Jorge Lauret
We give a simple procedure to construct explicit examples of nilmanifolds admitting an Anosov diffeomorphism, and show that a reasonable classification up to homeomorphism (or even up to commensurability) of such nilmanifolds would not be possible.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show by means of a counterexample that a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is not necessarily Anosov. The counterexample can bear higher smoothness up to $C^3$. Also we include a result from the 2006 Ph.D. thesis of T. Fisher: a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is Anosov itself provided that Holder exponents of the conjugacy and its inverse are sufficiently large.
Conformal hyperbolic distances and holonomies of Anosov diffeomorphisms
本文研究了实值Anosov微分同胚的传递性，通过共形双曲距离定义稳定/不稳定曲线的长度，证明其全局同伦性质。
We construct an example of planar Anosov diffeomorphisms without fixed points which is not topologically conjugate to a translation.
We show that there is no transitive Anosov diffeomorphism with the global product structure, which is homotopic to a product of pseudo-Anosov diffeomorphisms, on a product of two closed surfaces each of which has genus greater than or equal to two.
Let $M$ be an $n$-dimensional manifold supporting a quasi Anosov diffeomorphism. If $n=3$ then either $M={\mathbb T}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of ${\mathbb Z} ^6$. If $n=4$ then $Π_1(M)$ contains a copy of ${\mathbb Z} ^4$, provided that the diffeomorphism is not Anosov.
It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Under suitable hypothesis on Lyapunov exponents on the set of periodic points and the structure of the MME we get transitivity of $C^1-$Anosov diffeomorphisms.
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.
Weprovethattheasymptoticsofergodicintegralsalonganinvariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, are determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [GL] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1,e^{h_{top}} )$.
Quasi-Anosov diffeomorphism is a kind of important dynamical system due to R. Mane 1970s. In this paper, we present a criterion for such dynamics using ergodic theory.
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.
We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms.
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.
Codimension-one holomorphic Anosov diffeomorphisms
本文证明紧连通复流形上的余维一全纯Anosov微分同胚必双全纯共轭于复环面的双曲自同构，验证了Ghys猜想在余维一及复三维情形。
We study the effects that the constant periodic data condition have on topological entropy of Anosov diffeomorphisms. Under constant periodic data condition we prove that Anosov diffeomorphism has finitely many measures of maximal entropy and each one of them is absolutely continuous with respect to Lebesgue. From this, in the setting of $C^{\infty}-$Anosov diffeomorphisms satisfying constant periodic data, we provide a characterization of transitivity property via measures of maximal entropy.
We construct examples of $C^{1}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$ which are of Krieger type ${\rm III}_{1}$ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative $C^{1+α}$ Anosov diffeomorphisms have a smooth invariant measure does not hold true in the $C^{1}$ setting.
After more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the (rational) Lie algebra of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we…
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 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 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 short note we prove that if a symplectomorphism f is C1-stably shadowable, then f is Anosov. The same result is obtained for volume-preserving diffeomorphisms.
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.
The Anosov-Katok method is one of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties. To measure the complexity of systems with entropy zero, invariants like slow entropy have been introduced. In this article we develop several mechanisms facilitating computation of topological and measure-theoretic slow entropy of Anosov-Katok diffeomorphisms.
We show that various classes of closed manifolds with non-trivial higher homotopy groups do not support (transitive) Anosov diffeomorphisms. In particular we show that a finite product of spheres at least one of which is even-dimensional does not support transitive Anosov diffeomorphisms.
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。
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 …
Product Anosov diffeomorphisms and the two-sided limit shadowing property
本文用双侧极限跟踪性质刻画乘积阿诺索夫微分同胚，证明其等价于万有覆盖上提升的唯一双侧极限跟踪性质，并通过巴拿赫空间中映射的不动点关联跟踪轨道。
We prove that if the stable foliation and the unstable foliation of an Anosov diffeomorphism on a connected compact manifold are $C^3$, then the diffeomorphism has fixed points. This is a partial positive answer to a Smale conjecture for fixed points of Anosov diffeomorphisms.
Anosov diffeomorphisms of open surfaces
本文研究开曲面上Anosov微分同胚的存在性，在周期点稠密和几何一致条件下，证明其具有沿稳定与不稳定叶的Margulis测度系统。
This note illustrates the strategy of our paper on piecewise affine surface homeomorphisms by giving a new proof of the finite multiplicity of the maximum entropy measure of Anosov diffeomorphisms (here on surfaces). This approach avoids the explicit construction of Markov partitions and will be applied elsewhere to some non-uniformly hyperbolic diffeomorphisms.
