Henk Bruin
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.
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We develop a thermodynamic formalism for a class of diffeomorphisms of a torus that are "almost-Anosov". In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium states for a collection of non-Hölder continuous geometric potentials over almost Anosov systems with an indifferent fixed point, as well as prove exponential decay of correlations and the central limit theorem for these equilibrium measures.
The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.
We present an example of a C1 Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.
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.
We establish sharp bounds on the mixing rates of a class of two dimensional non-uniformly hyperbolic symplectic maps. This provides a primer on how to investigate such questions in a concrete example and, at the same time, it solves a controversy between previous rigorous results and numerical experiments.
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any vo…
Let $M$ be a closed $3$-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.
This paper is part 1 of a series of papers culminating in the result that measure preserving diffeomorphisms of the disc or 2-torus are unclassifiable. It also addresses another classical problem: which abstract measure preserving systems are realizable as smooth diffeomorphisms of a compact manifold? The main result of this paper gives new symbolic representations of the Anosov-Katok diffeomorphisms.
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。
From Markovian actions to Anosov flows
研究双叶平面上群作用与拓扑阿诺索夫流的关系，核心方法是证明群作用源于流的充要条件为保定向强马尔可夫作用，主要贡献是建立条件及证明矩形可提升为流的马尔可夫分割。
Realization of Anosov Diffeomorphisms on the Torus
本文证明环面上保面积阿诺索夫微分同胚的几何势压力函数集合与线性阿诺索夫自同构的Hölder势压力函数集合相同，并据此否定回答了Rodriguez Hertz关于相同压力函数是否蕴含$C^\infty$共轭的$C^{1+α}$版本问题。
We prove transitivity for volume preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^3$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.
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。
In this paper, we prove that for $\mathcal{C}^1$ generic volume-preserving Anosov diffeomorphisms of a compact Riemannian manifold, Livšic measurable rigidity theorem holds. We also prove that for $\mathcal{C}^1$ generic volume-preserving Anosov flows of a compact Riemannian manifold, Livšic measurable rigidity theorem holds.
We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume preserving case.
The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable foliation both to be $C^{1+Lip}$, and then following the same argument as in the paper one can conclude the proof. But this extra hypothesis seems implying that the flow has a contact structure. Then it will be only a result which improves the Foulon's proof on contract structure for $C^\infty$ regularity to $C^2$, not so interest.
We show a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.
On homeomorphisms of three-dimensional manifolds with pseudo-Anosov attractors and repellers
本文研究带有限曲面吸引/排斥子的三维流形保定向同胚，证明一类伪阿诺索夫型同胚的环境Ω-共轭性及其中中心化子的构成性质。
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.
In this paper we focused our study on Derived From Anosov diffeomorphisms (DA diffeomorphisms ) of the torus $\mathbb{T}^3,$ it is, an absolute partially hyperbolic diffeomorphism on $\mathbb{T}^3$ homotopic to an Anosov linear automorphism of the $\mathbb{T}^3.$ We can prove that if $f: \mathbb{T}^3 \rightarrow \mathbb{T}^3 $ is a volume preserving DA diffeomorphism homotopic to linear Anosov $A,$ such that the center Lyapunov exponent satisfies $λ^c_f(x) > λ^c_A > 0,$ with $x $ belongs to a positive volume set, then the center foliation of $f$ is non absolutely continuous. We construct a new…
We answer a question of Burns and Wilkinson, showing that there are open families of volume-preserving partially hyperbolic diffeomorphisms which are accessible and center bunched and neither dynamically coherent nor Anosov. We also show in the volume-preserving setting that any diffeomorphism which is partially hyperbolic and Anosov may be isotoped to a diffeomorphism which is partially hyperbolic and not Anosov.
Decay of correlations on Abelian covers of isometric extensions of volume-preserving Anosov flows
研究了封闭流形阿贝尔覆盖上保体积阿诺索夫流的各向同性扩展的关联函数的渐近衰减行为。
Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.
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。
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。
If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving $CC^{1+a}$, $a>0$, Anosov diffeomorphism without the usual Hopf argument.
Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms
本文研究二维环面上保面积Anosov微分同胚的有限周期Jacobian数据刚性，通过加权holonomy构造与Bowen等分布的一致有效版本，证明大周期数据匹配可推出指数级近似光滑共轭。
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。
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. One important corollary is that if F is a Reebless finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classe…
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 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 …
From pseudo-Anosov flows on graph manifolds to totally periodic flows
研究显示图流形上的伪Anosov流可通过构造部分Birkhoff截面，几乎等价于完全周期性伪Anosov流或悬垂Anosov流，结合先前工作证明图流形上的Transitive Anosov流几乎等价于悬垂Anosov流。
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.
Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessi…
We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.
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…
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…
Transverse surfaces and pseudo-Anosov flows
本文刻画了传递伪阿诺索夫流下的（几乎）横截曲面，证明了无完美契合时瑟斯顿范数最小化曲面几乎横截，并回答了Cooper-Long-Reid的问题。
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…
Oseledets regularity functions quantify the deviation of the growth associated with a dynamical system along its Lyapunov bundles from the corresponding uniform exponential growth. Precise degree of regularity of these functions is unknown. We show that for every invariant Lyapunov bundle of a volume preserving Anosov flow on a closed smooth Riemannian manifold, the corresponding Oseledets regularity functions are in $L^p(m)$, for some $p > 0$, where $m$ is the probability measure defined by the volume form. We prove an analogous result for essentially bounded cocycles over volume preserving A…
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…
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。
We prove that a C1-generic volume preserving diffeomorphism has a symbolic extension if and only if this diffeomorphism is partial hyperbolic. This result is obtained by means of good dichotomies. In particular, we prove Bonatti's conjecture in the volume preserving scenario. More precisely, in the complement of Anosov diffeomorphisms we have densely robust heterodimensional cycles.
We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In a previous paper [Ba-Fe1] we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. In the present paper we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow…
We prove that in a $C^1$-open and $C^k$-dense set of some classes of $C^k$ Anosov flows all Lyapunov exponents have multiplicity 1 with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Hölder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Hölder-continuous potentials. In the proof, we use and prove perturbative results for jets of flows to modify eigenvalues of certain Poincaré maps and, using a Markov partition, apply the simplicity criterion of Avila and Viana.
Drilling veering triangulations with applications to pseudo-Anosov flows
本文开发了一种转向三角剖分钻孔算法，用于构造转向母三角剖分，应用于伪阿诺索夫流的闭轨道钻孔，为Ghys关于传递阿诺索夫流的猜想提供了计算证据。
Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$
本文证明了三维环面上保守 Anosov 微分同胚的 $C^1$ 邻域内，任意 $C^2$ 微分同胚要么强稳定与强不稳定丛联合可积，要么其所有全支撑 $u$-Gibbs 测度均为 SRB 测度。
Let f be a volume-preserving diffeomorphism of a closed C1 n-dimensional Riemannian manifold M: In this paper, we prove the equivalence between the following conditions: (a) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak shadowing property, (b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the limit weak shadowing property, (c) f is Anosov.
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…
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。
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…
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.
Proof of the Verjovsky Conjecture
本文证明了Verjovsky猜想，即维数大于三流形上的余维一Anosov流均拓扑等价于双曲环面自同构的悬挂流，并借助时间变换下的联合可积性给出更一般的推导。
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov's classical theorem on ergodicity of Anosov diffeomorphisms to the wider class of weakly hyperbolic group actions. To state the second main result, suppose that A denotes a weakly hyperbolic action of a higher rank lattice (such as SL(n,Z) for n>2) by C^2 volume preserving diffeo…
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 $\{…
Let f be a volume-preserving diffeomorphism of a closed C^\infty n-dimensional Riemannian manifold M: In this paper, we prove the equivalence between the following conditions: (a) f belongs to the C1-interior of the set of volume-preserving diffeoeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods. (b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods. (c) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms whic…
Construction of Anosov flows on fibered hyperbolic 3-manifolds
该研究证明了纤维化双曲3流形上存在丰富的Anosov流，并给出了构造方法及多个实例。
For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $Π$ which maps into the space of invariant distributions in $\cap_{u<0} H^{u}(\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s>0} H^{s}(\mathcal{M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}=SM$ of a compact manifold, we apply this theory to study questions related to $X$-ray transform on symmetric tensors on …
Completing Prelaminations
本文研究圆周上横向预层状结构对何时可完备化或实现为平面双叶状结构端点，给出统一刻画并涵盖伪 Anosov 流等奇异情形。
In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences. The sequences of maps need not correspond to typical points of a random dynamical system. The methods in the proof rely on the stability of compositions of hyperbolic dynamical systems. We introduce the notion of sequential conjugacies and prove that these vary in a Lipschitz way with respect to the generating sequences of dynamical systems. As a consequence, we prove sta…
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