the point $p$ to $x_{\\star}$ is exponentially fast. More precisely, let $\\gamma$ be the orbit of $p$ and let $\\mathbb{R} \\ni t \\mapsto \\gamma(t)$ be the flow parametrization of $\\gamma$. Then, there exists uniform constants $C,\\theta > 0$ (independent of $\\gamma$) and $A_\\pm \\in \\mathbb{R}$ (depending on $\\gamma$) such that the following holds:\n\\begin{equation}\n\\label{equation:distance}\nd(\\gamma(A_\\pm \\pm n T_{\\star}), x_{\\star}) \\leq C e^{-\\theta n}.\n\\end{equation}\nThe points $\\gamma(A_\\pm)$ correspond to an arbitrary choice of points in $W^{s,u}_{\\delta_0}(x_{\\star})\\cap \\gamma$ (for some arbitrary $\\delta_0 > 0$ small enough). Homoclinic orbits have infinite length (except the orbit of $x_{\\star}$ itself) but it will be convenient to introduce a notion of \\emph{length} $T_\\gamma$ which we define to be equal to $T_\\gamma:=A_+-A_-$ (note that this is a highly non-canonical definition). We define the trunk to be equal to the central segment $\\gamma([A_-,A_+])$. In other words, the length of $\\gamma$ is equal to the length of its trunk. We also define the points: $x_n^\\pm := \\gamma(A_\\pm \\pm nT_{\\star})$. Note that another choice of values $A'_\\pm$ has to differ from $A_\\pm$ by $mT_{\\star}$ for some $m \\in \\mathbb{Z}$. Homoclinic orbits will play a key role as we shall see in due course. \n\n\\begin{lemma}\n\\label{lemma:dense}\nAssume that the flow is transitive. Then the set $\\mathcal{W}$ of points belonging to a homoclinic orbit in $\\mathcal{H}$ is dense in $\\mathcal{M}$.\n\\end{lemma}\n\n\n\\begin{proof}\nThis is a straightforward consequence of the shadowing Theorem \\ref{theorem:shadowing}: one concatenates a long segment $S$ of a transitive orbit with $\\gamma_{\\star}$ i.e. one applies Theorem \\ref{theorem:shadowing} with $... \\gamma_{\\star} \\gamma_{\\star} S \\gamma_{\\star} \\gamma_{\\star} ...$.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nIn the particular case of an Anosov geodesic flow on the unit tangent bundle, one can check that $\\mathcal{H}$ is in one-to-one correspondence with $\\pi_1(M)\/\\langle \\widetilde{\\gamma_{\\star}} \\rangle$, where $\\widetilde{\\gamma_{\\star}} \\in \\pi_1(M)$ is any element such that the conjugacy class of $\\widetilde{\\gamma_{\\star}}$ in $\\pi_1(M)$ corresponds\\footnote{Recall that the set of free homotopy classes $\\mathcal{C}$ is in one-to-one correspondence with conjugacy classes of $\\pi_1(M)$, see \\cite[Chapter 1]{Hatcher-02}.} to the free homotopy class $c \\in \\mathcal{C}$ whose unique geodesic representative is $\\gamma_{\\star}$. \n\\end{remark}\n\n\n\n\n\\subsubsection{Applications of the Ambrose-Singer formula}\n\n\nConsider a Hermitian vector bundle $\\E$ over $(\\mathcal{M}, g)$ equipped with a unitary connection $\\nabla = \\nabla^{\\E}$. If $x, y \\in \\mathcal{M}$ are at a distance less than the injectivity radius of $\\mathcal{M}$, denote by $C_{x \\to y}: \\E_x \\to \\E_y$ the parallel transport with respect to $\\nabla^{\\E}$ along the shortest geodesic from $x$ to $y$, by $C(x, t): \\E_x \\to \\E_{\\varphi_tx}$ the parallel transport along the flow and by $C_\\gamma$ the parallel transport along a curve $\\gamma$. For $U \\in C^\\infty(\\mathcal{M},\\mathrm{End}(\\mathcal{E}))$, we define, for all $x \\in \\mathcal{M}$, \n\\[\n\\|U\\|_x := \\Tr(U^*(x)U(x))^{1\/2},\n\\]\nand $\\|U\\|_{L^\\infty} := \\sup_{x \\in \\mathcal{M}} \\|U\\|_x$. In particular, if $U \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\mathcal{E}))$ is unitary, then $\\|U\\|_{L^\\infty} = \\sqrt{\\rk(\\E)}$. We record the following consequences of Lemma \\ref{lemma:ambrosesinger}:\n\n\n\n\\begin{lemma}\\label{lemma:ASgeometry}\nThe following consequences of the Ambrose-Singer formula hold:\n\\begin{enumerate}\n\t\\item Assume we are in the setting of Theorem \\ref{theorem:shadowing}: for some $C, \\varepsilon, T > 0$, let $x, p \\in \\mathcal{M}$ satisfy $d(\\varphi_t x, \\varphi_t p) \\leq C \\varepsilon e^{-\\theta \\min(t,T-t)}$ for all $t \\in [0, T]$. Then for any $0 \\leq T_1 \\leq T$:\n\t\\[\\|C(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\|_x \\leq \\frac{c_0C \\varepsilon}{\\theta} \\times \\|F_\\nabla\\|_{C^0},\\]\n\twhere $c_0 = c_0(X, g) > 0$ depends only on the flow $X$ and the metric.\n\t\n\t\\item Assume $\\gamma \\subset B(p, \\imath\/2)$ is a closed piecewise smooth curve at $p$ of length $L$. Then for some $C = C(g) > 0$ depending on the metric:\n\t\\[\\|C_\\gamma - \\id_p\\|_p \\leq C L \\times \\sup_{y \\in \\gamma} d(p, y) \\times \\|F_\\nabla\\|_{C^0}.\\]\n\t\n\t\\item Let $\\gamma: [0, L] \\to M$ be a unit speed curve based at $p$, and $\\nabla'$ be a second unitary connection on $\\E$, whose parallel transport along $\\gamma$ we denote by $C'_\\gamma$. Then:\n\t\\[\\|C_{\\gamma}^{-1}C'_{\\gamma} - \\id_p\\|_p \\leq L \\times \\|\\nabla - \\nabla'\\|_{C^0}.\\]\n\t\\end{enumerate}\n\\end{lemma}\nThe geometries appearing in (1), (2) and (3) are depicted in Figure \\ref{fig:ASgeometry} (A), (B) and (C), respectively.\n \n \\begin{figure}\n \\centering\n\\begin{subfigure}{0.39\\linewidth}\n \\centering\n\\begin{tikzpicture}[scale = 0.55, everynode\/.style={scale=0.5}]\n\\tikzset{cross\/.style={cross out, draw=black, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt},\ncross\/.default={1pt}}\n\n \n \t\n \n\t\t\\fill[pattern=vertical lines, pattern color=blue, draw = black] (0, 0) -- (0, 1.75) to[out = -35, in= -145, distance=75] (10, 1.75) -- (10, 0) -- (0,0);\n\t\t\n\t\t\n\t\t\\fill (0, 0) node[below left] {\\small $x$} circle (1.5pt);\n\t\t\\fill (10, 0) node[below right] {\\small $\\varphi_Tx$} circle (1.5pt);\n\t\t\\fill (0, 1.75) node[left] {\\small $p$} circle (1.5pt);\n\t\t\\fill (10, 1.75) node[right] {\\small $\\varphi_T p$} circle (1.5pt);\n\n\t\t\\draw[thick, ->] (5, 1.1) -- (5, 0.6) node[above left] {\\tiny $\\varphi_t p$};\n\t\t\\draw (5, 1.25) node[right] {$\\mathcal{O}(e^{-\\theta t})$};\n\t\t\\fill (5, 0.6) circle (1pt);\n\t\t\\draw[thick, ->] (5, -0.5) -- (5, 0) node[below left] {\\tiny $\\varphi_t x$};\t\t\n\t\t\\fill (5, 0) circle (1pt);\n\\end{tikzpicture}\n \\caption{\\small Shadowing homotopy.}\n \n \\end{subfigure}\n \\begin{subfigure}{0.29\\linewidth}\n \\centering\n \t\t\\begin{tikzpicture}[scale=0.8]\n \t\t\\draw (0, 0) circle (1.5);\n \t\t\t\\draw (0, 0) node[below]{$p$}--(-1, 1)--(1,1)--(0,0);\n \t\t\t\\fill (0, 0) circle (1pt) (1, 1) circle (1pt) (-1, 1) circle (1pt);\n \t\t\t\\draw[blue] (0, 0)--(-0.8, 1) (0, 0)--(-0.6,1) (0, 0)--(-0.4,1) (0, 0)--(-0.2,1) (0, 0)--(-0,1) (0, 0)--(0.2,1) (0, 0)--(0.4,1) (0, 0)--(0.6,1) (0, 0)--(0.8,1);\n \t\t\\draw (0.5, 0.4) node[right]{$\\gamma$};\n \t\t\\end{tikzpicture}\n \t\t \\caption{\\small Radial homotopy.}\n \n \\end{subfigure}\n \\begin{subfigure}{0.29\\linewidth}\n \\centering\n \t\t\\begin{tikzpicture}\n \t\t\t\\draw[thick, ->] (0, 0) node[below]{\\small $p = \\gamma(0)$}--(0.5, 0.5);\n \t\t\t\\draw[thick] (0.5, 0.5)--(1, 1) node[right]{\\small $\\gamma(L)$};\n \t\t\t\\fill (0, 0) circle(1pt) (1, 1) circle(1pt);\n \t\t\n \t\t\t\\draw[->] (0.5, 0.2)--(0.8, 0.5) node[below right]{\\small $C'_\\gamma$};\n \t\t\t\\draw[->] (0.5, 0.8)--(0.2, 0.5) node[above left]{\\small $C_\\gamma^{-1}$};\n \t\t\\end{tikzpicture}\n \t\t \t\t \\caption{\\small Straight-line homotopy.}\n \n \\end{subfigure}\n \\caption{\\small Presentation of the geometries considered in Lemma \\ref{lemma:ASgeometry}.}\n \\label{fig:ASgeometry}\n\\end{figure}\n\\begin{proof\n\tWe first prove (1). For $C, \\varepsilon$ small enough, for all $t \\in [0, T]$ we denote by $\\tau_t$ the unit speed shortest geodesic, of length $\\ell(t)$, from $\\varphi_tx$ to $\\varphi_t p$. Define a smooth homotopy $\\Gamma: [0, 1]^2 \\to \\mathcal{M}$ by setting:\n\t\\[\\Gamma(s, t) := \\tau_{tT_1}(s \\ell(t)),\\]\n\tand note that by assumption $\\ell(t) \\leq C\\varepsilon e^{-\\theta \\min(t, T - t)}$. We apply Lemma \\ref{lemma:ambrosesinger} to the homotopy $\\Gamma$ to obtain, after a rescaling of parameters $s$ and $t$:\n\t\\begin{multline*}\n\t\tC(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\\\\n\t\t= \\int_0^{T_1} \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, ds \\, dt.\n\t\\end{multline*}\n\tHere we recall $C_{\\uparrow}$ and $C_{\\rightarrow}$ are parallel transport maps obtained by parallel transport along curves as in Figure \\ref{fig:AS1}. Since $C_{\\uparrow}$ and $C_{\\rightarrow}$ are isometries, and since by compactness $|\\partial_t \\tau_t(s)| \\leq D$ for some $0 < D = D(X, g)$, we have:\n\t\\begin{multline*}\n\t\t\\|C(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\|_x\\\\\n\t\t \\leq CD\\varepsilon \\|F_{\\nabla}\\|_{C^0} \\int_0^T e^{-\\theta \\min(t, T - t)} dt \\leq \\frac{2D C}{\\theta} \\times \\varepsilon \\|F_\\nabla\\|_{C^0}.\n\t\\end{multline*}\n\t\n\tFor (2), we may assume by approximation that $\\gamma$ is smooth. Then taking the homotopy \n\t\\[\\Gamma(s, t) = \\exp_x( t \\exp_x^{-1} (\\gamma(sL))),\\]\n\tand applying Lemma \\ref{lemma:ambrosesinger}, we obtain by a rescaling of $s$ and writing $\\widetilde{\\Gamma}(s, t) = \\Gamma(s\/L, t)$:\n\t\\[C_\\gamma - \\id_x = \\int_0^L \\int_0^1 C_1(s, t)^{-1} F_\\nabla(\\partial_s \\widetilde{\\Gamma}, \\partial_t \\widetilde{\\Gamma}) C_2(s, t) \\,dt \\,ds.\\]\n\tThe estimate now follows by using $\\|\\partial_t \\widetilde{\\Gamma}\\| \\leq C d(x, \\gamma(s))$, where we introduce the positive constant $C = \\sup_{x \\in M} \\sup_{|y|_{g_x} < \\imath\/2}\\|d\\exp_x(y)\\|_{T_xM \\to T_{\\exp_x(y)}}$.\n\t\n\tFor the final item, denote by $C_t, C_t'$ the parallel transports along $\\gamma|_{[0, t]}$ with the connection $\\nabla, \\nabla'$, respectively. Then it is straightforward that $\\partial_t(C_t^{-1}C'_t) = C_t^{-1}(\\nabla- \\nabla')(\\dot{\\gamma}(t)) C_t'$, so\n\t\\[C_\\gamma^{-1} C_\\gamma' - \\id_p = \\int_0^L C_t^{-1}(\\nabla- \\nabla')(\\dot{\\gamma}(t)) C_t' \\,dt.\\]\n\tThe required estimate follows\n\\end{proof}\n\nWe also have the following result to which we will refer to as the \\emph{spiral Lemma}:\n\n\\begin{lemma}\n\\label{lemma:spiral}\nLet $x_{\\star} \\in \\mathcal{M}$ be a periodic point of period $T_{\\star}$ and let $x_0 \\in W^s_{\\mathrm{loc}}(x_{\\star})$. Define $x_n := \\varphi_{nT_{\\star}}x_0$ and write $q_n := C(x_{\\star},nT_{\\star})^{-1}C_{x_n \\rightarrow x_{\\star}} C(x_0, n T_{\\star})C_{x_{\\star} \\rightarrow x_0}$. Then:\n\\[\n\\rho(x_0) := \\lim_{n \\rightarrow +\\infty} q_n \\in \\mathrm{U}(\\E_{x_{\\star}})\n\\]\nexists. Moreover, there exist some uniform constants $C,\\theta > 0$ such that\n\\[\n|q_n - \\rho(x_0)| \\leq C e^{-\\theta n}.\n\\]\n\\end{lemma}\n\n\\begin{center}\n\\begin{figure}[htbp!]\n\\includegraphics{spiral.eps}\n\\caption{The spiral Lemma: the set $\\Omega_1$ corresponds to the area over which the integral in the Ambrose-Singer formula is computed for $n=1$.}\n\\label{fig:spiral}\n\\end{figure}\n\\end{center} \n\n\\begin{proof}\nApply the Ambrose-Singer formula as in the first item of the previous Lemma (same notations as in the previous proof):\n\\[\nq_n - \\mathbbm{1}_{\\E_{x_{\\star}}} = \\int_0^{nT_{\\star}} \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, ds \\, dt,\n\\]\nwhere $\\tau_t$ is the unit speed shortest geodesic of length $\\ell(t)$ from $\\varphi_t x_0$ and $\\varphi_t x_{\\star}$. Observe that this integral converges absolutely as (see \\eqref{equation:distance}):\n\\[\n\\int_0^{nT_{\\star}} \\norm{ \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, \\dd s \\, } \\dd t \\leq \\int_0^{n T_{\\star}} C \\|F_{\\nabla}\\|_{C^0} e^{-\\theta t} \\dd t < \\infty,\n\\]\nand thus the limit exists. Moreover, it is clear that the convergence is exponential.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of the exact Liv\\v{s}ic cocycle Theorem}\n\n\\label{section:weak-strong}\n\n\n\n\\subsubsection{Parry's free monoid}\n\nAs we shall see, Parry's free monoid is the key notion to understand the holonomy of unitary connections.\nWhereas flat connections up to gauge equivalence correspond to representations of the fundamental group up to conjugacy, in the setting of hyperbolic dynamics, we will show that \\emph{arbitrary} connections up to cocycle equivalence correspond to representations of Parry's free monoid. \nRecall from \\S\\ref{sssection:homoclinic} that $x_{\\star} \\in \\mathcal{M}$ is a periodic point of period $T_{\\star}$. Let $\\mathbf{G}$ be the free monoid generated by $\\mathcal{H}$ (homoclinic orbits to $x_{\\star}$), namely the formal set of words\n\\[\n\\mathbf{G} := \\left\\{\\gamma_1^{m_1} ... \\gamma_k^{m_k} ~|~k \\in \\mathbb{N}, m_1,...,m_k \\in \\mathbb{N}_0, \\gamma_1,...,\\gamma_k \\in \\mathcal{H}\\right\\},\n\\]\nendowed with the obvious monoid structure. The empty word corresponds to the identity element denoted by $\\mathbf{1}_{\\mathbf{G}}$. Note the periodic orbit corresponding to $x_{\\star}$ also belongs to the set of homoclinic orbits. We call $\\mathbf{G}$ \\emph{Parry's free monoid} as the idea (although not written like this) was first introduced in his work \\cite{Parry-99} (see also \\cite{Schmidt-99} for a related approach).\nThe main result of this paragraph is the following:\n\n\\begin{prop}\n\\label{proposition:representation0}\nLet $\\nabla^{\\E}$ be a unitary connection on the Hermitian vector bundle $\\E \\rightarrow \\mathcal{M}$. Then $\\nabla^{\\E}$ induces a representation\n\\[\n\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}}).\n\\]\n\\end{prop}\n\nFormally, this proposition could have also been stated as a definition.