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.
This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank k random perturbations of $C^2$ diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms: hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimen…
We prove that a $C^1-$generic symplectic diffeomorphism is either Anosov or the topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of the periodic points. We also prove that $C^1-$generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and finally we give examples of volume preserving diffeomorphisms which are not point of upper semicontinuity of entropy function in $C^1-$topology.
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。
We prove that every $C^2$ Anosov diffeomorphism in a compact and connected Riemannian manifold has a unique SRB and physical probability measure, whose basin of attraction covers Lebesgue almost every point in the manifold. Then, we use structural stability of Anosov diffeomorphisms to deduce that all $C^1$ Anosov diffeomorphisms on compact and connected Riemannian manifolds are transitive.
Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. Roughly, an Anosov family is a two-sided sequence of diffeomorphisms (or non-stationary dynamical system) with similar behavior to an Anosov diffeomorphisms. We show that the set consisting of Anosov families is an open subset of the set consisting of two-sided sequences of diffeomorphisms, which is endowed with the strong topology (or Whitney topology).
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…
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…
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 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.
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.
A cohomological approach to Ruelle-Pollicott resonances and speed of mixing of Anosov diffeomorphisms
通过定义各向异性de Rham上同调并证明其与标准上同调同构，揭示了Anosov微分同胚的Ruelle-Pollicott共振与动力系统在de Rham上同调上诱导作用的特征值之间的深刻联系，并由此得到混合速度的上同调界。
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.
We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.
There exist no contact Anosov diffeomorphisms
该文证明闭奇数维流形上任何 Anosov 微分同胚均无不变切触结构，否定其存在性。
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.
This monographic short book is intended to a brief introduction in the classic topic of the theory of Hiperbolic Dynamical Systems, for Spanish speaking students of an undergraduate course in Mathematics. We revisit the classic definition of Markov Partitions for Anosov diffeomorphisms on compact manifolds. We focus on the theorem of Sinai (Func. Anal. and its Applic. 2, 1968) stsating the existence of such partitions with arbitrarily small diameter. To prove the theorem of Sinai we restate the constructive method in Bowen, Lecture Notes in Math. 470, 1975. Finally we also relate the results, …
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.
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.
In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if the manifold is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to these manifolds, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions…
Entropy-minimizing diffeomorphisms of pseudo-Anosov type on K3 surfaces
本文在K3曲面上构造了无穷多个伪阿诺索夫型微分同胚，证明它们在同伦类中熵最小，且不保持任何复结构。
In this paper we mainly address the problem of disintegration of Lebesgue measure and measure of maximal entropy along the central foliation of (conservative) Derived from Anosov (DA) diffeomorphisms. We prove that for accessible DA diffeomorphisms of $\mathbb{T}^3$, atomic disintegration has the peculiarity of being mono-atomic (one atom per leaf). We further provide open and non-empty condition for the existence of atomic disintegration. Finally, we prove some new relations between Lyapunov exponents of DA diffeomorphisms and their linearization.
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.
Let $λ$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $λ$ with measure-theoretic entropy equal to topological entropy, we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of the circle.
We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.
Let $M$ be a closed oriented $C^\infty$ manifold and $f$ a $C^\infty$ Anosov diffeomorphism on $M$. We show that if $M$ is the two torus $T^2$, then $f$ is conjugate to a hyperbolic automorphism of $T^2$, either by a $C^\infty$ diffeomorphism or by a singular homeomorphism. We also show that for general $M$, if $f$ admits an absolutely continuous invariant measure $μ$, then $μ$ is a $C^\infty$ volume. The proofs are concatenations of well known results in the field.
We consider extensions of Anosov diffeomorphisms of an infranilmanifold by the real vector space R^ω. Our main result, based on the analogous theorem in finite dimensions proven by Nitica and Pollicott, is that any Holder cocycle satisfying an obvious obstruction induces a topologically transitive extension (topologically weak mixing, in fact). We show how to construct cocycles satisfying these conditions for any Anosov diffeomorphism, and then observe that unlike the finite dimensional case, where cocycles satisfying the obstruction are C^0-stably transitive, there can be no infinite-dimensio…
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 long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。