Totally non-symplectic Anosov actions on tori and nilmanifolds
David Fisher, Boris Kalinin, Ralf Spatzier
我们证明了环面和流形上高秩阿贝尔群的充分不可约全非辛Anosov作用与仿射作用光滑共轭。
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.
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Anosov automorphisms on certain classes of nilmanifolds
本文给出了与图$(k \geq 3)$相关联的$k$ -步诣零流形允许Anosov自同构的一个充要条件.我们还证明了Anosov自同构的不存在性的某些类的2-步骤和3-步骤的nilmanifold。
Global Rigidity of Higher Rank Anosov Actions on Tori and Nilmanifolds
我们证明了环面和流形上高秩阿贝尔群的充分不可约Anosov作用与仿射作用光滑共轭。
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.
Anosov diffeomorphisms on nilmanifolds up to dimension 8
经过三十多年，唯一已知的Anosov同构的例子是infranilmanifold的双曲自同构。同样重要的是要注意，Anosov自同构的存在是infranilmanifold上的一个非常强的条件。任何Anosov自同构决定了相应格的Mal'cev完备化的（有理）李代数的自同构，该格是双曲的和幺模的。这两个条件加在一起足够强，使这种有理幂零李代数（称为Anosov李代数）非常杰出的对象。本文对维数小于等于8的Anosov李代数进行了分类，并对维数小于等于8的Anosov李代数中的一个Anosov李同态进行了分类。作为推论，我们得到：如果一个维数n<9的infranilmanifold允许Anosov同态f，并且它不是一个环面或紧平坦流形（即被环面覆盖），则n=6或8，并且f的签名必然分别等于{3，3}或{4，4}。我们不得不研究一套所有的合理形式同构的许多真正的李代数，这是一个主题本身，它是治疗的一节完全独立的其余部分的文件。
Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restr…
Anosov Lie algebras and algebraic units in number fields
本文将数域中代数单位的基本性质应用于相应的Anosov李代数，研究了允许Anosov自同构的流形。我们确定Anosov李代数存在的障碍。13维Anosov李代数的情况下，作为一个说明的技术。（ii）每个8维Anosov李代数及其3维或5维导出代数都包含一个交换因子.
Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
次幂零流形是由单连通幂零李群$G$的商空间$Γ\backslash G$构成的流形，其中$Γ$是通过所谓的仿射映射在$G$上适当地不连续和共紧作用的离散群。流形$Γ\backslash G$被称为李群$G$的模型。这类流形被证明是唯一允许Anosov同构的闭流形。然而，这是远远不明显的，这些infra-nilmanifold实际上不承认Anosov naromorphism。本文对自由$c$ -步幂零李群上的次幂零流形完全解决了这个问题。
Examples of Anosov diffeomorphisms
我们给出了一个简单的过程来构造明确的例子，承认Anosov同胚的nilmanifold，并表明，一个合理的分类同胚（或甚至高达可分性），这样的nilmanifold将是不可能的。
We study 9 and 10-dimensional Anosov Lie algebras, by using the properties of very special algebraic numbers. We classify k-step complex Anosov Lie algebras for $k>2$ and in the two step case, we give an example in each possible type, or we give a non-existence result.
It is conjectured that every manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov diffeomorphism. The main theorem of this article gives a general method for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we give counterexamples to a corollary of the classification of low-dimensional nilmanifolds with Anosov diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about …
On the Topological Transitivity of Interval Extensions of Anosov Diffeomorphisms
本文证明了加厚幂零流形上一类解析部分双曲微分同胚（由Anosov微分同胚的区间扩张构造的斜积系统）的拓扑传递性，并提出了允许中心方向任意有限阶抛物退化的机制。
A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, whi…
Global rigidity of higher rank Anosov algebraic actions
我们证明了所有的$C^\infty$ Anosov $Z^r$ -行动的环面和nilmanifold没有秩一因子的行动，直到$C^\infty$共轭，行动的自同构。
Stable Lyapunov Spectrum Rigidity of Nilmanifold Endomorphisms
本文研究幂零流形阿诺索夫映射，在非可逆和不可约条件下，证明不变不稳定丛、拓扑共轭与常数周期稳定李雅普诺夫指数三者等价，即稳定李雅普诺夫谱刚性。
We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…
Expanding maps and non-trivial self-covers on infra-nilmanifolds
闭流形上的每一个扩张映射都拓扑共轭于亚零流形上的一个可扩映射，但不是每一个亚零流形都有一个可扩映射。本文给出了含有可扩映射的亚幂零流形的一个完整的代数刻划。我们表明，就像在Anosov同态的情况下，一个可扩映射的存在性只依赖于合理的holonomy表示的infra-nilmanifold。对于具有非平凡自覆盖的次幂零流形也给出了类似的刻画，这对应于确定哪些几乎-比伯巴赫群是余霍普夫群。这些结果提供了许多新的例子，下nilmanifold没有非平凡的自覆盖或扩张映射。
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Γ$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.