\n\n\\begin{proof}\nSince $\\mathbf{G}$ is a free monoid, it suffices to define $\\rho$ on the set of generators of $\\mathbf{G}$, namely for all homoclinic orbits $\\gamma \\in \\mathcal{H}$. For the neutral element we set $\\rho(\\mathbf{1}_{\\mathbf{G}}) = \\mathbbm{1}_{\\E_{x_{\\star}}}$. For the periodic orbit $\\gamma_{\\star}$ of $x_{\\star}$, we set $\\rho(\\gamma_{\\star}) := C(x_{\\star},T_{\\star})$.\n\nLet $\\gamma \\in \\mathcal{H}$ (and $\\gamma \\neq \\gamma_{\\star}$) and consider a parametrization $\\mathbb{R} \\ni t \\mapsto \\gamma(t)$. Following the notations of \\S\\ref{sssection:homoclinic}, we let $x_n^\\pm := \\gamma(A_\\pm \\pm nT_{\\star}), x_n^+ = \\varphi_{T_n}(x_n^-)$ for some $T_n = A_+-A_-+2n T_{\\star}$, where $T_\\gamma := A_+ - A_-$ (length of the trunk), and the points $(x_n^\\pm)_{n \\in \\mathbb{N}}$ converge exponentially fast to $x_{\\star}$ as $n \\rightarrow \\infty$. As we shall see, there is a small technical issue coming from the fact that $C(x_{\\star},T_{\\star})$ is not trivial and this can be overcome by considering a subsequence $k_n \\rightarrow \\infty$ such that\\footnote{For any compact metric group $G$, if $g \\in G$, there exists a sequence $k_n \\in \\mathbb{N}$ such that $g^{k_n} \\rightarrow_{n \\rightarrow \\infty} \\mathbf{1}_G$.}\n\\begin{equation}\\label{eq:k_n}\n\tC(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\E_{x_{\\star}}}, \\quad n \\to \\infty.\\end{equation}\n\n\n\nFor $n,m \\in \\mathbb{N}$, we define $\\rho_{m, n}(\\gamma) \\in \\mathrm{U}(\\E_{x_{\\star}})$ as follows: \n\\begin{equation}\\label{eq:rhomn}\n\\rho_{m,n}(\\gamma) := C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x_0^+, k_m T_{\\star}) C(x_0^-, T_\\gamma)C(x_{k_n}^-,k_nT_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-},\n\\end{equation}\nand we will write $\\rho_{n}(\\gamma) := \\rho_{n,n}(\\gamma)$.\n\n\\begin{lemma}\n\\label{lemma:convergence}\nThere exists $\\rho(\\gamma) \\in \\mathrm{U}(\\E_{x_{\\star}})$ such that:\n\\[\n\\rho_{m,n}(\\gamma) \\rightarrow_{n,m \\rightarrow \\infty} \\rho(\\gamma),\n\\]\nand $\\rho(\\gamma)$ does not depend in which sense the limit in $n,m$ is taken. \n\\end{lemma}\n\n\n\\begin{proof}\nWe have by construction:\n\\begin{equation}\n\\label{equation:cv0}\n\\begin{split}\n\\rho_{m,n}(\\gamma) & = C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x^+_0, k_m T_{\\star})C(x^-_0, T_\\gamma)C(x_{k_n}^-,k_n T_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-}\\\\\n& = \\left[C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x^+_0, k_m T_{\\star}) C_{x_{\\star} \\rightarrow x^+_0} C(x_{\\star},k_mT_{\\star})^{-1}\\right] \\\\\n& \\hspace{2cm} \\times C(x_{\\star},k_mT_{\\star}) C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} C(x_{\\star},k_nT_{\\star}) \\\\\n& \\hspace{2cm} \\times \\left[C(x_{\\star},k_nT_{\\star})^{-1}C_{x^-_0 \\rightarrow x_{\\star}}C(x_{k_n}^-,k_n T_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-}\\right],\n\\end{split}\n\\end{equation}\nwhere $T_\\gamma$ is independent of $n$ (trunk of $\\gamma$). For the middle term we have by \\eqref{eq:k_n}\n\\[\nC(x_{\\star},k_mT_{\\star}) C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} C(x_{\\star},k_nT_{\\star}) = C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} + o(1),\n\\]\nas $n,m$ go to $+\\infty$. Moreover the convergence of the terms between brackets follow from the spiral Lemma \\ref{lemma:spiral} (the convergence is exponentially fast).\n\\end{proof}\n\nThis concludes the proof.\n\n\\end{proof}\n\n\\begin{remark}\n\\rm\nFor $\\gamma \\in \\mathcal{H}$, \\eqref{equation:cv0} shows that $\\rho(\\gamma)$ does not depend on the choice of subsequence $(k_n)_{n \\in \\mathbb{N}}$ as long as it satisfies $C(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}$. However, $\\rho(\\gamma)$ does depend on the choice of trunk $[x_0^-x_0^+]$ for $\\gamma$ and another choice of trunk produces a $\\rho'(\\gamma)$ which differs from $\\rho(\\gamma)$ by:\n\\begin{equation}\n\\label{equation:differs}\n\\rho'(\\gamma) = C(x_{\\star},T_{\\star})^{m_L(\\gamma)} \\rho(\\gamma) C(x_{\\star},T_{\\star})^{m_R(\\gamma)},\n\\end{equation}\nwhere $m_L(\\gamma), m_R(\\gamma) \\in \\mathbb{Z}$. \n\\end{remark}\n\n\n\\subsubsection{Conjugate representations}\n\nWe introduce the submonoid $\\mathbf{G}^* := \\mathbf{G} \\setminus \\left\\{ \\gamma_{\\star}^k, k \\geq 1\\right\\}$ that is $\\mathbf{G}$ minus powers of $\\gamma_{\\star}$. Recall that the character of a representation $\\rho$ is defined by $\\chi_\\rho(\\bullet) := \\Tr(\\rho(\\bullet))$. This paragraph is devoted to proving the following:\n\n\\begin{prop}\n\\label{proposition:representation}\nLet $\\nabla^{\\E_{1,2}}$ be two unitary connections on the Hermitian vector bundles $\\E_1,\\E_2 \\rightarrow \\mathcal{M}$. Assume that the connections have trace-equivalent holonomies in the sense of Definition \\ref{definition:equivalence}. Then, the induced representations $\\rho_{1,2} : \\mathbf{G}^* \\rightarrow \\mathrm{U}({\\E_{1,2}}_{x_{\\star}})$ have the same character. In particular, this implies that they are isomorphic, i.e. there exists $p_{\\star} \\in \\mathrm{U}({\\E_2}_{x_{\\star}}, {\\E_1}_{x_{\\star}})$ such that: \n\\begin{equation}\\label{eq:repconj}\n\\forall \\gamma \\in \\mathbf{G}, ~~ \\rho_1(\\gamma) = p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}.\n\\end{equation}\n\\end{prop}\n\nFollowing Lemma \\ref{lemma:convergence}, we consider a subsequence $(k_n)_{n \\in \\mathbb{N}}$ such that $C_{1,2}(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}$. \n\n\\begin{proof}\nOnce we know that the representations have the same character, the conclusion is a straightforward consequence of a general fact of representation theory, see \\cite[Corollary 3.8]{Lang-02}. \nFor the sake of simplicity, we take $\\gamma = \\gamma_1 \\cdot \\gamma_2$, where $\\gamma_{1,2} \\in \\mathcal{H}$ (and both $\\gamma_{1,2}$ cannot be equal to $\\gamma_{\\star}$ at the same time since the word $\\gamma$ is in $\\mathbf{G}^*$) but the generalization to longer words is straightforward as we shall see and words of length $1$ are also handled similarly (one does not even need to concatenate orbits in this case). The empty word (corresponding to the identity element in $\\mathbf{G}^*)$ will also be dealt with separately. This proposition is based on the shadowing Theorem \\ref{theorem:shadowing} and the fact that one can concatenate orbits. But we will have to be careful to produce periodic orbits which are primitive.\n\nWe have by Lemma \\ref{lemma:convergence}:\n\\[\n\\rho_1(\\gamma) = \\rho_1(\\gamma_1) \\rho_1(\\gamma_2) = \\rho_{1 ; n,N}(\\gamma_1)\\rho_{1 ; n,n}(\\gamma_2) + o(1), \\quad n \\to \\infty,\n\\]\nwhere we use the convention $\\rho_{i;a,b}$ to denote the expression in \\eqref{eq:rhomn} with respect to $\\nabla^{\\E_i}$, for $i=1,2$. The term $N = N(n) \\geq n$ will ensure that a certain orbit is primitive as we shall see below. Let $x_{n}^\\pm(i)$ be the points on the orbit $\\gamma_i$ that are exponentially close to $x_{\\star}$, given by \\S\\ref{sssection:homoclinic}. Consider the concatenation of the orbits $S := [x_{k_{N}}^-(1) x_{k_{n}}^+(1)] \\cup [x_{k_n}^-(2) x_{k_n}^+(2)]$. Note that the starting points and endpoints of these segments are at distance at most $\\mathcal{O}(e^{-\\theta k_n})$. Thus by the shadowing Theorem \\ref{theorem:shadowing}, there exists a genuine periodic orbit $\\widetilde{\\gamma_n}$ and a point $y_n \\in \\widetilde{\\gamma_n}$ (of period $T'_n$) which $\\mathcal{O}(e^{-\\theta k_n})$-shadows the concatenation $S$ (here, if we have a longer word of length $k$, it suffices to apply the shadowing Theorem \\ref{theorem:shadowing} with $k$ segments). \n\nWe claim that $\\widetilde{\\gamma_n}$ is primitive for all $N$ large enough. Indeed, observe that $\\widetilde{\\gamma_n}$ can be decomposed into the following six subsegments:\n\n\\begin{center}\n\\begin{figure}[htbp!]\n\\includegraphics{primitive.eps}\n\\caption{The orbit $\\widetilde{\\gamma_n}$ is made of six segments: in the first segment (of length $k_{n}T_{\\star}$), it shadows the first portion $[x_{k_{n}}^-(2) x_{0}^-(2)]$ which wraps around $\\gamma_{\\star}$; in the second (of length $T_{\\gamma_2}$), it shadows the trunk $[x_{0}^-(2) x_{0}^+(2)]$, in the third (of length $k_{n}T_{\\star}$), it shadows the last portion of $[x_{0}^+(2) x_{k_{n}}^+(2)]$ which also wraps around $\\gamma_{\\star}$; then this process is repeated but for the second orbit $\\gamma_1$.}\n\\label{fig:primitive}\n\\end{figure}\n\\end{center} \n\nMoreover, the total length of $\\widetilde{\\gamma_n}$ is\n\\[\nT'_n = T_{\\gamma_2} + 2 k_n T_{\\star} + T_{\\gamma_1} + (k_{N}+k_n) T_{\\star} + \\mathcal{O}(e^{-\\theta k_n}).\n\\]\nTake $x \\in \\gamma_1 \\cup \\gamma_2$ with $x \\not \\in \\gamma_{\\star}$, and consider a small $\\varepsilon > 0$ such that $d(x, \\gamma_{\\star}) > 3\\varepsilon$. Let $n$ large enough so that for all $m \\geq n$ we have the tail $[x_m^-(1) x_n^-(1)] \\subset B(\\gamma_{\\star}, \\varepsilon)$, $\\widetilde{\\gamma_n}$ satisfies $d(\\widetilde{\\gamma_n}, x) < \\varepsilon$ and finally, so that the shadowing factor of Theorem \\ref{theorem:shadowing} satisfies $\\mathcal{O}(e^{-\\theta k_n}) < \\varepsilon$. Pick $N \\geq n$ such that $(k_{N} - k_n) T_{\\star} > T'_n\/2$. We argue by contradiction and assume that $\\widetilde{\\gamma_n} = \\gamma_0^k$ for some $k \\geq 2$ and $\\gamma_0 \\in \\mathcal{G}^\\sharp$, a primitive orbit.\n\nThis implies that there is a copy of $\\gamma_0$ in the central red segment of Figure \\ref{fig:primitive} which $\\mathcal{O}(e^{-\\theta k_n})$-shadows the orbit of $x_N^-(1)$ and this forces $\\widetilde{\\gamma_n} \\subset B(\\gamma_{\\star}, 2\\varepsilon)$. Thus $d(\\widetilde{\\gamma_n}, x) > \\varepsilon$, which is a contradiction.\n\n\nBy the first and second items of Lemma \\ref{lemma:ASgeometry}, we have:\n\\[\n\\rho_{1;n,N}(\\gamma_1)\\rho_{1;n,n}(\\gamma_2) = C_{1,y_n \\rightarrow x_{\\star}} C_1(y_n, T'_n)C_{1,y_n \\rightarrow x_{\\star}}^{-1} + \\mathcal{O}(e^{-\\theta k_n}).\n\\]\nBy assumption, we have $\\Tr(C_1(y_n, T'_n)) = \\Tr(C_2(y_n,T'_n))$. This yields:\n\\[\n\\begin{split}\n\\Tr(\\rho_1(\\gamma)) & = \\Tr(C_{1,y_n \\rightarrow x_{\\star}} C_1(y_n, T'_n)C_{1,y_n \\rightarrow x_{\\star}}^{-1} ) + o(1) \\\\\n& = \\Tr(C_1(y_n, T'_n)) + o(1) \\\\\n& = \\Tr(C_2(y_n, T'_n)) + o(1) = \\Tr(\\rho_2(\\gamma)) + o(1).\n\\end{split}\n\\]\nTaking the limit as $n \\rightarrow \\infty$, we obtain the claimed result about characters for all non-empty words $\\gamma \\in \\mathbf{G}^*$. \n\nIt remains to deal with the empty word. For that, take any $\\gamma \\in \\mathbf{G}^*$, and consider $n_i \\in \\mathbb{N}$, a subsequence such that $\\rho_1(\\gamma)^{n_i} \\rightarrow \\mathbbm{1}_{{\\E_1}_{x_{\\star}}}$ and $\\rho_2(\\gamma)^{n_i} \\rightarrow \\mathbbm{1}_{{\\E_2}_{x_{\\star}}}$. Then:\n\\[\n\\Tr\\left(\\rho_1(\\gamma)^{n_i}\\right) =\\Tr\\left(\\rho_2(\\gamma)^{n_i}\\right),\n\\]\nand taking the limit as $i \\rightarrow \\infty$ gives\n\\[\n\\Tr(\\rho_1(\\mathbf{1}_{\\mathbf{G}^*}))=\\rk(\\E_1)=\\rk(\\E_2) =\\Tr(\\rho_2(\\mathbf{1}_{\\mathbf{G}^*})).\n\\]\nBy the mentioned \\cite[Corollary 3.8]{Lang-02}, there is a $p_{\\star}$ satisfying \\eqref{eq:repconj} for $\\gamma \\in \\mathbf{G}^*$.\n\nIt is now straightforward to show \\eqref{eq:repconj} for all $\\gamma \\in \\mathbf{G}$. Applying \\eqref{eq:repconj} with $\\gamma_{\\star} \\gamma \\in \\mathbf{G}^*$, where $\\gamma \\in \\mathcal{H}\\setminus \\{\\gamma_{\\star}\\}$ is arbitrary, we get that:\n\\[\n\\begin{split}\n\\rho_1(\\gamma_{\\star} \\gamma)& = \\rho_1(\\gamma_{\\star}) \\rho_1(\\gamma) \\\\\n& = p_{\\star} \\rho_2(\\gamma_{\\star} \\gamma) p_{\\star}^{-1} = p_{\\star} \\rho_2(\\gamma_{\\star}) p_{\\star}^{-1} p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}.\n\\end{split}\n\\]\nSince $\\rho_1(\\gamma) = p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}$ (because $\\gamma \\in \\mathbf{G}^*$), we get that $\\rho_1(\\gamma_{\\star}) = p_{\\star} \\rho_2(\\gamma_{\\star}) p_{\\star}^{-1}$, that is $C_1(x_{\\star},T_{\\star}) = p_{\\star} C_2(x_{\\star},T_{\\star}) p_{\\star}^{-1}$ or equivalently $P(x_{\\star},T_{\\star})p_{\\star} = p_{\\star}$ (where $P$ denotes the parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to the mixed connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X$, as in \\eqref{equation:tp-mixed}), concluding the proof.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nAlthough the representations $\\rho_{1,2}$ depend on choices (namely on a choice of trunk for each homoclinic orbit $\\gamma \\in \\mathcal{H}$), the map $p_{\\star} \\in \\mathrm{U}(\\E_{x_{\\star}})$ does not. Indeed, taking $\\rho_{1,2}'$ two other representations (for some other choices of trunks), one gets by \\eqref{equation:differs}:\n\\[\n\\begin{split}\n\\rho'_1(\\gamma) & = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} \\rho_1(\\gamma) C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n& = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1} C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n& = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} p_{\\star} C_2(x_{\\star},T_{\\star})^{-m_L(\\gamma)} \\rho'_2(\\gamma) C_2(x_{\\star},T_{\\star})^{-m_R(\\gamma)} p_{\\star}^{-1} C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n& = \\left(P(x_{\\star},m_L(\\gamma)T_{\\star})p_{\\star}\\right) \\rho'_2(\\gamma) \\left(P(x_{\\star},m_R(\\gamma)T_{\\star})p_{\\star}\\right)^{-1} = p_{\\star} \\rho'_2(\\gamma) p_{\\star}^{-1},\n\\end{split}\n\\]\nsince $P(x_{\\star},T_{\\star})p_{\\star}=p_{\\star}$, that is $p_{\\star}$ also conjugates the representations $\\rho'_{1,2}$. Note that the map $p_{\\star}$ given by \\cite[Corollary 3.8]{Lang-02} is generally not unique. Nevertheless, if the representation is irreducible, it is unique modulo the trivial $\\mathbb{S}^{1}$-action.