Semyon Dyatlov, Maciej Zworski
The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C^\infty Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [arXiv:1203.0904] but our approach is different and is based on the study of the generator on the flow as a semiclassical differential operator.
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We complete the microlocal study of the geodesic X-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou and pursued by Guillarmou and the second author. We prove new stability estimates and clarify some properties of the operator $Π_m$, the generalized X-ray transform. These estimates rely on a refined version of the Livsic theorem for Anosov flows, especially on a new quantitative finite time Livsic theorem.
Isometric extensions of Anosov flows via microlocal analysis
本文利用 Pollicott-Ruelle 共振的微局部框架重新审视了保体积 Anosov 流等距扩张的遍历性与混合性质，并应用于证明负曲率 Riemannian 流形上标架流的遍历性。
FBI Transform in Gevrey Classes and Anosov Flows
本研究在无边紧致流形上构造解析FBI变换并验证其性质，将其应用于带Gevrey系数的Anosov流的Ruelle谱研究，证明了相关动力行列式的有限阶性。
Critical axis of Ruelle resonances for Anosov flow with a potential
本研究结合微局部分析与维数理论，研究带光滑实值势的Anosov流的最大实部共振，揭示其与Climenhaga提出的稳定流形上特殊测度族的关联，建立共振存在性与平衡测度混合性质的联系并实现测度的谱重构。
This errata fixes a mistake in the part of Giulietti, P.; Liverani, C.; Pollicott, M. Anosov flows and dynamical zeta functions. Ann. of Math. (2) {\bf 178} (2013), no. 2, 687--773, which proves a spectral gap for contact Anosov flows with respect to the measure of maximal entropy (Section 7). However, the first part of the paper, in which it is proved that the Ruelle zeta function is meromorphic, is unaffected.
Anosov Flows and Dynamical Zeta Functions
研究了紧致光滑流形上$\Cs^r$ Anosov流$r > 2$的Ruelle和Selberg zeta函数.我们证明了几个结果，最显著的是：（a）对于$\Cs^\infty$流，zeta函数在整个复平面上是亚纯的;（b）对于满足聚束条件的接触流，zeta函数在整个复平面上是亚纯的（例如，负曲率流形上的测测地流优于$\frac 19$ -pinched）zeta函数在拓扑熵处有一个极点，并且在其左侧的条带中是解析的;（c）在与（b）相同的假设下，我们得到了关于周期轨道数的精确结果。我们的论点是基于对各向异性流的适当巴拿赫空间上的转移算子的谱性质的研究。
Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows
该研究针对三维流形上的光滑伪阿诺索夫流，构造了推广Anosov情形的ζ函数，证明其亚纯延拓性质、Fried猜想版本及Dirichlet类数公式的拓扑类似物，为处理奇点采用了基于马尔可夫分形的C^∞方法。
We consider Anosov flows on closed 3-manifolds preserving a volume form $Ω$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[ι_{X}Ω]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $\textit{semisim…
We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than Hölder: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is als…
Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^κ$-Anosov actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle-Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of $\mathbb{C}^κ$, with $λ=0$ being always a leading resonance. The joint resonant states at $0$ give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to …
Generic twisted Pollicott--Ruelle resonances and zeta function at zero
本文研究了Anosov流在曲面上的扭曲Ruellezeta函数在零点处的零点阶数，并扩展了Fried的猜想到更广泛的表示集。
Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function
本文建立了伪阿诺索夫流与Heegaard Floer同调的新联系，构造链复形计算链环Floer同调，并证明其生成元对应流的闭多轨道，同时引入动力学分次范畴化zeta函数。
We use a simple argument to extend the microlocal proofs of meromorphicity of dynamical zeta functions to the nonorientable case. In the special case of geodesic flow on a connected non-orientable negatively curved closed surface, we compute the order of vanishing of the zeta function at the zero point to be the first Betti number of the surface.
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that…
We discuss about the conjectural cohomological theory of dynamical zeta functions in the case of general Anosov flows. Our aim is to provide a functional-analytic framework that enables us to justify the basic part of the theory rigorously. We show that the zeros and poles of a class of dynamical zeta functions, including the semi-classical (or Gutzwiller-Voros) zeta functions, are interpreted as eigenvalues of the generators of some transfer operators acting on the leaf-wise de Rham cohomology spaces of the unstable foliation.
