Smooth ergodic theory
英文摘要：
This is an expository article/encyclopedia entry explaining the history, techniques, and central results in the field of smooth ergodic theory.
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
相关论文
Ergodic theory for smooth one-dimensional dynamical systems
In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was done in the series of papers [BL1-BL5], but the approach to the general smooth case must be different.
math.DS
Unstable entropy in smooth ergodic theory
In this survey we recall basic notions of disintegration of measures and entropy along unstable laminations. We review some roles of unstable entropy in smooth ergodic theory including the so-called invariance principle, Margulis construction of measures of maximal entropy, physical measures and rigidity. We also give some new examples and pose some open questions.
math.DS
Smooth ergodic theory of $\mathbb{Z}^d$-actions
In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{Hölder}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action and establish controls on their local geometry. In the second part we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier--You…
math.DS
A hyperbolic diffeomorphism with countably many ergodic components near identity
We construct a smooth hyperbolic volume preserving diffeomorphism on a four dimensional compact Riemannian manifold which has countably many ergodic components and is arbitrarily close to the identity map.
math.DS
Nonstandard Smooth Realizations of Liouville Rotations
We augment the method of $C^\infty$ conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.
math.DS
Ergodic properties of equilibrium measures for smooth three dimensional flows
Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
math.DGmath.DS
更多相关论文收起更多论文
Some ergodic and rigidity properties of discrete Heisenberg group actions
The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimen…
math.DS
Ergodic optimization theory for Axiom A flows
In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $\mc C^{0,\a}$ ($\a\in(0,1]$) or $\mc C^1$ the minimizing measure is unique and is supported on a periodic orbit.
math.DS
光滑曲面微分同胚的Lyapunov指数与熵的不连续性
Discontinuity of Lyapunov exponents vs Entropy for smooth surface diffeomorphisms
CNRS Orsay（法国国家科学研究中心奥赛）
研究光滑曲面微分同胚中Lyapunov指数与熵的关系，通过简化论证展示最大熵测度附近指数的连续性性质。
math.DS
Smooth Anosov Katok Diffeomorphisms With Generic Measure
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.
math.DS
Dimension approximation in smooth dynamical systems
For a non-conformal repeller $Λ$ of a $C^{1+α}$ map $f$ preserving an ergodic measure $μ$ of positive entropy, this paper shows that the Lyapunov dimension of $μ$ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+α}$ diffeomorphism $f$ preserving a hyperbolic ergodic measure $μ$ of positive entropy, if $(f, μ)$ has only two Lyapunov exponents $λ_u(μ)>0>λ_s(μ)$, then the Hausdorff or lower box or upper box dimension of $μ$ can be approximated by the corresponding dimension of the horseshoes $\{Λ_n\}$. The same statement holds true if $…
math.DS
On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics
We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.
math-phmath.DSmath.MP
On manifolds supporting distributionally uniquely ergodic diffeomorphisms
A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].
math.DGmath.DS
Intrinsic metric on graded graphs,standardness,and invariant measures
We define a general notion of a smooth invariant (central) ergodic measure on the space of paths of an $N$-graded graph (Bratteli diagram). It is based on the notion of standardness of the tail filtration in the space of paths, and the smoothness criterion uses the so-called intrinsic metric which can be canonically defined on the set of vertices of these graphs. In many cases known to the author, like the Pascal graph, the Young graph, the space of configurations, all ergodic central measures are smooth (in this case, we say that the graph is smooth). But even in these cases, the intrinsic me…
math.DSmath.RT
遍历薛定谔算子的稠密现象：I. 谱、态密度积分与李雅普诺夫指数
Dense Phenomena for Ergodic Schrödinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
本文证明对紧致度量空间上同胚的遍历轨道采样得到的薛定谔算子，存在稠密采样函数集使谱间隙有限、态密度积分光滑且李雅普诺夫指数光滑为正，并肯定回答Walters的非一致上循环存在性问题。
math-phmath.DSmath.MP
Thermodynamic formalism for some systems with countable Markov structures
We study ergodic properties of certain piecewise smooth two-dimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correleations.
math.DS
Exponential mixing, KAM and smooth local rigidity
Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.
math.DS
Local rigidity of higher rank non-abelian action on tours
In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms and prove the existence of such action on any torus TN for any even N ? 6. We also give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.
math.DS
There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus
We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maxi…
math.DSmath.SPnlin.CD
关于 $r$-中和熵：熵公式及达到上确界的测度的存在性
On $r$-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum
本文研究光滑动力系统中的 $r$-中和局部熵，给出其与 Brin-Katok 局部熵及逐点维数的公式，并证明最大化测度的存在性、非唯一性及相关刚性。
math.DS
Nonhyperbolicity of invariant measures on maximal attractor
The article states that for every compact manifold M of dimension 4 or higher there is an area U in a set of smooth diffeomorphisms over M such that every map f from U has local maximal partially hyperbolic attractor and nonatomic ergodic invariant measure on it where one of Lyapunov exponents vanish. The result is first proved for skew products over horseshoe and then transferred onto smooth diffeomorphisms.
math.DS
