Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov
Carlos F. Álvarez
证明了具有紧致二维中心叶理的环面部分双曲系统遍历极大熵测度的双曲性。
我们证明了一类具有紧致二维中心叶理的$\nmathbb{T}^{d}$部分双曲微分同胚的遍历极大熵测度的双曲性。
We prove the hyperbolicity of ergodic maximal entropy measures for a class of partially hyperbolic diffeomorphisms of $\mathbb{T}^{d}$, which have a compact two-dimensional center foliation.
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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 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.
We prove that if $f$ is a partially hyperbolic diffeomorphism on 3-torus $\mathbb{T}^3$ and isotopic to Anosov, then $f$ is chain transitive.
We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.
The purpose of this article is to obtain dynamically coherence of partially hyperbolic diffeomorphisms in certain classes of Anosov diffeomorphisms on nilmanifolds, extending a result due to T. Fisher, R. Potrie and M. Sambarino [FPS] on the torus.
We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.
In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on $\mathbb{T}^4$, with 2-dimensional center foliation. To do so we propose to study the disintegration of measures along 1-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.
Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are invariant under the Hamiltonian flow. We show that a monotone Hamiltonian flow without conjugate point is Anosov if and only if these distributions are transversal. We also show that if the reduced curvature of the Hamiltonian system is non-positive, then the flow is Anosov if and only if the reduced curvature is negative somewhere along each trajecto…
Hyperbolicity, transitivity and the two-sided limit shadowing property
本文刻画了闭流形上具有双侧极限伪轨跟踪性质的微分同胚，证明其C^1内部及通有情形均为传递Anosov微分同胚，从而将Anosov传递性猜想归结为该性质与Anosov性质的等价性。
Invariance of entropy for maps isotopic to Anosov
研究同痕于Anosov的映射的熵不变性，证明具有简单中心丛、诱导作用双曲的部分双曲微分同胚的拓扑熵恒定，无简单性条件时构造出鲁棒传递反例。
Partial hyperbolicity and pseudo-Anosov dynamics
本文证明容许部分双曲微分同胚的双曲 3 维流形必容许 Anosov 流，并利用中心分支叶理和坍缩 Anosov 流完整分类相关微分同胚。
If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism $A$, it is topologically conjugate to $A$.
Generalized foliations for homeomorphisms isotopic to a pseudo-Anosov homeomorphism: a geometric realization of a result by Fathi
本研究给出Fathi一个结果的新几何证明，证明闭曲面上同痕于伪Anosov同胚的同胚都存在与伪Anosov同胚叶状结构性质相似的稳定、不稳定不变划分。
We analyze a class of deformations of Anosov diffeomorphisms: these $C^0$-small, but $C^1$-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum. We also establish expansiveness around those measures. This class of deformations contains many of the known nonhyperbolic robustly transitive diffeomorphisms. In particular, we show that it includes a class of no…
Entropy stability for diffeomorphisms isotopic to Anosov on $\mathbb{T}^d$
本文研究d维环面上与线性Anosov自同构同痕的部分双曲微分同胚的拓扑熵，证明一类DA映射的拓扑熵与线性映射一致，得到低振荡势的平衡态唯一性等结果，还构造出拓扑熵严格大于线性部分的动态相容微分同胚。
We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M, is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in [6].
We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.
This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.
Hyperbolicity of complements of orbits in Anosov flows
该研究证明三维Anosov流在满足稳定与不稳定叶状结构可定向时，其满周期轨道补集为双曲流形，还明确了可定向条件的必要性，并给出悬挂流与测地流的相关结果。
Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities
证明曲面同胚是cw_F-双曲的当且仅当它是奇点仅为脊柱（1-叉）的伪阿诺索夫同胚，并分类至拓扑共轭：要么共轭于环面上的阿诺索夫自同构，要么共轭于球面上的标准超椭圆商。
A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures
在保守部分双曲三维环境中，针对三类微分同胚建立双曲性与非双曲遍历测度的二分法，方法无需扰动，还独立于保守假设构造了非双曲遍历测度，可应用于更多情境。
We consider C1 Anosov diffeomorphisms on a compact Riemannian manifold. We define the weak pseudo-physical measures, which include the physical measures when these latter exist. We prove that ergodic weak pseudo-physical measures do exist, and that the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak*-closed convex hull of the ergodic weak pseudo-physical measures. In brief, we give in the C1-scenario of uniform hyperbolicity, a characterization of Pesin's Entropy Formula in terms of physical-like properties.
