Liverani Carlangelo
Exponential decay of correlations for $\Co^{(4)}$ Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.
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Effective Equidistribution for Contact Anosov flows in Dimension Three
本文研究了三维空间中接触Anosov流的闭合周期轨道包的有效等分布性质，证明了其以指数速率等分布。
The first half of this paper is concerned with the topology of the space $\AAA(M)$ of (not necessarily contact) Anosov vector fields on the unit tangent bundle $M$ of closed oriented hyperbolic surfaces $Σ$. We show that there are countably infinite connected components of $\AAA(M)$, each of which is not simply connected. In the second part, we study contact Anosov flows. We show in particular that the time changes of contact Anosov flows form a $C^1$-open subset of the space of the Anosov flows which leave a particular $C^\infty$ volume form invariant, if the ambiant manifold is a rational ho…
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalising related rigidity results of A.Katok and P. Foulon.
$C^0$-Contact Anosov flows
本文证明了常负曲率流形上测地流的光滑重参数化是接触 Anosov 流，由此给出一类新的指数混合 Anosov 流，并引入 $C^0$-接触概念，证明经典 Gray 稳定性定理在此设定下失效。
For the time-one map $f$ of a contact Anosov flow on a compact Riemann manifold $M$, satisfying a certain regularity condition, we show that given a Gibbs measure on $M$, a sufficiently large Pesin regular set $P_0$ and an arbitrary $δ\in (0,1)$, there exist positive constants $C$ and $c$ such that for any integer $n \geq 1$, the measure of the set of those $x\in M$ with $f^k(x) \notin P_0$ for at least $δn$ values of $k = 0,1, \ldots,n-1$ does not exceed $C e^{-cn}$.
For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $ε> 0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral …
We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.
An obstruction to fiberwise Anosov flows over 3-dimensional Anosov flows
该研究探讨了三维Anosov流作为纤维化Anosov流基流的障碍，证明了基流在特定条件下不存在，并推导出相关流的轨道等价性。
We show that for smooth contact Anosov flows in dimension 3, the resonant states associated to the first band of Ruelle resonances are distributions that are killed by the unstable derivative.
New examples of Anosov flows on higher dimensional manifolds which fibre over $3$-dimensional Anosov flows
本文通过悬置Bonatti-Langevin流有限覆盖的基本群作用，在维数$3+2n$（$n\geq 2$）中构造了非代数Anosov流的新例子。
We prove exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure for contact Anosov flows admitting Pesin sets with exponentially small tails. This is achieved by establishing strong spectral estimates for certain Ruelle transfer operators for such flows.
There exist no contact Anosov diffeomorphisms
该文证明闭奇数维流形上任何 Anosov 微分同胚均无不变切触结构，否定其存在性。
We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.
This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radius of the transfer operators acting on the so-called anisotropic Sobolev space exactly in terms of dynamical exponents. Also we provide a simplified proof by using the FBI transform.
Hyperbolic models of transitive topologically Anosov flows in dimension three
本文证明了闭3-流形上的传递拓扑Anosov流轨道等价于保持遍历光滑体积形式的光滑Anosov流。
We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated growth rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. Then, we study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving in terms of those theories. We finally use our study to show that the b…
Pseudo-Anosov Reeb flows and contact structures
本文提出伪Anosov接触结构的概念，证明其具有普遍紧致性和无 torsion 性，并探讨了其在伪Anosov流有限性猜想中的应用。
Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In this article, we prove that a new class of examples are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on three manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow in quasigeodesic, and in this article we provide a new method to …
Symplecto-rigidity and bootstrap for Anosov symplectomorphisms and contact Anosov flows
本文通过自举提升2-捏合接触Anosov流的共轭正则性，引入辛刚性概念，并证明de la Llave例子在Anosov辛同胚空间中具有刚性，从而建立一系列光滑刚性结果。
Uniqueness of gluings and virtual finiteness of pseudo-Anosov flows on graph manifolds
本文刻画了粘合得到的伪Anosov流的轨道等价条件，结合相关描述证明了图流形上可递伪Anosov流有限性猜想的虚版本。
Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
本文从接触与辛几何角度刻画三维 Anosov 流，建立研究其动力学的几何拓扑框架，并给出双接触结构唯一性及基于 Reeb 流的 Anosov 性判据。
A note on the density of periodic orbits of Anosov geodesic flow in manifolds of finite volume
该研究证明有限体积流形上Anosov测地流的周期轨道具有稠密性，且该结果可推广至非紧场景的保守Anosov流，丰富了动力系统相关理论。
We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher dimensional codimension-one Anosov flow is topologically transitive. Recently, Simic showed that any higher dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flow up to topological equivalence in h…
We give a new proof of the existence of compact surfaces embedded in $R^3$ with Anosov geodesic flows. This proof starts with a non-compact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.