\n\\end{remark}\n\n\n\\subsubsection{Proof of Theorem \\ref{theorem:weak}}\n\nWe can now complete the proof of Theorem \\ref{theorem:weak}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:weak}]\nLet $\\mathcal{W}$ be the set of all points belonging to homoclinic orbits in $\\mathcal{H}$. By Lemma \\ref{lemma:dense}, $\\mathcal{W}$ is dense in $\\mathcal{M}$ and we are going to define the map $p$ (which will conjugate the cocycles) on $\\mathcal{W}$ and then show that $p$ is Lipschitz-continuous on $\\mathcal{W}$ so that it extends naturally to $\\mathcal{M}$. The map $p$ is defined as the parallel transport of $p_{\\star}$ with respect to the mixed connection.\n\nBy assumptions, we have $C_i(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\E_*}$, and thus $P(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}} )}$ (where we use the notation $\\mathbbm{1}_{\\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}} )}(q) = q$ for $q \\in \\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}})$). Consider a point $x \\in \\gamma$, where $\\gamma \\in \\mathcal{H}$ is a homoclinic orbit and also consider a parametrization of $\\gamma$ as in \\S\\ref{sssection:homoclinic}. For $n \\in \\mathbb{N}$ large enough, consider the point $x_n^- \\in \\gamma$ (which is exponentially close to $x_{\\star}$) and write $x = \\varphi_{T^-_n}(x_n^-)$ for some $T^-_n > 0$. Define:\n\\[\np^-_n(x) := P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x}).\n\\]\n\n\\begin{lemma}\n\\label{lemma:p-minus}\nFix $\\gamma \\in \\mathcal{H}$. Then for all $x \\in \\gamma$, there exists $p_-(x) \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x} )$ such that $p^-_n(x) \\rightarrow_{n \\rightarrow \\infty} p_-(x)$. There exists $C > 0$ such that: $|p^-_n(x)-p_-(x)| \\leq C\/n$. Moreover, $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X p_- = 0$ on $\\gamma$. \n\\end{lemma}\n\nIn particular, this shows that $p_-$ is smooth in restriction to $\\gamma$ as $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X$ is elliptic on $\\gamma$.\n\n\\begin{proof}\nBy construction, the differential equation is clearly satisfied if the limit exists. Moreover, we have for some time $T_0$ (independent of $n$, $T^-_{k_n} = T_0 + k_n T_{\\star}$):\n\\[\n\\begin{split}\np^-_n(x) & = P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star}\\\\\n& = P(x_0^-, T_0)P(x_{k_n}^-,k_n T_{\\star}) P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\\\\n& = P(x_0^-, T_0) P_{x_{\\star} \\rightarrow x_0^-} P(x_{\\star},T_{\\star})^{k_n} \\left[P(x_{\\star},T_{\\star})^{-k_n}P_{x_0^- \\rightarrow x_{\\star}} P(x_{k_n}^-,k_n T_{\\star}) P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star}\\right].\n\\end{split}\n\\]\nBy assumption, the term outside the bracket converges as $n \\rightarrow \\infty$ and the term between brackets converges by the spiral Lemma \\ref{lemma:spiral}.\n\\end{proof}\n\nWe now claim the following:\n\n\\begin{lemma}\n\\label{lemma:stable}\nThere exists a uniform constant $C > 0$ such that the following holds. Assume that $x$ and $z$ belong to two homoclinic orbits in $\\mathcal{H}$ and $z \\in W^u_{\\mathrm{loc}}(x)$. Then:\n\\[\n\\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\leq C d(x,z).\n\\]\n\\end{lemma}\n\nBy the previous proofs, the point $x$ is associated to points $x^-_n$ on the homoclinic orbit and we will use the same notations for the point $z$ associated to the points $z^-_n$.\n\n\\begin{proof}\nThere is here a slight subtlety coming from the fact that the parametrizations of the homoclinic orbits $\\gamma$ were chosen in a non-canonical way (via a choice of $A_\\pm$). In particular, it is not true that the flowlines of $z_{k_n}^-$ and $x_{k_n}^-$ shadow each other; in other words, we might not have $T^-_{k_n}(z) = T^-_{k_n}(x)$ but we rather have $T^-_{k_n}(z) = T^-_{k_n}(x) + mT_{\\star}$ for some integer $m \\in \\mathbb{Z}$ depending on both $x$ and $z$.\n\nWe have:\n\\[\n\\begin{split}\n& \\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\\\\n& = \\|P_{x \\rightarrow z} p^-_n(x) - p^-_n(z)\\| + o(1) \\\\\n& = \\|P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} - P(z_{k_n}^-,T^-_{k_n}(z))P_{x_{\\star} \\rightarrow z_{k_n}^-} p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n}^-,T^-_{k_n}(z))^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n},mT_{\\star})^{-1}P_{x_{\\star} \\rightarrow z_{k_n-m}^- } \\\\\n& \\hspace{1.2cm} \\times \\left[P_{z_{k_n-m}^- \\rightarrow x_{\\star}}P(z_{k_n-m}^-,T^-_{k_n}(z)-mT_{\\star})^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-}\\right]p_{\\star} - p_{\\star}\\| + o(1).\n\\end{split}\n\\]\nApplying the first item of Lemma \\ref{lemma:ASgeometry}, we have that:\n\\[\n\\|P_{z_{k_n-m}^- \\rightarrow x_{\\star}}P(z_{k_n-m}^-,T^-_{k_n}(z)-mT_{\\star})^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-} - \\mathbbm{1}_{\\mathrm{End}(\\E_{x_{\\star}})}\\| \\leq Cd(x,z),\n\\]\nwhere the constant $C > 0$ is uniform in $n$. Moreover, observe that\n\\[\n\\lim_{n \\rightarrow \\infty} P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n},mT_{\\star})^{-1}P_{x_{\\star} \\rightarrow z_{k_n-m}^- } = P(x_{\\star},mT_{\\star})^{-1}.\n\\]\n\n\nHence:\n\\[\n \\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\leq C\\left(\\| P(x_{\\star},mT_{\\star})^{-1}p_{\\star} - p_{\\star}\\| + d(x,z) + o(1)\\right).\n \\]\nUsing that $P(x_{\\star},T_{\\star})p_{\\star}=p_{\\star}$, we get that the first term on the right-hand side vanishes. Taking the limit as $n \\rightarrow +\\infty$, we obtain the announced result.\n\\end{proof}\n\nNote that we could have done the same construction ``in the future\" by considering instead:\n\\[\np_+(x) = \\lim_{n \\rightarrow \\infty} P(x,T^+_{k_n})^{-1} P_{x_{\\star} \\rightarrow x_{k_n}^+} p_{\\star} \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x}),\n\\]\nwhere $x_n^+ := \\varphi_{T^+_n}(x)$ is exponentially closed to $x_{\\star}$ as in \\S\\ref{sssection:homoclinic}. A similar statement as Lemma \\ref{lemma:stable} holds with the unstable manifold being replaced by the stable one. We have:\n\n\\begin{lemma}\n\\label{lemma:equality}\nFor all $x \\in \\mathcal{W}$, $p_-(x) = p_+(x)$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from Proposition \\ref{proposition:representation}. Indeed, we have:\n\\[\n\\begin{split}\n\\|p_-(x)-p_+(x)\\| & = \\|P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} - P(x,T^+_{k_n})^{-1} P_{x_{\\star} \\rightarrow x_{k_n}^+} p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x,T^+_{k_n})P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x_{k_n}^-,T_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1),\\\\\n\\end{split}\n\\]\nwhere $T_n := T_n^- + T_n^+$. Observe that:\n\\[\n\\begin{split}\n& P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x_{k_n}^-,T_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} \\\\\n& \\hspace{2cm}= C_{1,x_{k_n}^+ \\rightarrow x_{\\star}} C_1(x_{k_n}^-,T_{k_n}) C_{1,x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\left(C_{2,x_{k_n}^+ \\rightarrow x_{\\star}} C_2(x_{k_n}^-,T_{k_n}) C_{2,x_{\\star} \\rightarrow x_{k_n}^-} \\right)^{-1} \\\\\n& \\hspace{2cm} = \\rho_{1,n}(\\gamma) p_{\\star} \\rho_{2,n}(\\gamma)^{-1} = \\rho_1(\\gamma)p_{\\star} \\rho_2(\\gamma)^{-1} + o(1) = p_{\\star} + o(1),\n\\end{split}\n\\]\nby Proposition \\ref{proposition:representation}. Hence $\\|p_-(x)-p_+(x)\\| = o(1)$, that is $p_-(x)=p_+(x)$.\n\\end{proof}\n\nWe can now prove the following lemma:\n\n\\begin{lemma}\n\\label{lemma:lipschitz}\nThe map $p_-$ is Lipschitz-continuous.\n\\end{lemma}\n\n\\begin{proof}\nConsider $x, y \\in \\mathcal{W}$ which are close enough. Let $z := [x,y] = W^{wu}_{\\mathrm{loc}}(x) \\cap W^{s}_{\\mathrm{loc}}(y)$ and define $\\tau$ such that $\\varphi_\\tau(z) \\in W^u_{\\mathrm{loc}}(x)$. Note that $|\\tau| \\leq Cd(x,y)$ for some uniform constant $C > 0$; also observe that the point $z$ is homoclinic to the periodic orbit $x_{\\star}$. We have:\n\\[\n\\begin{split}\n& \\|p_-(x)-P_{y \\rightarrow x}p_-(y)\\| \\\\\n& \\leq \\|p_-(x)-P_{z \\rightarrow x}p_-(z)\\| + \\|p_-(z)-p_+(z)\\| + \\|P_{z \\rightarrow x}p_+(z)-P_{y \\rightarrow x}p_+(y)\\| + \\|p_+(y)-p_-(y)\\| \\\\\n& \\leq \\|p_-(x)-P_{z \\rightarrow x}p_-(z)\\| + \\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\| \\\\\n& \\leq \\|p_-(x)-P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))\\| + \\|P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))- P_{z \\rightarrow x}p_-(z)\\| + \\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\| \n\\end{split}\n\\]\nwhere the terms disappear between the second and third line by Lemma \\ref{lemma:equality}. By Lemma \\ref{lemma:stable}, the first term is controlled by:\n\\[\n\\|p_-(x)-P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))\\| \\leq Cd(x,\\varphi_{\\tau}(z)) \\leq Cd(x,y).\n\\]\nAs to the second term, using the second item of Lemma \\ref{lemma:ASgeometry}, we have:\n\\[\n\\|P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))- P_{z \\rightarrow x}p_-(z)\\| = \\|P_{x \\rightarrow z} P_{\\varphi_{\\tau}(z) \\rightarrow x} P(z,\\tau)p_-(z)- p_-(z)\\| \\leq Cd(x,y).\n\\]\nEventually, the last term $\\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\|$ is controlled similarly to the first term by applying Lemma \\ref{lemma:stable} (but with the stable manifold instead of unstable).\n\\end{proof}\n\nAs $\\mathcal{W}$ is dense, $p_-$ extends to a Lipschitz-continuous map on $\\mathcal{M}$ which satisfies the equation $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X p_- = 0$ and by \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, this implies that $p_-$ is smooth. This concludes the proof of the Theorem.\n\n\\end{proof}\n\n\n\n\\subsubsection{Proof of the geometric properties}\n\nWe now prove Proposition \\ref{proposition:opaque}.\n\n\\begin{proof}[Proof of Proposition \\ref{proposition:opaque}]\nThe equivalence between (1) and (2) can be found in \\cite[Section 5]{Cekic-Lefeuvre-20}. If $\\mathcal{F} \\subset \\mathcal{E}$ is a non-trivial subbundle that is invariant by parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$, it is clear that $\\rho$ will leave the space $\\mathcal{F}_{x_{\\star}}$ invariant and thus is not irreducible. Conversely, if $\\rho$ is not irreducible, then there exists a non-trivial $\\mathcal{F}_{x_{\\star}} \\subset \\E_{x_{\\star}}$ preserved by $\\rho$. Let $\\pi_{\\star} : \\mathcal{E}_{x_{\\star}} \\rightarrow \\mathcal{F}_{x_{\\star}}$ be the orthogonal projection. For $x$ on a homoclinic orbit, define $\\pi(x) : \\E_x \\rightarrow \\E_x$ similarly to $p_-$ in Lemma \\ref{lemma:p-minus} by parallel transport of the section $\\pi_{\\star}$ with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$. Following the previous proofs (we only use $\\rho \\pi_\\star = \\pi_\\star \\rho$), one shows that $\\pi$ extends to a Lipschitz-continuous section on homoclinic orbits which satisfies $\\pi^2 = \\pi$ and $\\nabla^{\\mathrm{End}}_X \\pi = 0$. By \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, $\\pi$ extends to a smooth section i.e. $\\pi \\in C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))$. Moreover, $\\pi(x_{\\star})=\\pi_{\\star}$, hence $\\pi$ is the projection onto a non-trivial subbundle $\\mathcal{F} \\subset \\mathcal{E}$.\n\\end{proof}\n\n\n\nWe now prove Theorem \\ref{theorem:iso}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:iso}]\nThe linear map $\\Phi : \\mathbf{R}' \\rightarrow \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$ is defined in the following way. Consider $u_{\\star} \\in \\mathbf{R}'$ and define, as in Lemma \\ref{lemma:p-minus}, for $x$ on a homoclinic orbit, $u_-(x)$ as the parallel transport of $u_{\\star}$ from $x_{\\star}$ to $x$ along the orbit (with respect to the endomorphism connection $\\nabla^{\\mathrm{End}(\\E)}$). Similarly, one can define $u_+(x)$ by parallel transport from the future. The fact that $u_{\\star} \\in \\mathbf{R}'$ is then used in the following observation (see Lemma \\ref{lemma:equality}):\n\\[\n\\|u_-(x)-u_+(x)\\| = \\|\\rho(\\gamma)u_{\\star} \\rho(\\gamma)^{-1} - u_\\star\\| = 0.\n\\]\n(Note that $\\rho(\\gamma)u_{\\star} \\rho(\\gamma)^{-1}$ corresponds formally to the parallel transport of $u_{\\star}$ with respect to $\\nabla^{\\mathrm{End}(\\E)}$ from $x_{\\star}$ to $x_{\\star}$ along the homoclinic orbit $\\gamma$.) Hence, following Lemma \\ref{lemma:lipschitz}, we get that $u_-$ is Lipschitz-continuous and satisfies $\\nabla^{\\mathrm{End}(\\E)}_X u_- = 0$. By \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, it is smooth and we set $u_- := \\Phi(u_{\\star}) \\in \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$.\n\nAlso observe that this construction is done by using parallel transport with respect to the unitary connection $\\nabla^{\\mathrm{End}(\\E)}$. As a consequence, if $u_\\star, u_\\star' \\in \\mathbf{R}$ are orthogonal (i.e. $\\Tr(u_\\star^* u_\\star') = 0$), then $\\Phi(u_{\\star})$ and $\\Phi(u'_\\star)$ are also pointwise orthogonal. This proves that $\\Phi$ is injective.