In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on $\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point.
Bi-contact plugs and dynamical hyperbolicity
该研究发展三维双接触塞理论，构造Anosov流等结构稳定流，回答了Béguin-Bonatti-Yu的问题，拓展了相关构造并揭示了双接触塞的Liouville结构性质。
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows wher…
Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups
利用左一致性，研究局部紧群连续自同态的阴影性，对李群给出无穷小刻画：自同态有阴影性当且仅当其微分是双曲的，并推导出拓扑扩张、正扩张、拓扑Anosov等性质的等价性。
Chaos on surfaces and beyond: a new notion of dynamical hyperbolicity
提出强正递归性作为均匀双曲性的推广，证明所有正熵光滑曲面微分同胚满足该性质，并确保指数混合等定量性质，同时探讨高维情形。
Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy
本文证明具有联合连续稳定/不稳定叶层的cw-双曲同胚满足周期阴影性质，且在拓扑混合时满足周期规范性质，并讨论将Bowen技术应用于获得最大熵测度的困难，给出Walter伪Anosov微分同胚和Sierpiński地毯上同胚的唯一最大熵测度。
We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.
We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the exponents in the center bundle vanish. This establishes in full a result announced by R. Mañé in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.
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。
Pseudo-Anosov surfaces and Dynamics in 3-manifolds
本文确定了闭可定向3-流形允许伪Anosov曲面的条件，并在不可约情形下证明自同胚同痕于新引入的“部分伪Anosov”同胚。
Measures of maximal entropy that are SRB
本文研究$T^3$上DA-微分同胚的最大熵测度的SRB性质，给出其为SRB测度的充要条件（正Lyapunov指数和在支撑集周期轨道上与线性Anosov映射一致），并指出该刚性结果弱于Katok猜想且附示例。
We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in whi…
In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries, constructed and described explicitly by Benoist. These manifolds are Finsler geometries which have isometrically embedded flats, but also some hyperbolicity and an explicit geometric structure. We prove the geodesic flow of the quotient is topologically mixing and satisfies a nonuniform Anosov closing lemma, with applications to entropy and orbit counting. We also prove entropy-expansiveness for the geodesic flow of any compac…
We prove that C^1-robustly transitive diffeomorphisms on surfaces with boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces with boundary which are C^k-robustly transitive, with k greater or equal than 2. This class of diffeomorphisms are examples where a version of Palis' conjecture on surfaces with boundary, about homoclinic tangencies and uniform hyperbolicity, does not hold in the C^2-topology. This follows showing that blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become C^2-robustly topologically mixing diffeomorphisms on a surfaces with b…
Minimality of Strong Foliations of Anosov and Partially Hyperbolic Diffeomorphisms
本文研究了部分双曲微分同胚扩张不变叶状结构的极小性，引入了在$C^1$扰动下稳健的s-横截性，并证明了在弱扩张条件下强不稳定叶状结构的稳健极小性及其在环面和唯一Gibbs测度上的应用。
Hyperbolic isometries of the fine curve graph of higher genus surfaces
该研究针对闭双曲曲面上同痕于恒等映射的同胚，证明了其在精细曲线图上的双曲作用、相对于有限不变集同痕于pseudo-Anosov映射、遍历同调旋转集非空内部这三者的等价性。
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。
Periodic points of endperiodic maps
本文研究无限型曲面上无环端周期映射的周期点计数，证明纺伪Anosov及特定Handel–Miller映射在高周期时于同伦类内最小化周期点数，并通过反例说明低周期情形不然。
Essential factors for surface homeomorphisms
针对亏格至少为1的闭可定向曲面上同痕于伪阿诺索夫自同胚的自同胚，通过坍缩万有覆盖中迭代直径一致有界的极大连续统构造单调因子，该因子具强混沌特征且可保面积，其极大熵测度为全支撑伯努利测度。
The amount of nonhyperbolicity for partially hyperbolic diffeomorphisms
本研究针对一类具有一维中心的部分双曲微分同胚，分析其中心李雅普诺夫指数水平集的熵，构造零指数水平集并证明其熵的正最大值与有限时间指数一致收敛性，涵盖多种鲁棒传递系统包括动力学不一致的反常例子。
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