Yong Fang
We classify five dimensional Anosov flows with smooth decomposition which are in addition transversely symplectic. Up to finite covers and a special time change, we find exectly the suspensions of symplectic hyperbolic automorphisms of four dimensional toris, and the geodesic flows of three dimensional hyperbolic manifolds.
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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.
We construct a plethora of Anosov-Katok diffeomorphisms with non-ergodic generic measures and various other mixing and topological properties. We also construct an explicit collection of the set containing the generic points of the system with interesting values of its Hausdorff dimension.
Smooth Stable Foliations of Anosov Diffeomorphisms
本文研究 Anosov 微分同胚余维一稳定叶状结构的刚性，证明其正则性略高于 $C^2$ 时即可达到与微分同胚相同的光滑性。
Topological and smooth classification of Anosov maps on torus
本文研究了Anosov映射在 торus 上的拓扑与光滑分类，证明了拓扑共轭性与Lyapunov指数的关系，并指出光滑共轭类由Jacobian决定。
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 develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric $t$-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.
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 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 classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-preserving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue measure.
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 transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+β} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to …
We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth…
Smooth rigidity for 3-dimensional dissipative Anosov flows
本文证明三维耗散Anosov流在连续共轭下，除例外情形外，共轭或光滑或交换正负SRB测度，并由此建立局部刚性、改进经典定理并揭示Teichmüller空间的分层结构。
Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one
该研究提出基于匹配函数的Anosov微分同胚光滑刚性新方法，证明三维环面线性双曲自同构小邻域内极非代数余维一Anosov微分同胚光滑共轭等价于雅可比周期数据匹配，并推广至高维情形。
We obtain smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. Among other results, we prove that a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ and its linearization are smoothly conjugated since they have the same periodic data. Assuming that for a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ every point is regular (in Oseledec's Theorem sense), then we obtain again smooth conjugacy with its linearization. We also obtain some results on local rigidity of linear Anosov endomorphisms of $d-$torus, where $d…
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.
Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^{3}$
研究在 $\mathbb{T}^{3}$ 上的 Anosov diffeomorphisms 之间 $C^0 $ 共轭时，若周期数据匹配则共轭必为光滑的性质。
We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $η\in C^{\reg}(M,\Diff^r(N))$ for a compact ma…
$C^0$-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。
Starting from any pseudo-Anosov map $φ$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of Derived from pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus. This is done by perturbing $φ$ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure $μ$, supported by the non-trivial unique minimal component of the stable foliation of $f$, with respect to which $f$ is mixing. In the process, we construct a uniquely ergodic Ge…
Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity
本文研究二维环面上Anosov自同态的光滑共轭，通过叶层正则性给出充分条件，并刻画了特殊自同态与其线性化的共轭关系。
Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+ν}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same co…
In this paper we consider the space of smooth conjugacy classes of an Anosov diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov diffeomorphism is the 2-torus, and Franks and Manning showed that every such diffeomorphism is topologically conjugate to a linear example, and furthermore, the eigenvalues at periodic points are a complete smooth invariant. The question arises: what sets of eigenvalues occur as the Anosov diffeomorphism ranges over a topological conjugacy class? This question can be reformulated: what pairs of cohomology classes (one determined by the expand…
The Teichmüller Space of a 3-Dimensional Anosov Flow
本文研究三维阿诺索夫流的泰希米勒空间，通过光滑轨道等价类将其表示为两个函数空间的乘积，并证明了三维可传递阿诺索夫流轨道等价空间的路径连通性，同时展示了其与微分同胚群的同伦等价关系。
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。
On Lyapunov exponents properties of special Anosov endomorphisms on $\mathbb{T}^d$
本文比较 $\mathbb{T}^d$（$d\geq 3$）上特殊 Anosov 自同态及其线性化的 Lyapunov 指数，刻画弱不稳定叶状结构的绝对连续性，并证明高维存在保守却不 Lipschitz 共轭于线性化的光滑例子。
In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on $\mathbb{T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.
In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $Γ$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $α$ be a smooth $Γ$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $ρ$ of $α$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $α$ and $ρ$, on a finite-index subgroup of $Γ$. If $α$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a coro…
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.
We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilmanifolds with irreducible exotic smooth structures.
On the Absence of Anosov Factors for DA Local Diffeomorphisms
本文构造了一类与环面双曲自同态同伦但不与任何Anosov局部微分同胚半共轭的局部微分同胚，并证明二维环面上具有扩张方向的非可逆部分双曲$C^1$局部微分同胚若半共轭于Anosov局部微分同胚，则其自身必为Anosov。
Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity
研究三维保体积阿诺索夫流，证明若两流拓扑共轭则共轭光滑，除非为常屋顶悬挂流；并推出曲面负曲率加权标记长度谱刚性等新结果。
Smooth rigidity for higher dimensional contact Anosov flows
本研究将匹配函数技术用于满足聚束假设的高维接触Anosov流，推广了3维刚性结论，证明其C⁰共轭可推出高阶光滑共轭，还可复现实双曲流形的标记长度谱刚性结果。
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.