\n\nIt now remains to show the surjectivity of $\\Phi$. Let $u \\in\\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$. Following \\cite[Section 5]{Cekic-Lefeuvre-20}, we can write $u = u_R + i u_I$, where $u_R^*=u_R,u_I^*=u_I$ and $\\nabla^{\\mathrm{End}(\\E)}_X u_R = \\nabla^{\\mathrm{End}(\\E)}_X u_I = 0$. By \\cite[Lemma 5.6]{Cekic-Lefeuvre-20}, we can then further decompose $u_R = \\sum_{i=1}^p \\lambda_i \\pi_{\\mathcal{F}_i}$ (and same for $u_I$), where $\\lambda_i \\in \\mathbb{R}$, $p \\in \\mathbb{N}$ and $\\mathcal{F}_i \\subset \\E$ is a maximally invariant subbundle of $\\E$ (i.e. it does not contain any non-trivial subbundle that is invariant under parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to $\\nabla^{\\E}$), and $\\pi_{\\mathcal{F}_i}$ is the orthogonal projection onto $\\mathcal{F}_i$. Setting $(\\pi_{\\mathcal{F}_i})_\\star := \\pi_{\\mathcal{F}_i}(x_\\star)$, invariance of $\\mathcal{F}_i$ by parallel transport implies that $\\rho(\\gamma)(\\pi_{\\mathcal{F}_i})_\\star = (\\pi_{\\mathcal{F}_i})_\\star\\rho(\\gamma)$, for all $\\gamma \\in \\mathbf{G}$, that is $(\\pi_{\\mathcal{F}_i})_\\star \\in \\mathbf{R}'$. Moreover, we have $\\Phi((\\pi_{\\mathcal{F}_i})_\\star) = \\pi_{\\mathcal{F}_i}$. This proves that both $u_R$ and $u_I$ are in $\\mathrm{ran}(\\Phi)$. This concludes the proof.\n\\end{proof}\n\nIt remains to prove the results concerning invariant sections:\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:invariant-section}]\nUniqueness is immediate since $\\nabla^{\\E}_X u = 0$ implies that\n\\[\nX |u|^2 = \\langle \\nabla^{\\E}_X u, u \\rangle = \\langle u , \\nabla^{\\E}_X u \\rangle = 0,\n\\]\nthat is $|u|$ is constant. Now, given $u_\\star$ which is $\\mathbf{G}$-invariant, we can define $u_-(x)$ for $x$ on a homoclinic orbit $\\gamma$ by parallel transport of $u_\\star$ from $x_\\star$ to $x$ along $\\gamma$ with respect to $\\nabla^{\\E}$, similarly to Lemma \\ref{lemma:p-minus} and to the proof of Theorem \\ref{theorem:iso}. We can also define $u_+(x)$ in the same fashion (by parallel transport in the other direction). Then one gets that $\\|u_-(x)-u_+(x)\\| = \\|u_\\star - \\rho(\\gamma)u_\\star\\|=0$ and the same arguments as before show that $u_-$ extends to a smooth function in the kernel of $\\nabla^{\\E}_X$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:rank2}]\nThis is based on the following:\n\n\\begin{lemma}\nAssume that for all periodic orbits $\\gamma \\in \\mathcal{G}$, there exists $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$. Then for all $g \\in \\mathbf{G}$, there exists $u_g \\in \\E_{x_\\star}$ such that $\\rho(g)u_g = u_g$. \n\\end{lemma}\n\n\\begin{proof}\nRecall that by the construction of Proposition \\ref{proposition:representation}, each element $\\rho(g) \\in \\mathrm{U}(\\E_{x_\\star})$ can be approximated by the holonomy $C_{y_n \\to x_\\star}C(y_n,T'_n)C_{x_\\star \\to y_n}$ along a sequence of periodic orbits of points $y_n$ converging to $x_\\star$. Now, each $C(y_n,T'_n)$ has $1$ as eigenvalue by assumption and taking the limit as $n \\rightarrow \\infty$, we deduce that $1$ is an eigenvalue of $\\rho(g)$.\n\\end{proof}\n\nAs a consequence, we can write for all $g \\in \\mathbf{G}$, in a fixed orthonormal basis of $\\E_{x_\\star}$\t:\n\\[\n\\rho(g) = \\alpha_g \\begin{pmatrix} 1 & 0 \\\\ 0 & s(g) \\end{pmatrix} \\alpha_g^{-1},\n\\]\nfor some $\\alpha_g \\in \\mathrm{U}(\\E_{x_\\star})$ and $s(g)$ is an $(r-1) \\times (r-1)$ matrix. For $\\rk(\\E)=1$, the Lemma is then a straightforward consequence of Lemma \\ref{lemma:invariant-section} since the conjugacy $\\alpha_g$ does not appear. For $\\rk(\\E)=2$, one has the remarkable property that $s(g)$ is \\emph{still} a representation of $ \\mathbf{G}$ since $\\det \\rho(g) = s(g) \\in \\mathrm{U}(1)$. As a consequence, $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_\\star})$ has the same character as $\\rho' : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_\\star})$ defined by:\n\\[\n\\rho'(g) := \\begin{pmatrix} 1 & 0 \\\\ 0 & s(g) \\end{pmatrix}.\n\\]\nBy \\cite[Corollary 3.8]{Lang-02}, we then conclude that these representations are isomorphic, that is there exists $p_\\star \\in \\mathrm{U}(\\E_{x_\\star})$ such that $\\rho(g) = p_\\star \\rho'(g) p_\\star^{-1}$. If $u_\\star' \\in \\E_{x_\\star}$ denotes the vector fixed by $\\rho'(\\mathbf{G})$, then $u_\\star := p_\\star u'_\\star$ is fixed by $\\rho(\\mathbf{G})$. We then conclude by Lemma \\ref{lemma:invariant-section}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Pollicott-Ruelle resonances and local geometry on the moduli space of connections}\n\n\\label{section:geometry}\n\nThis section is devoted to the study of the moduli space of connections, with the point of view of Pollicott-Ruelle resonances. We will first deal with the opaque case and then outline the main distinctions with the non-opaque case. We consider a Hermitian vector bundle $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ endowed with a unitary connection over the Anosov Riemannian manifold $(M,g)$. Recall the notation of \\S \\ref{section:twisted}: we write $\\mathbf{X} = (\\pi^*\\nabla^{\\E})_X$, $\\RR_\\pm(z) = (\\pm \\mathbf{X} + z)^{-1}$ for its resolvent and $\\RR^\\pm_0$, $\\Pi_0^\\pm$ for the holomorphic parts and the spectral projector at zero, respectively.\n\n\\subsection{The Coulomb gauge}\n\nWe study the geometry of the space of connections (and of the moduli space of gauge-equivalent connections) in a neighborhood of a given unitary connection $\\nabla^{\\mathcal{E}}$ of regularity $C^s_*$ (for ${\\color{red}1 < }s < \\infty$\\footnote{It is very likely that the case $s=\\infty$ still works. This would require to use the Nash-Moser Theorem.}) such that $\\ker(\\nabla^{\\mathrm{End}(\\E)}) = \\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$. For the standard differential topology of Banach manifolds, we refer the reader to \\cite{Lang-99}. We denote by\n\\[\n\\mathcal{O}_s(\\nabla^{\\mathcal{E}}) := \\left\\{ \\nabla^{\\mathcal{E}} + p^{-1} \\nabla^{\\mathrm{End}(\\E)}p ~|~ p \\in C_*^{s+1}(M,\\mathrm{U}(\\E)),\\,\\, \\|p - \\mathbbm{1}\\|_{C^{s+1}_*} < \n\\delta\\right\\}\n\\]\nthe orbit of gauge-equivalent connections of $C^{s}_*$ regularity, where $\\delta > 0$ is small enough so that $\\mathcal{O}_s(\\nabla^{\\E})$ is a smooth Banach submanifold. We also define the slice at $\\nabla^{\\E}$ by\n\\[\n\\mathcal{S}_s(\\nabla^{\\E}) := \\nabla^{\\E} + \\ker (\\nabla^{\\mathrm{End}(\\E)})^* \\cap \\big\\{A \\in C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E)): \\|A\\|_{C_*^s} < \\delta\\big\\}.\n\\]\nNote that $\\mathbb{S}^1$ acts by multiplication freely and properly on $C_*^s(M, \\mathrm{U}(\\E))$ and hence we may form the quotient Banach manifold, denoted by $C_*^s(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$, which in particular satisfies\n\\begin{equation}\n\tT_{\\mathbbm{1}_{\\E}} \\Big(C_*^s(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1\\Big) = C_*^s(M, \\mathrm{End}_{\\mathrm{sk}}(\\E))\/\\big(\\mathbb{R}\\cdot (i\\mathbbm{1}_{\\E})\\big), \n\\end{equation}\nwhere we used the identification of tangent spaces given by the exponential map.\n\tNext, observe that the map $O: p \\mapsto p^*\\nabla^{\\E}$ is injective modulo the multiplication action of $\\mathbb{S}^1$ on $C^{s + 1}_*(M, \\mathrm{U}(\\E))$ and that it is an immersion at $p = \\mathbbm{1}$ with $\\dd_{\\mathbbm{1}} O (\\Gamma) = \\nabla^{\\mathrm{End}(\\E)} \\Gamma$. Therefore by equation \\eqref{eq:decomposition-tt}, $\\mathcal{O}_s(\\nabla^{\\E})$ and $\\mathcal{S}_s$ are smooth transverse Banach manifolds with:\n\t\\[T_{\\nabla^{\\E}} \\mathcal{O}(\\nabla^{\\mathrm{End}(\\E)}) = \\ran(\\nabla^{\\mathrm{End}(\\E)}), \\quad T_{\\nabla^{\\mathrm{End}(\\E)}} \\mathcal{S}_s = \\ker(\\nabla^{\\mathrm{End}(\\E)})^*.\\]\n\t\n\n\nWe will say that a connection $\\nabla_2^{\\E}$ is in the \\emph{Coulomb gauge} with respect to $\\nabla_1^{\\E}$ if $(\\nabla_1^{\\mathrm{End}(\\E)})^*(\\nabla_2^{\\E} - \\nabla_1^{\\E}) =0$. The following lemma shows that, near $\\nabla^{\\E}$, we may always put a pair of connections in the Coulomb gauge with respect to each other. It is a slight generalisation of the usual claim (see \\cite[Proposition 2.3.4]{Donaldson-Kronheimer-90}).\n\n\\begin{lemma}[Coulomb gauge]\n\\label{lemma:coulomb}\nLet $s > 1$. There exists $\\varepsilon = \\varepsilon(s, \\nabla^{\\E}) > 0$ and a neighbourhood $\\mathbbm{1}_{\\E} \\in \\mathcal{U} \\subset C_*^{s + 1}(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$ such that for any $A_i \\in C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A_i\\|_{C^s_*} < \\varepsilon$, after setting $\\nabla_i^{\\E} = \\nabla^{\\E} + A_i$ for $i = 1, 2$, there exists a unique $p_{A_1, A_2} \\in \\mathcal{U}$ such that $p_{A_1, A_2}^*\\nabla_2^{\\E} - \\nabla_1^{\\E} \\in \\ker (\\nabla_1^{\\mathrm{End}(\\E)})^*$. Furthermore, if $A_i$ are smooth, then $p_{A_1, A_2}$ is smooth.\nMoreover, the map \n\\[\n\\big(C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))\\big)^2 \\ni (A_1, A_2) \\mapsto \\phi(A_1, A_2) := p_{A_1, A_2}^*\\nabla_2^{\\E} \\in \\mathcal{S}_s(\\nabla_1^{\\E}),\n\\]\nis smooth. Setting $\\phi(A) := \\phi(0, A)$, we have:\n \\[\n \\dd \\phi|_{A = 0} = \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\E)})^*}.\n \\]\n\\end{lemma}\n\\begin{proof}\nNote that the exponential map $\\exp: C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i \\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}} \\to C^{s + 1}_*(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$ is well-defined and a local diffeomorphism at zero, so we reduce the claim to finding a neighbourhood $0 \\in \\mathcal{V} \\subset C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}$ and setting $p = p_{A_1, A_2} = \\exp(\\chi_{A_1, A_2})$ for $\\chi = \\chi_{A_1, A_2} \\in \\mathcal{V}$, that is $\\mathcal{U} = \\exp(\\mathcal{V})$. Define the functional\n\t\\begin{align*}\n\t\t&F: \\big(C^s_*(M, T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))\\big)^2 \\times C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}\\to C^{s - 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E))\\\\\n\t\t&\\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}, F(A_1, A_2, \\chi) :=\n\t\t(\\nabla_1^{\\mathrm{End}(\\E)})^*\\Big(\\exp(-\\chi) \\nabla^{\\mathrm{End}(\\E)} \\exp(\\chi) + \\exp(-\\chi) A_2 \\exp(\\chi) - A_1\\Big).\n\t\\end{align*}\n\tWe have $F$ well-defined, i.e. with values in skew-Hermitian endomorphisms, since $\\nabla^{\\E}$ is unitary, and integrating by parts $\\langle{F(A_1, A_2, \\chi), \\mathbbm{1}_{\\E}}\\rangle_{L^2} = 0$; note that $F$ is smooth in its entries. Next, we compute the partial derivative with respect to the $\\chi$ variable at $A_1 = A_2 = 0$ and $\\chi = 0$:\n\t\\[\\dd_{\\chi}F(0, 0, 0) (\\Gamma) = \\partial_t|_{t = 0} F(0, 0, t\\Gamma) = (\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)} \\Gamma.\\]\n\tThis derivative is an isomorphism on \n\t\\[(\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)}: C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i \\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}} \\to C^{s - 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}},\\]\n\tby the Fredholm property of $(\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)}$ and since $\\ker (\\nabla^{\\mathrm{End}(\\E)}) = \\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$ by assumption. The first claim then follows by an application of the implicit function theorem for Banach spaces. \n\t\n\tThe fact that $p$ is smooth if $(A_1, A_2)$ is, is a consequence of elliptic regularity and the fact that $C_*^s$ is an algebra, along with the Coulomb property:\n\t\\[(\\nabla_1^{\\mathrm{End}(\\E)})^* \\nabla_1^{\\mathrm{End}(\\E)} p = (\\nabla_1^{\\mathrm{End}(\\E)}p) p^{-1} \\bullet \\nabla_1^{\\mathrm{End}(\\E)} p + p(\\nabla_1^{\\mathrm{End}(\\E)})^* \\big(p^{-1} (A_1 - A_2) p\\big) \\in C_*^{s}\\]\n\timplies $p \\in C_*^{s+2}$. Bootstrapping we obtain $p_{A_1, A_2} \\in C^\\infty$. Here $\\bullet$ denotes the operation of taking the inner product on the differential form side and multiplication on the endomorphism side.\n\t\n\tEventually, we compute the derivative of $\\phi(A)$. Write $p_A := p_{0, A}$ and $\\chi_A := \\chi_{0, A}$, where $\\chi_A$ is orthogonal to $i \\mathbbm{1}_{\\E}$ with respect to the $L^2$-scalar product, so that by definition\n\t\\begin{equation}\n\t\\label{equation:phi}\n\t\\phi(A) = \\nabla^{\\E} + p_A^{-1} \\nabla^{\\mathrm{End}(\\E)} p_A + p_A^{-1} A p_A.\n\t\\end{equation}\n\tBy differentiating the relation $F(A, \\chi_A) := F(0, A,\\chi_A)=0$ at $A=0$, we obtain for every $\\Gamma \\in C^s_*(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$:\n\t\\begin{align*}\n\t0 &= \\dd_A F|_{A=0,\\chi=0}(\\Gamma) + \\dd_\\chi F|_{A=0,\\chi=0}(\\dd \\chi_A|_{A=0}(\\Gamma))\\\\\n\t&= (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma + (\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)} \\dd \\chi_A|_{A=0}(\\Gamma),\n\t\\end{align*}\n\tthat is $\\dd \\chi_A|_{A=0}(\\Gamma) = - [(\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)}]^{-1} (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma$. \t\n\t Observe that $\\dd p_A|_{A=0} = \\dd \\chi_A|_{A=0}$ via the exponential map and by \\eqref{equation:phi}, we obtain:\n\t\\[\n\t\\dd \\phi|_{A=0}(\\Gamma) = \\nabla^{\\mathrm{End}(\\E)} \\dd \\chi_A|_{A=0}(\\Gamma) + \\Gamma = \\Gamma - \\nabla^{\\mathrm{End}(\\E)}[(\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)}]^{-1} (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma.\n\t\\]\n\tWe then conclude by \\eqref{equation:projection}\n\\end{proof}\n\nIn particular, the proof also gives that the map\n\\[\nC^{s}_*(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\ni A \\mapsto \\phi(A) \\in \\mathcal{S}_s := \\mathcal{S}_{s}(\\nabla^{\\E}),\n\\]\nis constant along orbits of gauge-equivalent connections (by construction)\n\n\n\n\n\\subsection{Resonances at $z=0$: finer remarks}\n\\label{ssection:finer}\n\n\n\\begin{lemma}\\label{lemma:symmetricspectrum}\nThe Pollicott-Ruelle resonance spectrum of $\\mathbf{X}$ is symmetric with respect to the real axis.\n\\end{lemma}\n\n\\begin{proof}\nIf $z_0$ is a resonance associated to $-\\mathbf{X}$ i.e. a pole of $z \\mapsto \\RR_+(z)$ then $\\overline{z}_0$ is a resonance associated to $+\\mathbf{X}$ i.e. a pole of $z \\mapsto \\RR_-(z)$ by \\eqref{equation:adjoint}. Let $u \\in \\mathcal{H}_+^s$ be a non-zero resonant state such that $-\\mathbf{X} u = z_0 u$, for some $s > 0$. Let $R : (x,v) \\mapsto (x,-v)$ be the antipodal map. By inspecting the construction of the anisotropic Sobolev space in \\cite{Faure-Sjostrand-11}, we see that we may assume $R^* : \\mathcal{H}_\\pm^s \\rightarrow \\mathcal{H}_\\mp^s$ is an isomorphism.\\footnote{Simply replace the degree function $m$ in the construction by $\\frac{m - R^*m}{2}$, which then implies that $R^*m = -m$.}\nThen $-R^* \\mathbf{X} u = \\mathbf{X} R^* u = z_0 R^* u$ and $R^*u \\in \\mathcal{H}_-^s$. Thus $R^*u$ is a resonant state associated to the resonance $z_0$. So both $z_0$ and $\\overline{z}_0$ are resonances for $+\\mathbf{X}$ and the same holds for $-\\mathbf{X}$.\n\\end{proof}\n\nConsider a contour $\\gamma \\subset \\mathbb{C}$ such that $-\\mathbf{X}$ has no resonances other than zero inside or on $\\gamma$. By continuity of resonances (see \\cite{Bonthonneau-19}), there is an $\\varepsilon > 0$ such that for all skew-Hermitian $1$-forms $A$ with $\\|A\\|_{C_*^s} < \\varepsilon$ the operator $-\\mathbf{X}_A := (\\pi^*(\\nabla^{\\E} + A))_X$ has no resonances on $\\gamma$. Here we need to take $s$ large enough (depending on the dimension), so that the framework of microlocal analysis applies.\n\nIn the specific case where $\\dim \\ker(\\mathbf{X}|_{\\mathcal{H}_+}) = 1$, we denote by $\\lambda_A$ the unique resonance of $-\\mathbf{X}_A$ within $\\gamma$. Note that the map $A \\mapsto \\lambda_A$ is $C^3$-regular for $\\varepsilon > 0$ small enough.\n\n\n\\begin{lemma}\n\\label{lemma:diff-2}\nAssume that $\\dim \\ker(\\mathbf{X}|_{\\mathcal{H}_+}) = 1$. Then $\\lambda_A \\in \\mathbb{R}$ and for $\\Gamma \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$:\n\\[\n\\dd \\lambda_A|_{A=0} = 0, ~~ \\dd^2 \\lambda_A|_{A=0}(\\Gamma,\\Gamma) = - \\langle \\Pi \\pi_1^*\\Gamma u_0, \\pi_1^*\\Gamma u_0 \\rangle_{L^2},\n\\]\nwhere $u_0$ is a resonant state associated to $A=0$ and $\\|u_0\\|_{L^2}=1$.\n\\end{lemma}\n\n\\begin{proof}\nBy the symmetry property of Lemma \\ref{lemma:symmetricspectrum} and continuity of resonances we know $\\lambda_A \\in \\mathbb{R}$. Also, observe that $u_0$ is either pure odd or pure even with respect to $v$ (i.e. $R^*u_0 = u_0$ or $R^*u_0 = -u_0$) because $R^*$ keeps $\\ker (\\mathbf{X})$ fixed and $\\ker (\\mathbf{X})$ is assumed to be one dimensional.\n\nFor the second claim, it is sufficient to start with the equality $-\\mathbf{X}_{\\tau \\Gamma} u_{\\tau \\Gamma} = \\lambda_{\\tau \\Gamma} u_{\\tau \\Gamma}$, where $\\Gamma \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$, $\\tau \\in (-\\delta, \\delta)$ is small enough so that $\\tau \\mapsto \\lambda_{\\tau \\Gamma}$ and $\\tau \\mapsto u_{\\tau \\Gamma} \\in \\mathcal{H}_+$ are $C^3$, and to compute the derivatives at $\\tau=0$. Observe that $\\dot{\\mathbf{X}}_0 = \\pi_1^*\\Gamma,\\, \\ddot{\\mathbf{X}}_0 = 0$. We obtain: $-\\dot{\\mathbf{X}}_0 u_0 -\\mathbf{X}_0 \\dot{u}_0 = \\dot{\\lambda}_0 u_0$ and taking the $L^2$ scalar product with $u_0$, we find $\\dot{\\lambda}_0 = 0$, using that $u_0$ is either pure odd or pure even. Thus $\\dot{u}_0- \\Pi_0^+ \\dot{u}_0 = -\\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0$. Then, taking the second derivative at $\\tau = 0$, we get: $-2 \\pi_1^*\\Gamma \\dot{u}_0 - \\mathbf{X}_0 \\ddot{u}_0 = \\ddot{\\lambda}_0 u_0$ and taking once again the scalar product with $u_0$, we find $\\ddot{\\lambda}_0 = - 2 \\langle \\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0, \\pi_1^*\\Gamma u_0 \\rangle_{L^2}$. It is then sufficient to observe that by symmetry (using $(\\mathbf{R}_0^+)^* = \\mathbf{R}_0^-$ and $\\ddot{\\lambda}_0 \\in \\mathbb{R}$)\n\\[\n\\langle \\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0, \\pi_1^*\\Gamma u_0 \\rangle_{L^2} = \\langle \\mathbf{R}_0^- \\pi_1^*\\Gamma u_0, \\pi_1^*\\Gamma u_0 \\rangle_{L^2}.\n\\]\nThis proves the result.\n\\end{proof}\n\n\n\\subsection{P-R resonance at $0$ of the mixed connection: opaque case}\n\n\\label{ssection:pollicott-ruelle-dea}\n\nWe now further assume that $\\mathbf{X} := (\\pi^* \\nabla^{\\mathrm{End}(\\E)})_X$ has the resonant space at $0$ spanned by $\\mathbbm{1}_{\\E}$. This condition is known as the \\emph{opacity} of the connection $\\pi^*\\nabla^{\\E}$. When $(M,g)$ is Anosov, this is known to be a generic condition, see \\cite[Theorem 1.6]{Cekic-Lefeuvre-20}.\n\n\nAs in \\S\\ref{sssection:connection-induced}, we assume that $s \\gg 1$ (so that standard microlocal analysis applies) and we introduce the mixed connection induced by $\\nabla_1^{\\E} = \\nabla^{\\E} + A_1$ and $\\nabla_2^{\\E} = \\nabla^{\\E} + A_2$, namely\n\\[\n\\nabla^{\\Hom(\\nabla_1^{\\E}, \\nabla_2^{\\E})} u = \\nabla^{\\mathrm{End}(\\E)} u + A_2u - u A_1,\n\\]\nand we set $\\mathbf{X}_{A_1, A_2} := (\\pi^* \\nabla^{\\Hom(\\nabla_1^{\\E}, \\nabla_2^{\\E})})_X$ and $z \\mapsto \\RR_{\\pm}(z, A_i)$ for the resolvents. The operator $\\mathbf{X} := \\mathbf{X}_{0, 0}$ has the resonant space at $z=0$ spanned by $\\mathbbm{1}_{\\E}$ \nand we denote by $\\lambda_{A_1, A_2}$ the unique resonance in $\\left\\{\\Re(z) \\leq 0\\right\\}$ close to $0$, namely the unique pole of $\\RR_{\\pm}(z,A_i)$ close to $0$. For $\\|A_1\\|_{C_*^s}, \\|A_2\\|_{C_*^s}$ small enough we know that the map $(A_1, A_2) \\mapsto \\lambda_{A_1, A_2}$ is $C^3$ (see \\S \\ref{ssection:finer}) and by Lemma \\ref{lemma:diff-2} that $\\lambda_{A_1, A_2} \\in \\mathbb{R}$. In fact $\\lambda_{A_1, A_2}$ descends to the moduli space: if $p_i^*(\\nabla^{\\E} + A_i') = \\nabla^{\\E} + A_i$ for $\\|A_i'\\|_{C_*^s}$ small enough, then using \\eqref{equation:lien} we get\n\\begin{equation}\\label{eq:mixedunitaryequiv}\n\t\\mathbf{X}_{A_1', A_2'}u = (p_2)^{-1} \\cdot \\mathbf{X}_{A_1, A_2}(p_2 u (p_1)^{-1}) \\cdot p_1, \\quad u \\in \\mathcal{H}_+.\n\\end{equation}\nHere we used that $\\mathcal{H}_+$ is stable under multiplication by $C_*^s$ for $s$ large enough; hence $\\mathbf{X}_{A_1, A_2}$ and $\\mathbf{X}_{A_1', A_2'}$ have equal P-R spectra and so $\\lambda_{A_1, A_2} = \\lambda_{A_1', A_2'}$.\n\n\n\nNext, we need a uniform estimate for the $\\Pi_1^{\\mathrm{End}(\\E)}$ operator in a neighbourhood of $\\nabla^{\\E}$:\n\n\\begin{lemma}\\label{lemma:Pi_1uniform}\n\tAssume $\\Pi_1^{\\mathrm{End}(\\E)}$ is $s$-injective. There are constants $\\varepsilon, C > 0$ depending only on $\\nabla^{\\E}$ such that for all skew-Hermitian $1$-forms with $\\|A\\|_{C_*^s} < \\varepsilon$:\n\t\\[\n\\forall f \\in H^{-1\/2}(M, T^*M \\otimes \\mathrm{End}(\\mathcal{E})), ~~~ \\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} f, f \\rangle_{L^2} \\geq C \\|\\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A})^*}f\\|^2_{H^{-1\/2}}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\n\tObserve firstly that the left hand side of the inequality vanishes on potential tensors by \\eqref{eq:Pi_mproperties} and hence it suffices to consider $f \\in \\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*$. Then we obtain:\n\t\\begin{align*}\n\t\t\t\\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} &f, f \\rangle_{L^2} = \\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})} f, f \\rangle_{L^2} + \\langle (\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}) f, f \\rangle_{L^2}\\\\\n\t\t\t&\\geq C_0 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*}f\\|_{H^{-1\/2}}^2 - \\|\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}\\|_{H^{-1\/2} \\to H^{1\/2}} \\|f\\|_{H^{-1\/2}}^2\\\\\n\t\t\t&\\geq \\frac{1}{4} C_0 \\|f\\|_{H^{-1\/2}}^2 - \\frac{1}{2} C_0 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E})})^*} - \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A})^*}\\|_{H^{-1\/2} \\to H^{-1\/2}} \\|f\\|_{H^{-1\/2}}^2\\\\\n\t\t\t&\\geq \\frac{1}{8}C_0 \\|f\\|_{H^{-1\/2}}^2.\n\t\\end{align*}\n\tIn the second line, we used Lemma \\ref{lemma:generalized-xray} (item 3) with a constant $C_0$. In the next line we used that the map $A \\mapsto \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} \\in \\Psi^{-1}$ is continuous; the proof of this fact is analogous to the proof of \\cite[Proposition 4.1]{Guillarmou-Knieper-Lefeuvre-19} and we omit it. Thus for $\\varepsilon > 0$ small enough $\\|\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}\\|_{H^{-1\/2} \\to H^{1\/2}} \\leq C_0\/4$. Similarly in the last line we used that $A \\mapsto \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E) + A})^*} \\in \\Psi^0$ is continuous which follows from standard microlocal analysis from \\eqref{equation:projection}, so again we choose $\\varepsilon > 0$ small enough so that $\\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E})})^*} - \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*}\\|_{H^{-1\/2} \\to H^{-1\/2}} \\leq C_0\/4$. The claim follows by setting $C = C_0\/8$.\n\\end{proof}\n\nRecall that $\\lambda_{A_1, A_2} \\leq 0$ in the following:\n\n\\begin{lemma}\n\\label{lemma:borne-lambda-a-1}\nAssume that the generalized X-ray transform $\\Pi^{\\mathrm{End}(\\E)}_1$ defined with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$ is $s$-injective. For $s \\gg 1$ large enough, there exist constants $\\varepsilon, C > 0$ such that for all $A_i \\in C^s(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A_i\\|_{C^s_*} < \\varepsilon$ for $i = 1, 2$, we have:\n\\[\n0 \\leq \\|\\phi(A_1, A_2) - \\nabla_1^{\\E}\\|^2_{H^{-1\/2}(M,T^*M\\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq C |\\lambda_{A_1, A_2}|.\n\\]\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWe introduce two functionals in the vicinity of $\\nabla^{\\E}$, for small enough $\\varepsilon > 0$: \n\\begin{align*}\n\tF_1, F_2 &: \\big(C^{s_0}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{\\|A\\|_{C_*^{s_0}} < \\varepsilon\\}\\big)^2 \\to \\mathbb{R},\\\\\n\t F_1(A_1, A_2) &:=\\lambda_{A_1, A_2}, \\quad F_2(A_1, A_2) := -\\|\\phi(A_1, A_2)-\\nabla_1^{\\E}\\|^2_{H^{-1\/2}}.\n\\end{align*}\nThey are well-defined and restrict as $C^{3}$-regular maps on $\\mathcal{S}_{s_0}$ for some $s_0 \\gg 1$ large enough by Lemma \\ref{lemma:coulomb} and the discussion above. Moreover, using Lemma \\ref{lemma:diff-2}, we have for all $A$:\n\\[F_1(A, A) = F_2(A, A) = 0 \\quad \\mathrm{and}\\quad \\dd F_1|_{(A, A)} = \\dd F_2|_{(A, A)} = 0.\\] \nWe will compare the second partial derivatives in the variable $A_2$ at a point $(A, A)$. Given $\\Gamma \\in T_{\\nabla^{\\E}} \\mathcal{S}_{s_0} \\simeq \\ker (\\nabla^{\\mathrm{End}(\\E)})^*$, we have by Lemma \\ref{lemma:coulomb}:\n\\begin{equation}\\label{eq:F_2}\n\\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma) = -2\\|\\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\Gamma\\|^2_{H^{-1\/2}}\n\\end{equation}\nBy Lemma \\ref{lemma:diff-2}, we have\n\\[\n\\dd^2_{A_2} F_1|_{(A, A)}(\\Gamma,\\Gamma) = -c^2 \\langle \\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)} \\pi_1^* \\Gamma \\mathbbm{1}_{\\E}, \\pi_1^* \\Gamma \\mathbbm{1}_{\\E} \\rangle_{L^2} = - c^2 \\langle \\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)}_1 \\Gamma, \\Gamma \\rangle_{L^2},\n\\]\nfor some constant $c > 0$, where $\\Pi^{\\mathrm{End}(\\nabla^{\\E} +A)}$ denotes the $\\Pi$ operator with respect to the endomorphism connection induced by $\\nabla^{\\E} + A$. We used here that the orthogonal projection to the resonant space $\\mathbb{C} \\mathbbm{1}_{\\E}$ of $\\mathbf{X}_{A, A}$ at zero vanishes, because $\\langle{\\pi_1^*\\Gamma, \\mathbbm{1}_{\\E}}\\rangle_{L^2} = 0$ as $\\pi_1^*\\Gamma$ is odd. For $\\varepsilon > 0$ small enough, by Lemma \\ref{lemma:Pi_1uniform} we know $\\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)}_1$ is $s$-injective and there is a constant $C' = C'(\\nabla^{\\E}) > 0$ such that:\n\\begin{equation}\\label{eq:F_1}\n\\dd^2_{A_2} F_1|_{(A, A)}(\\Gamma,\\Gamma) \\leq - C' \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*}\\Gamma\\|^2_{H^{-1\/2}} = C'\/2 \\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma).\n\\end{equation}\nAs a consequence, writing $G(A_2) := F_1(A, A_2) - C'\/4 \\times F_2(A, A_2)$, we have $G(A)=0, \\dd G|_{A_2 = A} = 0$ and by \\eqref{eq:F_1}, \\eqref{eq:F_2}\n\\[\\dd^2 G|_{A_2 = A}(\\Gamma,\\Gamma) \\leq C'\/4 \\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma) = -C'\/2 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\Gamma\\|^2_{H^{-1\/2}}.\\]\nIf we now Taylor expand the $C^3$-map $\\mathcal{S}_{s_0} \\ni A_2 \\mapsto G(A_2)$ at $A_2 = A$, we obtain:\n\\begin{align*}\nG(A + \\Gamma) &= \\dfrac{1}{2} \\dd^2 G|_{A_2=A}(\\Gamma, \\Gamma) + \\mathcal{O}(\\|\\Gamma\\|_{C_*^{s_0}}^3)\\\\\n &\\leq -C'\/4 \\|\\Gamma\\|_{H^{-1\/2}}^2 + C'\/4 \\|(\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*} - \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*})\\Gamma\\|_{H^{-1\/2}}^2 + C''\\|\\Gamma\\|^3_{C_*^{s_0}}\\\\\n &\\leq -C'\/8\\|\\Gamma\\|_{H^{-1\/2}}^2 + C''\\|\\Gamma\\|_{C_*^{s_0}}^3.