Partially hyperbolic lattice actions on 2-step nilmanifolds
本文研究高阶秩格在2步幂零流形上包含部分双曲元素的作用，在技术假设下证明其必为仿射映射，从而推广了Brown等人的刚性结果。
Algebraic Anosov actions of Nilpotent Lie groups
本文对闭流形上幂零李群的代数Anosov作用进行了分类，推广了P. Tomter的结果.我们证明了它们都是零-悬挂在零流形上Z^k的Anosov作用或（修改的）Weyl chamber作用上。我们检查的有效性广义Verjovsky猜想在这个代数背景下。指出了代数Anosov作用与一般实李群中的Cartan子代数之间的密切关系。
On the density of Birkhoff sums for Anosov diffeomorphisms
本文证明，对于幂零流形上的Anosov微分同胚，若沿周期轨道的Hölder连续观测的Birkhoff和存在正负两侧的值，则所有周期点的Birkhoff和之集在实数中稠密。
Invariant tori for a class of affined Anosov mappings with quasi-periodic forces
本文证明对一类带拟周期力的仿射Anosov映射，存在唯一仅依赖于系统的正整数m，使度为m的不变环面数量的指数增长率等于拓扑熵。
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…
Let $f$ be an Anosov diffeomorphism of the $n$-dimensional torus ${\mathbb{T}}^n$ and $τ$ a continuous self-mapping of ${\mathbb{T}}^n$ commuting with $f$. We prove that $τ$ is surjective if and only if the restriction of $τ$ to each homoclinicity class of $f$ is injective.
Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov
我们证明了部分双曲型超同构的$d$维环面同位素的Anosov超同构，其中的合痕包含在一组部分双曲型超同构，是动态相干的。此外，我们还证明了一个\textit{global stability result}，即每个上述的部分双曲双同态是线性双曲双同态的\textit{leaf-conjugate}。作为结果，我们得到的内在遍历性和测量等价的部分双曲型同构与一维中心方向是同位素Anosov同构通过这样的路径。
Geometric Continued Fractions as Invariants in the Topological Classification of Anosov Diffeomorphisms of Tori
我们展示了如何从组合几何版本的分析数论，即几何连分数，出现在经典的光滑动力学，即在问题上的拓扑分类Anosov的环面同态。
Weakly hyperbolic actions of Kazhdan groups on tori
本文研究了弱双曲群作用的遍历性和刚性性质。受经典的Anosov同态定理的启发，我们得到了两个主要结果：第一，闭流形上的所有C^2保体积弱双曲作用是遍历的。这一结果将Anosov关于Anosov群同态遍历性的经典定理推广到更广泛的弱双曲群作用类。为了说明第二个主要结果，假设A表示高秩格（如SL（n，Z），n>2）在环面上的C^2保体积双同态的弱双曲作用。如果A被环面的万有覆叠空间R^n的作用覆盖，则A在环面的第一同调上诱导弱双曲表示。这一结果可以被看作是曼宁的贡献Anosov Mannos的弗兰克斯/曼宁分类的一个模拟。托丽。我们得出的结论与Zimmer的程序分类遍历体积保持行动的高秩格。这两个结果都依赖于Rauch和Talyor的一个新的正则性结果，该结果将相对于绝对连续叶理的叶子切向测量的索博列夫函数类与全局索博列夫函数类联系起来。
We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gouëzel and Liverani. By extending our previous work in [8], we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key applicatio…
Transitive centralizers and fibered partially hyperbolic systems
本文证明了光滑微分同胚中心化子的刚性结果，分类了具有传递中心化子的微分同胚，并在幂零流形上 Anosov 微分同胚等距扩张的扰动及 3 维部分双曲系统中给出了中心化子的 Lie 群结构与分类。
In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on th…
Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori
设$L$是$\mathbb T^d$，$d\ge3$的双曲自同构.研究了$L$的小$C^1$ -邻域$\mathcal U$中的光滑共轭问题.主要结果建立了具有相同周期特征值的两个Anosov系统共轭性的$C^{1+ν}$正则性。我们假设这些系统是$C^1$ -接近于具有简单实谱的不可约线性双曲自同构$L$，并且它们在某些中间叶理上满足自然传递性假设.我们详细介绍了在$\mathbb T^4$上的两个Anosov系统的de la Llave的例子，具有相同的常数周期特征值数据，只有Hölder共轭。我们表明，这些例子用尽所有可能的方法来扰动$C^{1+ν}$共轭类，而不改变周期性的特征值数据。此外，我们推广这些例子，以大多数的可约toral自同构，以及某些产品的$\mathbb T^4$ $C^1$ -接近原来的例子。
Pre-Lagrangian tori transverse to an Anosov flow
针对Anosov流横截嵌入环面是否在$\mathbb{R} \times M$中产生拉格朗日量的问题，研究发现存在拓扑障碍导致答案通常否定，仅当诱导叶状结构无平行紧叶时答案为肯定。
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 $\{…
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…
Tori with hyperbolic dynamics in 3-manifolds
设M是闭的可定向不可约3-流形，f是M上的一个单同态.我们称一个嵌入的2-环面T为Anosov环面，如果它是不变的，并且f在π_1（T）上的诱导作用是双曲的。我们证明了只有少数不可约3-流形允许Anosov环面：（1）3-环面，（2）-id的映射环面，（3）2-环面的双曲自同构的映射环面。这在3-流形的部分双曲动力系统的上下文中具有后果：如果存在不变的中心不稳定叶理，则它不可能具有紧叶[19]。这导致了具有一维中心丛的非动态相干部分双曲单同态的第一个例子[19]。
Geometric Anosov flows of dimension five with smooth distributions
我们分类五维Anosov流光滑分解，这是另外的横向辛。直到有限覆盖和特殊的时变，我们精确地发现了四维托里斯的辛双曲自同构的悬置，以及三维双曲流形的测地流。
We prove that toric symplectic manifolds admit Hamiltonian pseudo-rotations with a finite, and in a sense minimal, number of ergodic measures. The set of ergodic measures of these pseudo-rotations consists of the measure induced by the symplectic volume form and the Dirac measures supported at the fixed points of the torus action. Our construction relies on the conjugation method of Anosov and Katok.
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…
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。
Measure of maximal entropy for minimal Anosov actions
本文研究闭流形上最小Anosov作用的极大熵测度，证明其符合Ruelle-Taylor共振理论，并得出Bowen型公式及周期环面计数结果。
Some ergodic and rigidity properties of discrete Heisenberg group actions
研究离散Heisenberg群$\H$的光滑作用的遍历性和刚性性质。建立了任意$C^\infty$紧致黎曼流形$M$的切空间的李雅普诺夫指数分解，并证明了中心元的所有李雅普诺夫指数均为零.证明了如果一个$\H$群作用包含一个Anosov元，则在一定条件下，该元的中心元是有穷级的.特别地，在任何流形上都不存在忠实的余维一Anosov Heisenberg群作用，在环面上也不存在忠实的余维二Anosov Heisenberg群作用。此外，我们表现出光滑的局部刚性较高的秩遍历$\H$行动的toral自同构，使用推广的KAM（Kolmogorov-Arnold-Moser）迭代计划。
Local rigidity of higher rank partially hyperbolic algebraic actions
本研究结合表示论、调和分析与KAM迭代，完整解决高阶部分双曲代数作用局部分类问题，证明多类阿贝尔遍历代数作用的$C^\infty$局部刚性，首次得到非可达部分双曲作用及带非阿贝尔幂零根基扭空间的相关新结果。
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