Perturbations of time-one maps of transitive Anosov flows are studied. We show that most perturbations have no intermingled basins of hyperbolic physical measures.
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。
We show that a topologically mixing $C^\infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.
A {\em sectional-Anosov flow} is a vector field on a compact manifold inwardly transverse to the boundary such that the maximal invariant set is sectional-hyperbolic (in the sense of \cite{mm}). We prove that any $C^2$ transitive sectional-Anosov flow has a unique SRB measure which is stochastically stable under small random perturbations.
We prove that every sectional Anosov flow (or, equivalently, every sectional-hyperbolic attracting set of a flow) on a compact manifold has a periodic orbit. This extends the previous three-dimensional result obtained in [Existence of periodic orbits for singular-hyperbolic sets].
We introduce the matching functions technique in the setting of Anosov flows. Then we observe that simple periodic cycle functionals (also known as temporal distance functions) provide a source of matching functions for conjugate Anosov flows. For conservative codimension one Anosov flows $φ^t\colon M\to M$, $\dim M\ge 4$, these simple periodic cycle functionals are $C^1$ regular and, hence, can be used to improve regularity of the conjugacy. Specifically, we prove that a continuous conjugacy must, in fact, be a $C^1$ diffeomorphism for an open and dense set of codimension one conservative Ano…
We show that all generalized Pollicott-Ruelle resonant states of a topologically transitiv $C^\infty$-Anosov flow with an arbitrary $C^\infty$ potential, have full support.
For any $C^r$ contact Anosov flow with $r\ge 3$, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all $C^r$ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow.
A Smörgåsbord of (bi)contact structures, Reeb flows and pseudo-Anosov flows
本文为会议论文集撰写，围绕同一切触结构的两个不同Reeb流动力学关联问题，整理相关结果并提出更多问题与猜想，旨在邀请读者研究特定切触结构族。
We provide a new criterion for the existence of a global cross section to a volume-preserving Anosov flow. The criterion is expressed in terms of expansion and contraction rates of the flow and is more general than the previous results of similar kind.
Construction and upper bound on the minimum genus of an embedded surface with Anosov geodesic flow
研究在\(R^3\)中构造测地线流为阿诺索夫的光滑紧致曲面，通过确定其亏格给出具有阿诺索夫测地线流的嵌入曲面最小亏格的非精确上界，得到可物理实现的阿诺索夫系统。
We consider the Lagrange and the Markov dynamical spectra associated with a conservative Anosov flow on a compact manifold of dimension $3$ (including geodesic flows of negative curvature and suspension flows). We show that for a large set of real functions and typical conservative Anosov flows, both the Lagrange and Markov dynamical spectra have a non-empty interior.
Rigidity from the strong unstable foliation for higher-dimensional Anosov and partially hyperbolic flows
本文推广Burniol Clotet的刚性结果至高维Anosov流及其紧致李群等距扩张，证明强不稳定叶理等价蕴含时间重标度共轭。
A sectional-Anosov flow on a manifold M is a C^1 vector field inwardly transverse to the boundary for which the maximal invariant is sectional-hyperbolic. We prove that every attractor of every vector field C^1 close to a transitive sectional-Anosov flow with singularities on a compact manifold has a singularity. This extends the three-dimensional result obtained in [Morales, C.A., The explosion of singular-hyperbolic attractors].
Goodman surgery and projectively Anosov flows
本文将Goodman手术推广至射影Anosov流范畴，，证明了任何接触Anosov流均可通过手术构造新流，并在双曲3-流形上生成新例子，同时将双接触手术与经典接触手术关联并推导出紧性结果。
We consider the class of partially hyperbolic diffeomorphisms $f:M\to M$ obtained as the discretization of topological Anosov flows. We show uniqueness of minimal unstable lamination for these systems provided that the underlying Anosov flow is transitive and not orbit equivalent to a suspension. As a consequence, uniqueness of quasi-attractors is obtained. If the underlying Anosov flow is not transitive we get an analogous finiteness result provided that the restriction of the flow to any of its attracting basic pieces is not a suspension. A similar uniqueness result is also obtained for cert…
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.
Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds
本文在曲率满足特定假设的非紧流形上证明了 Anosov 测地流的 Ruelle 不等式，并在有限体积非紧流形上得到了相应的 Pesin 公式。
We establish exponential decay of correlations of all orders for locally $G$-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable foliations of the base Anosov flow are jointly $C^1$. This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by Dolgopyat's infinitesimal transitivity group, from which we obtain contraction properties for a class of 'twisted' symbolic transfer operators.
Transitive Anosov flows on non-compact manifolds
本文研究非紧三维流形上Anosov流的传递性，给出同调条件保证传递流在无限覆盖中的传递性，并构造满足悬垂流同伦刻画的传递流，回答相关问题的流版本。
We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving \cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.
We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector fi…
The purpose of this paper is to establish limit laws for volume preserving almost Anosov flows on $3$-three manifolds having a neutral periodic of cubic saddle type. In the process, we derive estimates for the Dulac maps for cubic neutral saddles in planar vector fields.
We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an R-covered transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.
Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line
本文证明线上双曲型作用的刚性，并由此给出$\mathbb{R}$-覆盖Anosov流的谱刚性定理，刻画轨道等价，进而提供自轨道等价判据及接触Anosov流有限性，附录中建立轨道等价与接触结构同构的对应。
For any $C^\infty$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^\infty$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the volume measure is the measure with maximal entropy. We also show that the the metric entropy with respect to the volume measure and the topological entropy of suspension flow over Anosov diffeomorphisms on torus achieve all possible values. Our results fit into two programs related to entropy rigidity and flexibility of Anosov systems.
We obtain limit theorems (Stable Laws and Central Limit Theorems, both Gaussian and non-Gaussian) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The proofs of the limit theorems for these flows are applications of corresponding theorems in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.
Oriented Birkhoff sections of Anosov flows
本文通过三种独立证明展示：在定向3流形上的Anosov流存在正Birkhoff截面当且仅当流是正扭曲的R覆盖。
Adapting tools that we introduced in [19] to study Anosov flows, we prove that the trace formula conjectured by Dyatlov and Zworski in [12] holds for Anosov flows in a certain class of regularity (smaller than $\mathcal{C}^\infty$ but larger than the class of Gevrey functions). The main ingredient of the proof is the construction of a family of anisotropic Hilbert spaces of generalized distributions on which the generator of the flow has discrete spectrum.
Let $(M, ξ)$ be a compact contact 3-manifold and assume that there exists a contact form $α_0$ on $(M, ξ)$ whose Reeb flow is Anosov. We show this implies that every Reeb flow on $(M, ξ)$ has positive topological entropy. Our argument builds on previous work of the author (http://arxiv.org/abs/1410.3380) and recent work of Barthelmé and Fenley (http://arxiv.org/abs/1505.07999). This result combined with the work of Foulon and Hasselblatt (http://www.tufts.edu/as/math/Preprints/FoulonHasselblattLegendrian.pdf) is then used to obtain the first examples of hyperbolic contact 3-manifolds on which …
Surgery on Anosov flows using bi-contact geometry
本文利用双接触几何定义了一类新的 Anosov 流 Dehn 手术，揭示了 Goodman 手术产生 Anosov 流的条件，并证明了沿简单闭测地线进行 Foulon-Hasselblatt Legendrian 手术是生成接触 Anosov 流的充要条件。
We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniforml…
In this paper, we describe a new approach to the problem of classification of transitive Anosov flows on 3-manifolds up to orbital equivalence. More specifically, generalizing the notion of Markov partition, we introduce the notion of Markovian family of rectangles in the bifoliated plane of an Anosov flow. We show that any transitive Anosov flow admits infinitely many Markovian families, each one of which can be canonically associated to a finite collection of combinatorial objects, called geometric types. We prove that any such geometric type describes completely the flow up to Dehn-Goodman-…
Using the factorization method, a method pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification over the moduli space of translation surfaces, we show that generalized $u$-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.
For Anosov flows obtained by suspensions of Anosov diffeomorphisms on surfaces, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3-manifold. These result generalize the similar ones of Sullivan and Ferreira-Pinto for 1-dimensional expanding dynamics and also a result of Ferreira-Pinto for 2-dimensional hyperbolic dynamics.
In this article, we give a quasi-final classification of quasiconformal Anosov flows. We deduce a very interesting differentable rigidity result for the orbit foliations of hyperbolic manifold of dimension at least three.
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