\n\\end{align*}\nIn the second line we introduced a uniform constant $C'' = C''(\\nabla^{\\E}) > 0$ using the $C^3$-regular property and $\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*} \\Gamma = \\Gamma$. For the last line, we observed that $A \\mapsto \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\in \\Psi^0$ is a continuous map by equation \\eqref{equation:projection} and hence by standard microlocal analysis the $H^{-1\/2} \\to H^{-1\/2}$ estimate is arbitrarily small for $\\varepsilon$ small enough. This estimate holds for all $\\|A\\|_{C_*^{s_0}}, \\|\\Gamma\\|_{C_*^{s_0}} < \\varepsilon\/2$.\n\nChoosing $s \\gg s_0$ and assuming that $A \\in C_*^s(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C^s_*} < \\varepsilon$, there is a $C'''(\\nabla^{\\E}) > 0$, such that for $\\varepsilon > 0$ with $C''' \\varepsilon \\leq C'\/16$, we then obtain by interpolation:\n\\[\nG(A + \\Gamma) \\leq -C'\/8 \\|\\Gamma\\|^2_{H^{-1\/2}} + \\underbrace{C''' \\|\\Gamma\\|_{C^s_*}}_{\\leq C'\/16} \\|\\Gamma\\|_{H^{-1\/2}}^2 \\leq - C'\/16 \\|\\Gamma\\|_{H^{-1\/2}}^2 \\leq 0.\n\\]\nAfter taking $\\varepsilon > 0$ small enough, the statement holds with $C=C'\/2$.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{remark}\n\\rm\nIt was proved in \\cite{Guillarmou-Knieper-Lefeuvre-19} that there exists a metric $G$ on the moduli space of isometry classes (of metrics with negative sectional curvature) which generalizes the usual Weil-Petersson metric on Teichm\\\"uller space in the sense that, in the case of a surface, the restriction of $G$ to Teichm\\\"uller space is equal to the Weil-Petersson metric. We point out that the operator $\\Pi_1$ also allows to define a metric $G$ at a generic point $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$, similarly to \\cite{Guillarmou-Knieper-Lefeuvre-19}. Indeed, if $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$, taking a representative $\\nabla^{\\E} \\in \\mathfrak{a}_0$, one has $T_{\\mathfrak{a}_0}\\mathbb{A}_{\\E} \\simeq \\ker (\\nabla^{\\mathrm{End}})^*$ and thus, given $\\Gamma \\in \\ker (\\nabla^{\\mathrm{End}})^*$, one can consider:\n\\[\nG_{\\mathfrak{a}_0}(\\Gamma,\\Gamma) := \\langle \\Pi_1 \\Gamma, \\Gamma \\rangle_{L^2(M,T^*M\\otimes\\mathrm{End}(\\E))} \\geq c \\|\\Gamma\\|^2_{H^{-1\/2}},\n\\]\nfor some constant $c>0$. Lemma \\ref{lemma:Pi_1uniform} shows that the constant $c$ is locally uniform with respect to $\\mathfrak{a}_0$.\n\\end{remark}\n\n\n\\subsection{P-R resonance at $0$ of the mixed connection: non-opaque case}\n\\label{ssection:non-opaque}\n\n\nThe aim of this paragraph is to deal with neighbourhoods of connections that are not necessarily opaque, and only assume $\\Pi_1^{\\mathrm{End}(\\E)}$ is injective. In other words, we do not want to assume the resonant space of $-(\\pi^*\\nabla^{\\mathrm{End}(\\E)})_X$ at zero is spanned by $\\mathbbm{1}_{\\E}$ necessarily.\n\nNext, as in \\S \\ref{ssection:pollicott-ruelle-dea}, we introduce the mixed connection with respect to $\\nabla^{\\E} + A$ and $\\nabla^{\\E}$, denoted by $\\nabla^{\\Hom(\\nabla^{\\E} + A, \\nabla^{\\E})}$, and set $\\mathbf{X}_A :=(\\pi^*\\nabla^{\\Hom(\\nabla^{\\E} + A, \\nabla^{\\E})})_X$. We assume $s \\gg 1$. As before, consider a contour $\\gamma \\subset \\mathbb{C}$ around zero such that $\\mathbf{X} := \\mathbf{X}_0$ has only the resonance zero within $\\gamma$ and $\\varepsilon > 0$ such that $-\\mathbf{X}_A$ has no resonances on $\\gamma$ for all $\\|A\\|_{C_*^s} < \\varepsilon$. We introduce\n\\[\\Pi_A^+ := \\frac{1}{2\\pi i} \\int_\\gamma (z + \\mathbf{X}_A)^{-1} dz, \\quad \\lambda_A := \\Tr(-\\mathbf{X}_A\\Pi_A^+).\\]\nThis generalises the quantity studied in \\S \\ref{ssection:finer}, where it was assumed that the multiplicity of $\\mathbf{X}$ at zero is equal to one. By \\cite{Bonthonneau-19} it follows that $A \\mapsto \\Pi_A^+$ is $C^3$-regular and hence $A \\mapsto \\lambda_A$ is also $C^3$-regular. As in \\eqref{eq:mixedunitaryequiv}, we observe that the operators $-\\mathbf{X}_A$ and $-\\mathbf{X}_{A'}$ are unitarily equivalent on $\\mathcal{H}_+$ whenever $\\nabla^{\\E} + A$ and $\\nabla^{\\E} + A'$ are gauge equivalent; hence $\\lambda_A = \\lambda_{A'}$ and so $\\lambda_A$ descends to the local moduli space\n\n\n\nNote also that $\\re(\\lambda_A) \\leq 0$, since all resonances of $-\\mathbf{X}_A$ lie in the half-plane $\\{\\re z \\leq 0\\}$ and this gives us hope that $\\re(\\lambda_A)$ controls the distance between the connections. Assume that the resonant space of $-\\mathbf{X}$ at zero is spanned by smooth $L^2$-orthonormal resonant states $\\{u_i\\}_{i = 1}^p$. We have the following generalisation of Lemma \\ref{lemma:diff-2}:\n\n\\begin{lemma}\n\\label{lemma:variations}\nFor $A \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C_*^s} < \\varepsilon$, we have $\\lambda_A \\in \\mathbb{R}$ and the following perturbation formulas hold:\n\\[\n\\dd \\lambda_A|_{A=0} = 0, ~~ \\dd^2 \\lambda_A|_{A=0}(\\Gamma,\\Gamma) = - \\sum_{i = 1}^p \\langle \\Pi u_i \\pi_1^*\\Gamma, u_i \\pi_1^*\\Gamma\\rangle_{L^2}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\tThe fact that $\\lambda_A$ is real follows from the symmetry of the spectrum of $-\\mathbf{X}_A$ shown in Lemma \\ref{lemma:symmetricspectrum}. The first derivative formula is obvious as $\\lambda_A \\leq 0$; the second one follows from minor adaptations of \\cite[Lemma 5.9]{Cekic-Lefeuvre-20}, where the analogous case of endomorphisms was considered.\n\\end{proof}\n\nNext, by a straightforward adaptation of the Lemma \\ref{lemma:coulomb}, we obtain the existence of $\\varepsilon > 0$, such that for all $A$ with $\\|A\\|_{C_*^s} < \\varepsilon$, there is a smooth map $A \\mapsto \\phi(A) \\in \\mathcal{S}_s$ that sends $\\nabla^{\\E} + A$ to Coulomb gauge with respect to $\\nabla^{\\E}$, that is it satisfies $\\phi(A) - \\nabla^{\\E} \\in \\ker (\\nabla^{\\mathrm{End}(\\E)})^*$.\n\\begin{remark}\\rm\n\tWe cannot get the analogous statement to Lemma \\ref{lemma:coulomb} for parameters $(A_1, A_2)$, because the range of $F(A_1, A_2, \\chi)$ equals $\\ker(\\nabla_1^{\\mathrm{End}(\\E)})^\\perp$ and this is not uniform in $A_1$ (i.e. $\\ker \\nabla_1^{\\mathrm{End}(\\E)}$ changes as we move $A_1$); the space $\\mathbb{A}_{\\E}$ is not a smooth manifold at reducible connections.\n\\end{remark}\n\n\n\n\n\n\nIn the following lemma, we will need to assume that $\\pi_{1*} \\Pi_0^+ = 0$. Equivalently, this means that the resonant states $u_i \\in \\ker(\\mathbf{X}|_{\\mathcal{H}_+})$ satisfy $\\pi_{1*} u_i = 0$, for $i = 1, \\dotso, p$, i.e. the degree $1$ Fourier mode of all the $u_i$ vanishes.\n\n\\begin{lemma}\n\\label{lemma:maininequalitygeneral}\nAssume that the generalized X-ray transform $\\Pi^{\\mathrm{End}(\\E)}_1$ defined with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$ is $s$-injective and additionally that $\\pi_{1*} \\Pi_0^+ = 0$. For $s \\gg 1$ large enough, there exist constants $\\varepsilon, C > 0$ such that for all $A \\in C^s(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C^s_*} < \\varepsilon$:\n\\[\n0 \\leq \\|\\phi(A)-\\nabla^{\\E}\\|^2_{H^{-1\/2}(M,T^*M\\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq C |\\lambda_A|.\n\\]\n\\end{lemma}\n\\begin{proof}\nThis is straightforward from the proof of Lemma \\ref{lemma:borne-lambda-a-1}. With the same functionals $F_1(A) = \\lambda_A$ and $F_2(A) = -\\|\\phi(A) - \\nabla^{\\E}\\|^2_{H^{-1\/2}}$, the only slight difference is the computation of $\\dd^2 F_1$. Pick an $L^2$-orthonormal basis $u_1, \\dotso, u_p$ of the resonant space of $-\\mathbf{X}$ at zero such that $u_1 = c\\mathbbm{1}_{\\E}$, where $c$ is a fixed constant. By Lemmas \\ref{lemma:variations} and \\ref{lemma:relations-resolvent}, we have\n\\[\n\\dd^2 F_1|_{A=0}(\\Gamma,\\Gamma) = -\\sum_{i = 1}^p \\langle \\Pi u_i \\pi_1^* \\Gamma, u_i \\pi_1^* \\Gamma \\rangle_{L^2} \\leq -c^2 \\langle \\Pi_1^{\\mathrm{End}(\\E)} \\Gamma, \\Gamma \\rangle_{L^2}.\n\\]\nNote importantly that we have used $\\Pi_0^+ \\pi_1^*\\Gamma = 0$. This follows from the expression for the projector $\\Pi_0^+ = \\sum_{i = 1}^p \\langle{\\bullet, u_i}\\rangle_{L^2} u_i$ and $\\pi_{1*}u_i = 0$ for all $i$. This suffices to run the proof in the same manner. \n\\end{proof}\n\n\n\n\n\\section{Injectivity of the primitive trace map}\n\nWe can now prove the main results stated in the introduction.\n\n\n\n\\label{section:proofs}\n\n\n\n\n\\subsection{The local injectivity result}\n\n\\label{ssection:proof-injectivity}\n\nWe now prove the injectivity result of Theorem \\ref{theorem:injectivity}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:injectivity}]\nWe fix a regularity exponent $N \\gg 1$ large enough so that the results of \\S\\ref{section:geometry} apply. We fix $\\nabla^{\\mathcal{E}}$, a smooth unitary connection on $\\E$ and assume that it is \\emph{generic} (see page \\pageref{def:generic}). \nBy mere continuity, the same properties hold for every connection $\\nabla^{\\E} + A$ such that $\\|A\\|_{C^N_*} < \\varepsilon$, where $\\varepsilon > 0$ is small enough depending on $\\nabla^{\\E}$.\n\nConsider two smooth unitary connections $\\nabla_i^{\\E} = \\nabla^{\\E} + A_i$ such that $\\|A_i\\|_{C^N_*} < \\varepsilon$ for $i = 1, 2$. Assume that $\\mathcal{T}^\\sharp(\\nabla_1^{\\E})=\\mathcal{T}^\\sharp(\\nabla_2^{\\E})$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} yields the existence of a smooth map $p \\in C^\\infty(SM,\\mathrm{U}(\\E))$ such that:\n\\[\n\\pi^* \\nabla^{\\Hom(\\nabla_1, \\nabla_2)}_X u = 0\n\\]\nthat is $u$ is a resonant state for the operator $\\mathbf{X}_{A_1, A_2}$ associated to the eigenvalue $0$. Assumptions \\textbf{(A)} and \\textbf{(B)} allow us to apply Lemma \\ref{lemma:borne-lambda-a-1}. We therefore obtain:\n\\[\n\\lambda_{A_1, A_2} = 0 \\leq - C \\|\\phi(A_1, A_2) - \\nabla_1^{\\E}\\|^2_{H^{-1\/2}(M,T^*M \\otimes \\mathrm{End}(\\E))} \\leq 0,\n\\]\nwhere $C = C(\\nabla^{\\E}) > 0$ only depends on $\\nabla^{\\E}$. Hence $\\phi(A_1, A_2) = p_{A_1, A_2}^*\\nabla_2^{\\E} = \\nabla_1^{\\E}$. In other words, the connections are gauge-equivalent.\n\\end{proof}\t\n\nNext, we discuss a version of local injectivity in a neighbourhood of a connection which is non-opaque. We will say a map $f: X \\to Y$ of topological spaces is \\emph{weakly locally injective at $x_0 \\in X$} if there exists a neighbourhood $U \\ni x$ such that $f(x) = f(x_0)$ for $x \\in U$ implies $x = x_0$. This notion appears in non-linear inverse problems where the linearisation is not continuous, see \\cite[Section 2]{Stefanov-Uhlmann-08}. We have:\n\n\\begin{theorem}\\label{thm:weaklocal}\n\tIf $N \\gg 1$ and $[\\nabla^{\\E}] \\in \\mathbb{A}_{\\E}$ is such that the generalised $X$-ray transform $\\Pi_1^{\\mathrm{End}(\\E)}$ is $s$-injective, as well as $\\pi_{1*} \\ker(\\pi^*\\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty}) = 0$, then the primitive trace map $\\mathcal{T}^\\sharp$ is weakly locally injective at $[\\nabla^{\\E}]$ in the $C^N$-quotient topology.\n\\end{theorem}\n\\begin{proof}\n\tThe proof is analogous to the proof of Theorem \\ref{theorem:injectivity}, by using the results of \\S \\ref{ssection:non-opaque}. We omit the details.\n\\end{proof}\n\nWe shall see below (see Lemma \\ref{lemma:Pi_1inj}) that flat connections have an injective generalised $X$-ray transform $\\Pi_1^{\\mathrm{End}(\\E)}$ and satisfy the additional condition that $\\ker (\\pi^*\\nabla^{\\E})_X|_{C^\\infty}$ consists of elements of degree zero (but might not be opaque). The previous Theorem therefore shows that the primitive trace map is weakly locally injective near such connections. Let us state this as a corollary for the trivial connection, as it partially answers an open question of Paternain \\cite[p33, Question (3)]{Paternain-13}.\n\n\n\n\\begin{corollary}\n\tLet $\\E = M \\times \\mathbb{C}^r$ be the trivial Hermitian vector bundle equipped with the trivial flat connection $d$. Then there exists a neighbourhood $\\mathcal{U} \\ni [d]$ in $\\mathbb{A}_{\\E}$ with $C^N$-quotient topology such that $[d]$ is the unique gauge class of transparent connections in $\\mathcal{U}$.\n\\end{corollary}\t\n\n\\subsection{Global injectivity results}\n\nWe now detail some cases in which Theorem \\ref{theorem:injectivity} can be upgraded.\n\n\n\\subsubsection{Line bundles}\n\n\\label{sssection:line}\n\nWe let $\\mathcal{T}^\\sharp_1$ be the restriction of the total primitive trace map \\eqref{equation:trace-total} to line bundles. The moduli space of all connections on line bundles $\\mathbb{A}_1$ carries a natural Abelian group structure using the tensor product.\nWhen restricted to line bundles, the primitive trace map $\\mathcal{T}^\\sharp_1$ takes value in $\\ell(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$, namely the set of sequences indexed by primitive free homotopy classes. We have:\n\n\\begin{lemma}\nThe map $\\mathcal{T}^\\sharp_1 : \\mathbb{A}_1 \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$ is a multiplicative group homomorphism.\n\\end{lemma}\n\\begin{proof}\n\tLeft as an exercise to the reader.