Yong Fang
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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On uniformly quasiconformal Anosov diffeomorphisms with two dimensional distributions
本研究证明具有二维不稳定分布的传递一致$u$-拟共形Anosov微分同胚有全局稳定和乐，由此去除Kalinin-Sadovskaya定理的额外假设，推导出该类微分同胚与下环面仿射Anosov微分同胚$C^\infty$共轭。
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.
We prove global smooth classification results for TNS totally Anosov Z^k actions on general compact manifolds, under each one of the following conditions: joint integrability, resonance-free or Lyapunov pinching condition. Unlike the previous results, we do not require any uniform quasiconformality or pinching condition of action elements on coarse Lyapunov distributions, nor do we have any restriction on the dimension of coarse Lyapunov distributions. The main novelty is in proving a new standard form of the derivative cocycle for any TNS totally Anosov Z^k action on general manifold. A main …
We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow $ψ_{t}$ of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volume-preserving flow that is smoothly orbit equivalent to $ψ_{t}$. Our techniques apply more generally to give an essentially complete classification of smooth, volume-preserving, dynamically coherent partially hyper…
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.
On quasiconformal non-equivalence of gasket Julia sets and limit sets
本文研究Julia集与极限集的拟共形不等价性，证明了任何Julia集与Apollonian gasket以及几何有限Kleinian群的gasket极限集拟共形不同。
We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is $3$-dimensional.
Self orbit equivalences on the Anosov flows on Dehn surgeries on the figure-eight knot
本文证明在八字结上大次数Dehn手术所得流形上，标准Anosov流的自轨道等价群同构于$\mathbb{Z}_2\oplus\mathbb{Z}_2$，且映射类群每个元素均可由自轨道等价实现。
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-…
We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples constructed in a paper by Bonatti, Gogolev, Hammerlindl and the third author belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations.…
Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows
本文推广了Goodman手术操作，证明对传递伪Anosov流进行水平Goodman手术可产生几乎等价的流，并得出相关等价与结构稳定性结果。
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。
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 …
Almost equivalence of suspension Anosov flows
本文通过构造 genus 一的 Birkhoff 截面，证明了由正迹超几何矩阵生成的悬吊阿诺索夫流之间几乎等价，并给出了距离的显式界。
We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $π_1(M)$-orbit of tree gives a complete invariant of orbit equivalence c…
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…
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。
Markovian families for pseudo-Anosov flows
本文提出用双叶平面上的 Markov 族、几何类型与循环等有限组合对象，对三维流形上的伪 Anosov 流进行轨道等价分类。
Exotic codimension one Anosov flows
本文在闭双曲三维流形上的某些圆丛中构造了外斯阿诺夫流，构造了Verjovsky猜想的反例，并利用伪外斯阿诺夫拟测地流相关的Cannon-Thurston映射，得到了某些四维流形上无穷多个轨道不等价的外斯阿诺夫流。
On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry
该论文通过接触几何证明闭双曲三维流形上Anosov流模同痕轨道等价类数量有限，还给出其数量显式界，并控制非双曲三维流形上Anosov流的动力学。
A uniqueness theorem for transitive Anosov flows obtained by gluing hyperbolic plugs
本文证明了通过粘合 hyperbolic plugs 构建的传递 Anosov 流的轨道等价类与粘合映射的精确选择无关，其唯一性结果基于一种编码程序并遵循 T. Barbot 引入的策略。
The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem of counting periodic orbits in an amenable cover $X$ to counting in a maximal abelian subcover $X^{\mathrm{ab}}$. In this way, we obtain an equivalence for the Gurevič entropy: $h(X)=h(X^{\mathrm…
We introduce a notion of abelian cohomology in the context of smooth flows. This is an equivalence relation which is weaker than the standard cohomology equivalence relation for flows. We develop Livshits theory for abelian cohomology over transitive Anosov flows. In particular, we prove an abelian Livshits theorem for homologically full Anosov flows. Then we apply this theorem to strengthen marked length spectrum rigidity for negatively curved surfaces. We also present an application to rigidity of contact Anosov flows. Some new results on homologically full Anosov flows are also given.
Anosov flows with the same periodic orbits
本文研究闭三维流形上传递伪Anosov流，当轨道空间含扇贝区域树时，分类具有相同自由同伦数据的流，并构造非轨道等价的显式例子。
Orbital equivalence classes of finite coverings of geodesic flows
本文研究闭 3-流形上测地流有限覆盖的 Anosov 流分类，证明当覆盖指数 n 为奇数时只有一个轨道等价类，n 为偶数时恰有两个轨道等价类。
Given a (transitive or non-transitive) Anosov vector field $X$ on a closed three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$ along two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $Λ_1,\dots,Λ_m$ of the connected components $V_1,\dots,V_m$ of $M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each $Λ_i$ is a compact invariant locally maximal transitive set for $X$, 2, the collection $\{…
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov f…
The path closure of figure eight knot complement space, $N_0$, supports a natural DA (derived from Anosov) expanding attractor. Using this attractor, Franks-Williams constructed the first example of non-transitive Anosov flow on the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map along their boundaries, named by Franks-Williams manifold. In this paper, our main goal is to classify expanding attractors on $N_0$ and non-transitive Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding attractor is the unique expanding attractor supported by $N_…
The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen's version of the measure of maximal entropy. These results ar…
Topological invariance of Liouville structures for taut foliations and Anosov flows
研究闭定向三维流形上紧叶状结构相关的刘维尔结构的拓扑不变性，通过特定方法证明其为拓扑不变量，还涉及阿诺索夫流弱叶状结构及轨道等价情况，证明结合平滑方案和沃格尔结果，附录用于构造新例子助于分类。
Rigidity of the unstable foliation
本文研究了transitive Anosov流在3流形上不稳定性叶的刚性问题，证明了不稳定性叶等价性下流的拓扑共轭性，扩展了Ratner关于负曲率表面水平流的刚性定理。
Reconstructing flows from the orbit space
本文给出简单条件，使双叶平面上的群作用源于伪阿诺索夫流或阿诺索夫流的诱导作用，并用于轨道等价唯一性证明及从织机空间恢复扩张流。
Smooth orbit equivalence rigidity for dissipative geodesic flows
本文研究了耗散测地流在光滑轨道等价下的刚性性质，证明了具有负 thermostat 曲率的 Gaussian thermostats 在同伦于恒等映射下，其背景度量通过光滑微分同胚同伦于恒等映射而 conformally 等价。
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