We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
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。
Smooth Models of Fibered Partially Hyperbolic Systems
本文研究了纤维化部分双曲微分同胚在拓扑障碍消失且同调极小扩张主导纤维畸变时可同伦于具有 $C^{\infty}$-中心纤维化的系统，并探讨了 Anosov 微分同胚到丛的光滑提升障碍。
We consider smooth partially hyperbolic volume preserving Z^k actions on smooth manifolds, with uniformly compact center foliation. We show that under certain irreducibility condition on the action, bunching and uniform quasiconformality conditions, the action is a smooth fiber bundle extension of an Anosov action, or the center foliation is pathological. We obtain several corollaries of this result. For example, we prove a global dichotomy result that any smooth conservative circle extension over a maximal Cartan action is either essentially a product of an action by rotations and a linear An…
We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livshitz Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the "correct" direction in top degree. In Part I we study our "base case": geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invari…
We show how an object from the combinatorially geometric version of the analytical number theory, namely geometric continued fractions, appears in the classical smooth dynamics, namely in the problem on the topological classification of Anosov diffeomorphisms of tori.
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. Moreov…
We study the Ruelle and Selberg zeta functions for $\Cs^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being: (a) for $\Cs^\infty$ flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g. geodesic flows on manifolds of negative curvature better than $\frac 19$-pinched) the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our argume…
In this paper, we consider codimension one Anosov actions of IR^k, k ? 1, on closed connected orientable manifolds of dimension n+k with n? 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of IR^k admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k=1, we show that, under some assumptions abou…
The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimen…
In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute…
We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a finite cover of such an action is smoothly conjugate to an action by toral automorphisms.
Rigidity of stable Lyapunov exponents and integrability for Anosov maps
本文研究了环面上不可约 Anosov 映射的稳定 Lyapunov 指数刚性与可积性，证明了一维及特定高维稳定丛下可积性与 Lyapunov 指数相等的等价关系，并得出拓扑共轭在稳定丛上光滑的结论。
We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to …
We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let $L\colon\mathbb T^3\to\mathbb T^3$ be a hyperbolic automorphism of the 3-torus with real spectrum and let $f$ be a $C^1$ small perturbation of $L$. Then $f$ is smoothly ($C^\infty$) conjugate to $L$ if and only if obstructions to $C^1$ conjugacy given by the eigenvalues at periodic points of $f$ vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya for conformal automorphisms this completes the local rigidity program for hyperbolic automorphis…
Path Connectivity of Anosov Metrics on Surfaces
研究了闭曲面上Anosov度量的路径连通性，通过共形变形保持Anosov性质并连接正负曲率度量。
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundle…
Regularity of Non-stationary Stable Foliations of Toral Anosov Maps
本文研究满足共同锥条件的二维及高维环面Anosov映射序列，证明范数一致有界时非平稳稳定叶层的正则性结论，推广经典结果并给出反例，为Sturmian哈密顿算子谱维数猜想提供研究基础。
For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $μ$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $μ$, we strengthen this to show that the group of $μ$-preserving diffeomorphism is virtually cyclic.
Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts.
Fractal Weyl law for the Ruelle spectrum of Anosov flows
本文研究 Anosov 流上 Ruelle 共振的分布规律，证明共振密度满足分形 Weyl 定律，并利用基于适配度量的波包变换构造各向异性 Sobolev 空间以处理非光滑动力学分布，同时给出共振波前集和转移算子可逆性的更精确结果。
The Livšic equation on differential forms over Anosov flows and applications
本文研究Anosov流上微分形式Livšic方程的可解性，证明非对称条件下中间阶方程在连续范畴中唯一可解，并应用于保体积非对称Anosov流，给出其强丛和唯一可积与拓扑共轭于Anosov微分同胚悬挂流的等价刻画。
Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow
研究在C^s拓扑下接近体积保持3维阿诺索夫流时间-1映射的微分同胚，证明其动力学性质，包括混合性、唯一物理测度及无周期点的示例。
Marked length spectrum rigidity for Anosov surfaces
本文证明了亏格≥2的光滑闭可定向曲面上，具有相同标记长度谱和Anosov测地流的两个度量必通过同位于恒同的等距映射等距，方法结合微局部分析与复曲线几何。
We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume pre…
On global rigidity of transversely holomorphic Anosov flows
本文研究了横向全纯Anosov流的全局刚性，证明了其在特定条件下可积分到复流形，并给出了拓扑transitive流的轨道等价结果。
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