\n\\end{proof}\n\n\n\\begin{remark}\n\\rm\nThere also exists a group homomorphism for higher rank vector bundles by taking the determinant instead of the trace. More precisely, writing $\\mathbb{A}_r$ for the set of all unitary connections on all possible Hermitian vector bundles of rank $r$ (up to isomorphisms), one can set\n\\begin{equation}\n\\label{equation:det}\n\\det{}^\\sharp : \\mathbb{A} \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1)),\n\\end{equation}\nby taking the determinant of the holonomy along each closed primitive geodesic. This map is also a group homomorphism (where the group structure on $\\bigsqcup_{r \\geq 0} \\mathbb{A}_r$) is also obtained by tensor product. Nevertheless, the determinant map \\eqref{equation:det} cannot be injective as all trivial bundles (of different ranks) have same image.\n\\end{remark}\n\n\n\nWe have the following result, mainly due to Paternain \\cite{Paternain-09}:\n\n\n\\begin{prop}[Paternain]\n\\label{proposition:line}\nLet $(M,g)$ be a smooth Anosov $n$-manifold. If $n \\geq 3$, then the restriction of the primitive trace map to line bundles\n\\begin{equation}\n\\label{equation:trace-line}\n\\mathcal{T}_1^\\sharp : \\mathbb{A}_1 \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nis globally injective. Moreover, if $n = 2$ then:\n\\[\n\\ker \\mathcal{T}^\\sharp_1 = \\left\\{ ([\\kappa^{\\otimes n}], [{\\nabla^{\\mathrm{LC}}}^{\\otimes n}]), n \\in \\mathbb{Z}\\right\\},\n\\]\nwhere $\\kappa \\rightarrow M$ denotes the canonical line bundle and $\\nabla^{\\mathrm{LC}}$ is connection induced on $\\kappa$ by the Levi-Civita connection.\n\\end{prop}\n\n\n\nObserve that on surfaces, the trivial line bundle $\\mathbb{C} \\times M \\rightarrow M$ (with trivial connection) and the canonical line bundle $\\kappa \\rightarrow M$ (with the Levi-Civita connection) both have trivial holonomy but are not isomorphic. This explains the existence of a non-trivial kernel for $n=2$. We will need this preliminary lemma:\n\n\\begin{lemma}\n\\label{lemma:iso}\nLet $(M,g)$ be a smooth closed Riemannian manifold of dimension $\\geq 3$ and let $\\pi : SM \\rightarrow M$ be the projection. Let $\\mathcal{L}_1 \\rightarrow M$ and $\\mathcal{L}_2 \\rightarrow M$ be two Hermitian line bundles. If $\\pi^*\\mathcal{L}_1 \\simeq \\pi^*\\mathcal{L}_2$ are isomorphic, then $\\mathcal{L}_1 \\simeq \\mathcal{L}_2$ are isomorphic.\n\\end{lemma}\n\n\\begin{proof}\nThe topology of line bundles is determined by their first Chern class. As a consequence, it suffices to show that $c_1(\\mathcal{L}_1) = c_1(\\mathcal{L}_2)$. By assumption, we have $c_1(\\pi^*\\mathcal{L}_1) = \\pi^* c_1(\\mathcal{L}_1) = c_1(\\pi^*\\mathcal{L}_2) = \\pi^* c_1(\\mathcal{L}_2)$ and thus it suffices to show that $\\pi^* : H^2(M,\\mathbb{Z}) \\rightarrow H^2(SM,\\mathbb{Z})$ is injective when $\\dim(M) \\geq 3$. But this is then a mere consequence of the Gysin exact sequence \\cite[Proposition 14.33]{Bott-Tu-82}. \n\\end{proof}\n\n\nWe can now prove Proposition \\ref{proposition:line}:\n\n\\begin{proof}[Proof of Proposition \\ref{proposition:line}]\nAssume that $\\mathcal{T}^\\sharp_1(\\mathfrak{a}_1) = \\mathcal{T}^\\sharp_1(\\mathfrak{a}_2)$, where $\\mathfrak{a}_1 \\in \\mathbb{A}_{\\mathcal{L}_1}$ and $\\mathfrak{a}_2 \\in \\mathbb{A}_{\\mathcal{L}_2}$ are two classes of connections defined on two (classes of) line bundles. By Theorem \\ref{theorem:weak}, we obtain that the pullback bundles $\\pi^*\\mathcal{L}_1$ and $\\pi^*\\mathcal{L}_2$ are isomorphic, hence $\\mathcal{L}_1 \\simeq \\mathcal{L}_2$ are isomorphic by Lemma \\ref{lemma:iso}. Up to composing by a first bundle (unitary) isomorphism, we can therefore assume that $\\mathcal{L}_1 = \\mathcal{L}_2 =: \\mathcal{L}$. Let $\\nabla^\\mathcal{L}_1 \\in \\mathfrak{a}_1$ and $\\nabla^\\mathcal{L}_2 \\in \\mathfrak{a}_2$ be two representatives of these classes. They satisfy $\\mathcal{T}^\\sharp(\\nabla^{\\mathcal{L}}_1) = \\mathcal{T}^\\sharp(\\nabla^{\\mathcal{L}}_2)$. Combing Theorem \\ref{theorem:weak} with \\cite[Theorem 3.2]{Paternain-09}, the primitive trace map $\\mathcal{T}^\\sharp_{\\mathcal{L}}$ is known to be globally injective for connections on the same fixed bundle. Hence $\\nabla^{\\mathcal{L}}_1$ and $\\nabla^{\\mathcal{L}}_2$ are gauge-equivalent.\n\nFor the second claim, $\\mathrm{x} = ([\\mathcal{L}],\\mathfrak{a})$. If $\\mathcal{T}^\\sharp_1(\\mathrm{x}) = (1,1,...)$ (i.e. the connection is transparent), then by Theorem \\ref{theorem:weak}, one has that $\\pi^*\\mathcal{L} \\rightarrow SM$ is trivial. By the Gysin sequence \\cite[Proposition 14.33]{Bott-Tu-82}, this implies that $c_1(\\mathcal{L})$ is divisible by $2g-2$, where $g$ is the genus of $M$ (see \\cite[Theorem 3.1]{Paternain-09}), hence $[\\mathcal{L}] = [\\kappa^{\\otimes n}]$ for some $n \\in \\mathbb{Z}$. Moreover, the Levi-Civita connection on $\\kappa^{\\otimes n}$ is transparent and by uniqueness (see \\cite[Theorem 3.2]{Paternain-09}), this implies that $\\mathfrak{a} = [{\\nabla^{\\mathrm{LC}}}^{\\otimes n}]$.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nThe target space in \\eqref{equation:trace-line} is actually $\\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$ (sequences indexed by $\\mathcal{C}^\\sharp$ and taking values in $\\mathrm{U}(1)$) which can be seen as a subset of $\\mathrm{U}(\\ell^\\infty(\\mathcal{C}^\\sharp))$, the group of unitary operators of the Banach space $\\ell^\\infty(\\mathcal{C}^\\sharp)$ (equipped with the sup norm). Then $\\mathcal{T}_1^{\\sharp}$ is a group homomorphism and Proposition \\ref{proposition:line} asserts that\n\\[\n\\mathcal{T}_1^{\\sharp} : \\mathbb{A}_1 \\rightarrow \\mathrm{U}(\\ell^\\infty(\\mathcal{C}^\\sharp))\n\\]\nis a faithful unitary representation of the Abelian group $\\mathbb{A}_1$.\n\\end{remark}\n\n\nWe end this paragraph with a generalization of Proposition \\ref{proposition:line}. There is a natural submonoid $\\mathbb{A}' \\subset \\mathbb{A}$ which is obtained by considering sums of lines bundles equipped with unitary connections, that is:\n\\[\n\\mathbb{A}' := \\left\\{ \\mathrm{x}_1 \\oplus ... \\oplus \\mathrm{x}_k ~\\mid~ k \\in \\mathbb{N}, \\mathrm{x}_i \\in \\mathbb{A}_1\\right\\}.\n\\]\nWe then have the following:\n\n\\begin{theorem}\n\\label{theorem:sum}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$. Then the restriction of the primitive trace map to $\\mathbb{A}'$:\n\\[\n\\mathcal{T}^{\\sharp} : \\mathbb{A}' \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp)\n\\]\nis globally injective.\n\\end{theorem}\n\n\n\\begin{proof}\nWe consider $\\mathcal{L} := \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_k$ and $\\mathcal{J} = \\mathcal{J}_1 \\oplus ... \\oplus \\mathcal{J}_{k'}$, two Hermitian vector bundles over $M$, equipped with the respective connections $\\nabla^{\\mathcal{L}_1} \\oplus ... \\oplus \\nabla^{\\mathcal{L}_k}$ and $\\nabla^{\\mathcal{J}_1} \\oplus ... \\oplus \\nabla^{\\mathcal{J}_{k'}}$ and we assume that they have same image by the primitive trace map. Fixing a periodic point $(x_\\star,v_\\star)$ and applying Proposition \\ref{proposition:representation}, we obtain that $k=k'$ and the existence of two isomorphic representations $\\rho_{\\mathcal{L}} : \\mathbf{G} \\to \\mathrm{U}(\\pi^*\\mathcal{L}_{(x_\\star,v_\\star)})$ and $\\rho_{\\mathcal{J}} : \\mathbf{G} \\to \\mathrm{U}(\\pi^*\\mathcal{J}_{(x_\\star,v_\\star)})$, where $\\mathbf{G}$ denotes Parry's free monoid at $(x_\\star, v_\\star)$. Since these representations are sums of $1$-dimensional representations, there is a unitary isomorphism $p_\\star : \\pi^*\\mathcal{L}_{(x_\\star,v_\\star)} \\to \\pi^*\\mathcal{J}_{(x_\\star,v_\\star)}$ such that for each $i \\in \\left\\{1,...,k\\right\\}$, there exists $\\sigma(i) \\in \\left\\{1,...,k\\right\\}$ with $p^{(i)}_\\star := p_\\star|_{\\pi^*\\mathcal{L}_{i, (x_\\star, v_\\star)}}$ is a representation isomorphism $p^{(i)}_\\star: \\pi^* \\mathcal{L}_{i, (x_\\star,v_\\star)} \\to \\pi^* \\mathcal{J}_{\\sigma(i), (x_\\star,v_\\star)}$.\n\nNow, following the arguments of Lemma \\ref{lemma:p-minus}, we parallel-transport $p^{(i)}_\\star$ along the homoclinic orbits with respect to the pullback of the mixed connection $\\pi^*\\nabla^{\\mathrm{Hom}(\\nabla^{\\mathcal{L}_i}, \\nabla^{\\mathcal{J}_{\\sigma(i)}})}_X$ (induced by the connections $\\nabla^{\\mathcal{L}_i}$ on $\\mathcal{L}_i$ and $\\nabla^{\\mathcal{J}_{\\sigma(i)}}$ on $\\mathcal{J}_{\\sigma(i)}$); the Lipschitz-regularity of the obtained section follows, as in Lemma \\ref{lemma:lipschitz}, from the fact that $p_\\star^{(i)} \\rho_{\\mathcal{L}_i}(g) = \\rho_{\\mathcal{J}_{\\sigma(i)}}(g) p_\\star^{(i)}$ for all $g \\in \\mathbf{G}$.\nUsing the regularity result of \\cite{Bonthonneau-Lefeuvre-20}, we thus obtain a unitary section\n\\[\np^{(i)} \\in C^\\infty(SM,\\pi^*\\mathrm{Hom}(\\mathcal{L}_i, \\mathcal{J}_{\\sigma(i)}))\n\\]\nconjugating the parallel transports along geodesic flowlines with respect to the connections $\\pi^* \\nabla^{\\mathcal{L}_i}$ and $\\pi^* \\nabla^{\\mathcal{J}_{\\sigma(i)}}$. In particular, the existence of such $p^{(i)}$ ensures that $\\mathcal{T}^{\\sharp}_1(\\mathcal{L}_i,\\nabla^{\\mathcal{L}_i}) = \\mathcal{T}^{\\sharp}_1(\\mathcal{J}_{\\sigma(i)},\\nabla^{\\mathcal{J}_{\\sigma(i)}})$. We then conclude by Proposition \\ref{proposition:line}, showing that each pair $(\\mathcal{L}_i, \\nabla^{\\mathcal{L}_i})$ is isomorphic to $(\\mathcal{J}_{\\sigma(i)}, \\nabla^{\\mathcal{J}_{\\sigma(i)}})$, for $i = 1, \\dotso, k$.\n\\end{proof}\n\n\n\n\\subsubsection{Flat bundles}\n\n\\label{sssection:flat}\n\nWe discuss the particular case of flat vector bundles. It is well-known that the data of a vector bundle equipped with a unitary connection (modulo isomorphism) is equivalent to a unitary representation of the fundamental group (modulo inner automorphisms of the unitary group). More precisely, given $\\rho \\in \\Hom(\\pi_1(M),\\mathrm{U}(r))$, one can associate a Hermitian bundle $\\E \\rightarrow M$ equipped with a flat unitary connection $\\nabla^{\\E}$ by the following process: let $\\widetilde{M}$ be the universal cover of $M$; consider the trivial bundle $\\mathbb{C}^r \\times \\widetilde{M}$ equipped with the flat connection $d$ and define the relation $(x,v) \\sim (x',v')$ if and only if $x' = \\gamma(x), v' = \\rho(\\gamma)v$, for some $\\gamma \\in \\pi_1(M)$; then $(\\E,\\nabla^{\\E})$ is obtained by taking the quotient $\\mathbb{C}^r \\times \\widetilde{M}\/\\sim$. Changing $\\rho$ by an isomorphic representation $\\rho' = p \\cdot \\rho \\cdot p^{-1}$ (for $p \\in \\mathrm{U}(r)$) changes the connection by a gauge-equivalent connection and this process gives a one-to-one correspondence between the moduli spaces. \n\n\nFor $r \\geq 0$, we let\n\\[\n\\mathcal{M}_r := \\mathrm{Hom}(\\pi_1(M),\\mathrm{U}(r))\/\\sim,\n\\]\nbe the moduli space of unitary representations of the fundamental group, where two representations are equivalent $\\sim$ whenever they are isomorphic. The space $\\mathcal{M}_r$ is called the \\emph{character variety}, see \\cite{Labourie-13} for instance. For $r=0$, it is reduced to a point; for $r=1$, it is given by $\\mathcal{M}_1 = \\mathrm{U}(1)^{b_1(M)}$, where $b_1(M)$ denotes the first Betti number of $M$. Given $\\mathrm{x} \\in \\mathcal{M}_r$, we let $\\Psi(\\mathrm{x}) = (\\E_{\\mathrm{x}}, \\nabla^{\\E_{\\mathrm{x}}})$ be the data of a Hermitian vector bundle equipped with a unitary connection (up to gauge-equivalence) described by the above process. The primitive trace map $\\mathcal{T}^\\sharp$ can then be seen as a map:\n\\[\n\\mathcal{T}^\\sharp : \\bigsqcup_{r \\geq 0} \\mathcal{M}_r \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp), \\quad \\mathcal{T}^\\sharp(\\mathrm{x}) := \\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}}})\n\\]\nwhere the right-hand side is understood by \\eqref{equation:trace}. We then have the following:\n\n\n\\begin{prop}\n\\label{proposition:flat}\nLet $(M, g)$ be an Anosov manifold of dimension $\\geq 2$. Then the primitive trace map\n\\[\n\\mathcal{T}^\\sharp : \\bigsqcup_{r \\geq 0} \\mathcal{M}_r \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp), \n\\]\nis globally injective. Moreover, given $\\mathrm{x}_0 = ([\\E_0],[\\nabla^{\\E}_0]) \\in \\mathcal{M}_r$, the primitive trace map is weakly locally injective (in the sense of Theorem \\ref{thm:weaklocal}) near $\\mathrm{x}_0$ in the space $\\mathbb{A}_{[\\E_0]}$ of all unitary connections on $[\\E_0]$.\n\\end{prop}\n\nThe previous Proposition \\ref{proposition:flat} will be strengthened below when further assuming that $(M,g)$ has negative curvature (see Lemma \\ref{lemma:small-curvature}): we will show that the primitive trace map is globally injective on connections with \\emph{small} curvature.\nThe first part of Proposition \\ref{proposition:flat} could be proved by purely algebraic arguments; nevertheless, we provide a proof with dynamical flavour, which is more in the spirit of the present article. We need a preliminary result:\n\n\\begin{lemma}\\label{lemma:Pi_1inj}\nAssume $(M,g)$ is Anosov and $\\nabla^{\\E}$ is a flat and unitary connection on the Hermitian vector bundle $\\E \\rightarrow M$. If $\\mathbf{X} := (\\pi^*\\nabla^{\\E})_X$, then:\n\\begin{itemize}\n\\item If $\\mathbf{X} u = f$ with $f=f_0 + f_1 \\in C^\\infty(M,(\\Omega_0 \\oplus \\Omega_1) \\otimes \\E)$ and $u \\in C^\\infty(SM,\\pi^*\\E)$, then $f_0 = 0$ and $u$ is of degree $0$. \n\\item In particular, smooth invariant sections $u \\in \\ker \\mathbf{X}|_{C^\\infty(SM,\\pi^*\\E)}$ are of degree $0$.\n\\item The operator $\\Pi_1^{\\E}$ is s-injective.