The spectral concentration for damped waves on compact Anosov manifolds
本文研究紧致 Anosov 流形上阻尼波的特征值分布，利用 Anosov 测地流的中偏差原理证明大多数特征值虚部以对数速度集中于阻尼函数平均值附近，并将其应用于扭曲 Selberg zeta 函数零点的集中性。
Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows
对于紧致Riemann流形上的Anosov流，我们研究了局部稳定流形上的球直径$B^s(x,\ep)$在Lyapunov正则点$x$处沿流的衰减率.我们证明，这种衰减率是相似的所有足够小的值的$ε> 0$。从这个和\cite{kn:St1}的主要结果，我们得到强谱估计的Ruelle转移算子接触Anosov流Lipschitz局部稳定holonomy映射。这些特别适用于严格负曲率的紧致局部对称流形上的测地流。众所周知，这种谱估计在某些相关领域有着深远的意义，例如在研究Ruelle zeta函数和偏微分算子的解析性质，闭轨道计数函数的渐近性等方面。
Nielsen zeta function, 3-manifolds and asymptotic expansions in Nielsen theory
本文证明了可定向三维闭流形（特殊的Haken流形或Seifert流形）上保向同胚的Nielsen zeta函数是一个有理函数或一个有理函数的根.在伪Anosov同胚曲面的情况下，我们计算了扭共轭类的个数或范数至多为$x$的Nielsen不动点类的个数的渐近性.
Perturbative BF theory in axial, Anosov, gauge
该研究通过轴向规范条件，将接触和阿诺索夫向量场的扭曲鲁elleζ函数与BF理论的微扰配分函数联系起来，揭示了两者在复参数下的等价关系。
We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flo…
Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing
本文通过构造非空间断域，证明无挠射影 Anosov 子群是具有唯一基本双曲集的局部齐次切触 Axiom A 流的单值群，并利用谱估计和亚纯延拓结果建立了指数混合等动力学性质。
Dynamical torsion for contact Anosov flows
本文引入“动力学挠率”这一新对象，证明其与 Turaev 挠率的比值在非循环表示空间上局部常数，并将 Fried 猜想从双曲 3-流形扩展至任意奇数维双曲流形。
Can quantum dynamics emerge from classical chaos?
该综述以Anosov测地流这一经典混沌模型为对象，揭示其离散Pollicott-Ruelle谱可通过微局部分析呈现，当最右谱带存在能隙时会主导类量子的有效动力学，阐明了量子动力学从经典混沌关联函数中涌现的机制。
Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $\dim(M) \geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study …
Resonant forms at zero for dissipative Anosov flows
本文研究 $3$-流形上耗散 Anosov 流在零处的共振微分形式，引入 helicity 概念并提供包含 horocyclic 不变性的理论，同时计算了 quasi-Fuchsian flows 等实例的共振 $1$-形式及 Ruelle zeta 函数的消失阶数。
Gibbs measures for contact Anosov flows are all exponentially mixing
通过增强Dolgopyat方法，证明接触Anosov流的所有吉布斯测度对Hölder观测具有指数衰减相关性。
A Hofer-like Metric on the Space of Anosov Flows
本文在Anosov向量场空间上构造Hofer型度量，证明其完备性并用于定量稳定性分析，连接线性响应并引入拓扑Anosov流，为Anosov动力学研究提供几何框架。
We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $τ>0$, that its zeros are contained in the union of the $τ$-neighborhood of the imaginary axis, $|\Re(s)|<τ$, and the region $\Re(s)<-χ_0+τ$, up to finitely many exceptions, where $χ_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.
The growth rate of Floer homology and symplectic zeta function
本文主要研究了紧致曲面的辛同胚的渐近不变量，它定义为辛同胚的迭代序列的辛弗洛尔同调的总维数序列的增长率。证明了渐近不变量与渐近Nielsen数和渐近绝对莱夫谢茨数一致。我们还证明了渐近不变量与辛同胚的伪Anosov分量的最大扩张一致，其对数与拓扑熵一致。这意味着辛zeta函数具有正的收敛半径，这也建立了弗洛尔同调与三维流形几何之间的联系。
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat \cite{D1} and further developed in \cite{St2} is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which pro…
Spectra of Ruelle transfer operators for Axiom A flows (Revised)
对于满足某些附加条件的基本集上的公理A流，我们证明了Ruelle转移算子的强谱估计，类似于Dolgopyat（1998）对紧致曲面上的测地流（对于一般势）和紧致流形上的传递Anosov流（对于Sinai-Bowen-Ruelle势）的强谱估计。这里我们讨论一般势。这些结果在一些相关领域有着重要的意义，如研究Ruelle zeta函数和偏微分算子的解析性质，闭轨道计数函数，Hölder连续势的关联衰减等.
We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maxi…
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