\n\\end{itemize}\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nThe proof is based on the twisted Pestov identity for flat connections. \n\n\\begin{lemma}[Twisted Pestov identity]\nLet $u\\in H^2(SM,\\pi^*\\E)$. Then\n\\[\n\\|\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u \\|_{L^2}^2 = \\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u, \\nabla^{\\E} _{\\mathbb{V}} u \\rangle_{L^2} + (n -1) \\| \\mathbf{X} u \\|_{L^2}^2.\n\\]\n\\end{lemma}\n\nFor the notation, see \\S \\ref{sssection:twisted-fourier}; for a proof, we refer to \\cite[Proposition 3.3]{Guillarmou-Paternain-Salo-Uhlmann-16}. An important point is that the following inequality holds for Anosov manifolds:\n\\[\n\\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u, \\nabla^{\\E} _{\\mathbb{V}} u \\rangle_{L^2} \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u\\|^2_{L^2},\n\\]\nwhere $C > 0$ is independent of $u$, see \\cite[Theorem 7.2]{Paternain-Salo-Uhlmann-15} for the case of the trivial line bundle (the generalization to the twisted case is straightforward). We thus obtain:\n\\begin{equation}\n\\label{equation:inegalite}\n\\|\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u \\|_{L^2}^2 \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u\\|^2_{L^2} + (n -1) \\| \\mathbf{X} u \\|_{L^2}^2.\n\\end{equation}\n\nBy assumption, $\\mathbf{X} u = f_0 + f_1 \\in C^\\infty(M, (\\Omega_0 \\oplus \\Omega_1) \\otimes \\E)$. Observe that this equation can be split into odd\/even parts, namely: $\\mathbf{X} u_{\\mathrm{even}} = f_1, \\mathbf{X} u_{\\mathrm{odd}} = f_0$, and $u_{\\mathrm{even},\\mathrm{odd}} \\in C^\\infty(SM,\\pi^*\\E)$ have respective even\/odd Fourier components. Applying \\eqref{equation:inegalite} with $u_{\\mathrm{odd}}$, we obtain $f_0 = 0$, $\\mathbf{X} u_{\\mathrm{odd}} = 0$ and $\\nabla^{\\E}_{\\mathbb{V}} u_{\\mathrm{odd}} = 0$, that is $u_{\\mathrm{odd}}$ is of degree $0$ but $0$ is even so $u_{\\mathrm{odd}} = 0$. As far as $u_{\\mathrm{even}}$ is concerned, observe that $\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u_{\\mathrm{even}} = \\nabla^{\\E} _{\\mathbb{V}} f_1$ and:\n\\[\n\\|\\nabla^{\\E} _{\\mathbb{V}} f_1\\|_{L^2}^2 = \\langle -\\Delta_{\\mathbb{V}}^{\\E} f_1, f_1 \\rangle_{L^2} = (n-1) \\|f_1\\|^2_{L^2}.\n\\]\nHence, applying the twisted Pestov identity with $u_{\\mathrm{even}}$, we obtain:\n\\[\n0 = \\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}} \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}}, \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}} \\rangle_{L^2} \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u_{\\mathrm{even}}\\|^2_{L^2},\n\\]\nthat is $u_{\\mathrm{even}}$ is of degree $0$. This proves the first point and the second point is a direct consequence of the first point.\n\nFor the last point, consider the equation $\\mathbf{X} u = \\pi_1^*f$. By the first point, $u$ is of degree zero, so $u = \\pi_0^*u'$ for some $u' \\in C^\\infty(M, \\E)$. Hence by \\eqref{eq:pullback} we get $f = \\nabla^{\\E}u'$ and the conclusion follows by Lemma \\ref{lemma:x-ray}.\n\\end{proof}\n\n\nWe can now prove Proposition \\ref{proposition:flat}.\n\n\\begin{proof}\nWe prove the first part of Proposition \\ref{proposition:flat}. We assume that $\\mathcal{T}^\\sharp(\\mathrm{x}_1) =\\mathcal{T}^\\sharp(\\mathrm{x}_2)$ or, equivalently, that $\\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}_1}}) = \\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}_2}})$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} implies that the bundles $\\pi^*\\E_{\\mathrm{x}_1}$ and $\\pi^*\\E_{\\mathrm{x}_2}$ are isomorphic and yields the existence of a section $p \\in C^\\infty(SM,\\mathrm{U}(\\pi^*\\E_{\\mathrm{x}_2}, \\pi^*\\E_{\\mathrm{x}_1}))$ such that: $C_{\\mathrm{x}_1}(x,t) = p(\\varphi_t x)C_{\\mathrm{x}_2}(x,t)p(x)^{-1}$ for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$, which is equivalent to\n\\[\n\\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}_X p = 0,\n\\]\nwhere $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}$ is the mixed connection induced by $\\nabla^{\\E_{\\mathrm{x}_2}}$ and $\\nabla^{\\E_{\\mathrm{x}_1}}$ on $\\mathrm{Hom}(\\E_{\\mathrm{x}_2}, \\E_{\\mathrm{x}_1})$. Observe that by \\eqref{equation:induced-curvature}, the curvature of $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}$ vanishes as both curvatures $F_{\\nabla^{\\E_{\\mathrm{x}_{1,2}}}}$ vanish. Applying Lemma \\ref{lemma:Pi_1inj} with $\\mathbf{X} := \\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}_X$ acting on the pullback bundle $\\pi^*\\mathrm{Hom}(\\E_{\\mathrm{x}_2}, \\E_{\\mathrm{x}_1})$, we get that $p$ is of degree $0$, which is equivalent to the fact that the connections are gauge-equivalent.\n\nAs to the second part of Proposition \\ref{proposition:flat}, by Theorem \\ref{thm:weaklocal} it is a straightforward consequence of the s-injectivity of $\\Pi_1^{\\mathrm{End}(\\E_0)}$ and the fact that elements of $\\ker(\\pi^*\\nabla^{\\mathrm{End}(\\E_0)})_X|_{C^\\infty})$ are of degree zero, which follows from Lemma \\ref{lemma:Pi_1inj} (items 2 and 3).\n\\end{proof}\n\n\n\\subsubsection{Negative sectional curvature}\n\n\\label{sssection:negative}\n\nWe now assume further that the Riemannian manifold $(M,g)$ has negative sectional curvature. We introduce the following condition:\n\n\\begin{definition}\nWe say that the pair of connections $(\\nabla^{\\E_1},\\nabla^{\\E_2})$ satisfies the \\emph{spectral condition} if the mixed connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1},\\nabla^{\\E_2})}$ has no non-trivial twisted CKTs. \n\\end{definition}\n\nThis condition is symmetric in the pair $(\\nabla^{\\E_1},\\nabla^{\\E_2})$. Observe that by \\eqref{equation:lien}, the previous condition is invariant by changing one of the two connections by $p^* \\nabla^{\\E_i}$, for some vector bundle isomorphism $p$, and thus this condition descends to the moduli space. We then define\n\\begin{equation}\\label{eq:setS}\n\\mathbf{S} \\subset \\mathbb{A} \\times \\mathbb{A},\n\\end{equation}\nthe subspace of all pairs of equivalence classes of connections satisfying the spectral condition. The set $\\mathbf{S}$ is open and dense (for the $C^N_*$-topology, $N \\gg 1$) as shown in Appendix \\ref{appendix:ckts}. Moreover, it also contains all pairs of connections with \\emph{small curvature}, that is if\n\\[\n\\Omega_\\varepsilon := \\left\\{ \\mathrm{x} = ([\\E],[\\nabla^{\\E}]) \\in \\mathbb{A}, ~~~ \\|F_{\\nabla^{\\E}}\\|_{L^\\infty(M,\\Lambda^2 T^*M \\otimes \\mathrm{End}(\\E))} < \\varepsilon \\right\\} \\subset \\mathbb{A},\n\\]\nthen we have the following:\n\n\n\\begin{lemma}\n\\label{lemma:small-curvature}\nLet $(M,g)$ be a negatively-curved Riemannian manifold of dimension $\\geq 2$ and let $-\\kappa < 0$ be an upper bound for the sectional curvature. There exists $\\varepsilon(n,\\kappa) > 0$ such that:\n\\[\n\\Omega_{\\varepsilon(n,\\kappa)} \\times \\Omega_{\\varepsilon(n,\\kappa)} \\subset \\mathbf{S}.\n\\]\nOne can take $\\varepsilon(n,\\kappa) =\\dfrac{\\kappa \\sqrt{n-1}}{4}$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWe start by a preliminary discussion. Given a Hermitian vector bundle $\\E \\to M$ with metric $\\langle \\bullet, \\bullet \\rangle$, a unitary connection $\\nabla^{\\E}$, we introduce, following \\cite[Section 3]{Guillarmou-Paternain-Salo-Uhlmann-16}, an operator $\\mathcal{F}^{\\E} \\in C^\\infty(SM,\\mathcal{N} \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ (recall that $\\mathcal{N}$ is the normal bundle, see \\S\\ref{sssection:line-bundle}) defined by the equality:\n\\begin{equation}\n\\label{equation:twisted-curvature}\n \\langle \\mathcal{F}^{\\E}(x,v) e, w \\otimes e' \\rangle := \\langle F_{\\nabla^{\\E}}(v,w)e,e' \\rangle,\n\\end{equation}\nwhere $F_{\\nabla^{\\E}}$ is the connection of $\\nabla^{\\E}$, and $(x,v) \\in SM, e, e' \\in \\E_x, w \\in \\mathcal{N}(x,v)$, and the metric on the left-hand side is the natural extension of the metric $\\langle \\bullet, \\bullet \\rangle$ on $\\E$ to $\\mathcal{N} \\otimes \\E$ by tensoring with the metric $g$. A straightforward computation shows that:\n\\begin{equation}\n\\label{equation:bad-bound}\n\\|\\mathcal{F}^{\\E}\\|_{L^\\infty(SM, \\mathcal{N} \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq \\|F_{\\nabla^{\\E}}\\|_{L^\\infty(M,\\Lambda^2 T^*M \\otimes \\mathrm{End}(\\E))}. \n\\end{equation}\nNow, let $\\nabla^{\\E_1}$ and $\\nabla^{\\E_2}$ be two unitary connections, $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$ be the mixed connection and $\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}$ be the operator induced by the mixed connection as in \\eqref{equation:twisted-curvature}. Observe that by \\eqref{equation:induced-curvature} and \\eqref{equation:bad-bound}, we get:\n\\begin{equation}\n\\label{equation:bad-bound-2}\n\\begin{split}\n\\|\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}\\|_{L^\\infty} \\leq \\|F_{\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}}\\|_{L^\\infty} \\leq \\|F_{\\nabla^{\\E_1}}\\|_{L^\\infty} + \\|F_{\\nabla^{\\E_2}}\\|_{L^\\infty} < 2\\varepsilon(n,\\kappa).\n\\end{split}\n\\end{equation}\nBy \\cite[Theorem 4.5]{Guillarmou-Paternain-Salo-Uhlmann-16}, if $m \\geq 1$ satisfies:\n\\begin{equation}\n\\label{equation:ckts}\nm(m+n-2) \\geq 4 \\dfrac{\\|\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}\\|^2_{L^\\infty}}{\\kappa^2},\n\\end{equation}\nthen there are no twisted CKTs of degree $m$ (for the connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$). Now, the choice of $\\varepsilon(n,\\kappa) > 0$ combined with \\eqref{equation:bad-bound-2} guarantees that \\eqref{equation:ckts} is satisfied for any $m \\geq 1$.\n\\end{proof}\n\n\n\n\nWe then have the following statement:\n\n\\begin{prop}\n\\label{proposition:negative}\nLet $(M,g)$ be a negatively-curved Riemannian manifold of dimension $\\geq 2$. Let $(\\mathfrak{a}, \\mathfrak{a}') \\in \\mathbf{S}$ such that $\\mathcal{T}^\\sharp(\\mathfrak{a}) = \\mathcal{T}^\\sharp(\\mathfrak{a}')$. Then $\\mathfrak{a} = \\mathfrak{a}'$.\n\\end{prop}\n\nIn other words, two connections satisfying the spectral condition and whose images by the primitive trace map are equal, are actually gauge-equivalent.\n\n\\begin{proof}\nConsider two representatives $\\nabla^{\\E_1} \\in \\mathfrak{a}$ and $\\nabla^{\\E_2} \\in \\mathfrak{a}'$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} provides a section $p \\in C^\\infty(SM,\\mathrm{U}(\\pi^*\\E_2,\\pi^*\\E_1))$ such that:\n\\[\n\\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{2}}, \\nabla^{\\E_{1}})}_X p = 0.\n\\]\nBy assumption, $(M,g)$ has negative curvature and thus $p$ has finite Fourier degree by \\cite[Theorem 4.1]{Guillarmou-Paternain-Salo-Uhlmann-16}. Moreover, since $ \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{2}}, \\nabla^{\\E_{1}})}$ has no non-trivial twisted CKTs, $p$ is of degree $0$ (see \\cite[Theorem 5.1]{Guillarmou-Paternain-Salo-Uhlmann-16}). This shows that the connections are gauge-equivalent.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Topological results}\n\n\\label{sssection:topology}\n\nIn this section we prove a global \\emph{topological} uniqueness result for the primitive trace map. \n\n\\begin{prop}\\label{proposition:topology}\n\tLet $(M, g)$ be an orientable Anosov manifold. If $\\mathrm{x}_i = ([\\E_i], [\\nabla^{\\E_i}]) \\in \\mathbb{A}$ for $i = 1, 2$, then $\\mathcal{T}^\\sharp(\\mathrm{x}_1) = \\mathcal{T}^\\sharp(\\mathrm{x}_2)$ implies:\n\t\n\t\\begin{itemize}\n\t\t\t\\item If $\\dim M$ is odd or more generally the Euler characteristic $\\chi(M) = 0$ vanishes, then $\\E_1 \\simeq \\E_2$ are isomorphic as vector bundles.\n\t\t\t\\item If $\\dim M = 2d$ for some $d \\in \\mathbb{N}$ and $\\chi(M) \\neq 0$, then\n\t\t\t\t\\begin{itemize}\n\t\t\t\t\t\\item The Chern classes satisfy $c_i(\\E_1) = c_i(\\E_2)$ for $i = 1, \\dotso, d - 1$; also $c_d(\\E_1) - c_d(\\E_2) \\in H^{2d}(M; \\mathbb{Z}) \\cong \\mathbb{Z}$ is a multiple of $\\chi(M)$.\n\t\t\t\t\t\\item If the even cohomology ring $H^{\\mathrm{even}}(M; \\mathbb{Z})$ is torsion-free, and the rank of the bundles is less than $d$ or more generally $c_d(\\E_1) = c_d(\\E_2)$, then $\\E_1$ and $\\E_2$ are stably isomorphic, i.e. there is an $m \\geq 0$ such that $\\E_1 \\oplus \\mathbb{C}^m \\simeq \\E_2 \\oplus \\mathbb{C}^m$\n\t\t\t\t\\end{itemize}\n\t\\end{itemize}\t\n\\end{prop}\n\\begin{proof}\n\tAs a direct consequence of Theorem \\ref{theorem:weak}, from $\\mathcal{T}^\\sharp(\\mathrm{x}_1) = \\mathcal{T}^\\sharp(\\mathrm{x}_2)$ we obtain that $\\pi^*\\E_1 \\simeq \\pi^*\\E_2$ are isomorphic.\n\t\n\tIf $(M, g)$ has a vanishing Euler characteristic, there is a non-vanishing vector field $V \\in C^\\infty(M, TM)$ (see \\cite[Chapter 11]{Bott-Tu-82}), that we normalise to unit norm using the metric $g$ and hence see as a section of $SM$. Then since $\\pi \\circ V = \\id_M$, we get\n\t\\[\\E_1 \\simeq V^*\\pi^*\\E_1 \\simeq V^*\\pi^* \\E_2 \\simeq \\E_2,\\]\n\tcompleting the proof of the first item. \n\t\n\tThe first point of the second item is immediate after an application of the Gysin exact sequence \\cite[Proposition 14.33]{Bott-Tu-82} for the sphere bundle $SM$. The second point follows from the first one and the fact that the Chern character gives an isomorphism between the rational $K$-theory and even rational cohomology, see \\cite[Proposition 4.5]{Hatcher-17}.\n\\end{proof}\n\n\n\nIt is not known to the authors if further results hold as to the injectivity of $\\pi^* : \\mathrm{Vect}(M) \\rightarrow \\mathrm{Vect}(SM)$ in even dimensions ($\\dim M \\geq 4$). \n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}