Anders Johansson, Thomas Jordan, Anders Öberg, Mark Pollicott
We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville--Pomeau map.
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Johansson, Jordan, Öberg and Pollicott ( Israel J. Math.(2010)) has studied the multifractal analysis of a class of one-dimensional non-uniformly hyperbolic systems, by introducing some new techniques, we extend the results to the case of high dimension.
This article is devoted to the study of the multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.
We study the multifractal analysis of dimension spectrum for almost additive potential in a class of one dimensional non-uniformly hyperbolic dynamic systems and prove that the irregular set has full Hausdroff dimension.
In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches.
Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding interval maps with finitely many branches
针对含抛物不动点的有限分支非一致扩张区间映射，论文运用热力学形式细化其性质，并加强了 Birkhoff 平均重分形谱的条件变分原理。
We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.
Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches
本文研究了具有可数无穷多个分支的非一致扩张Rényi区间映射的Birkhoff平均的多重分形谱，实质强化了条件变分公式，发展了热力学形式，并从Hausdorff维度分析了Khinchin指数，肯定回答了相关猜想。
Multifractal analysis of homological growth rates for hyperbolic surfaces
该研究对双曲曲面上定向测地线同调增长率开展多重分形分析，给出了由富克斯群广义庞加莱指数表示的水平集豪斯多夫维数公式，并借助符号动力学等方法证明了维数谱的解析性。
For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables, in terms of the Hausdorff dimension of invariant probability measures. Using our results, we are able to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers, answering in particular a question of Pollicott. Moreover, we establish formulas for multi-cusp winding spectra for the Bowen-Series maps associat…
On multifractal analysis and large deviations of singular-hyperbolic attractors
本文针对奇异双曲吸引子（含几何 Lorenz 吸引子），利用马蹄逼近性质证明水平集的稠密性与变分原理、非正则集的 Baire 通用性及满拓扑熵，并建立弱 Gibbs 测度的 level-2 大偏差界。
Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase …
In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph di…
Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces
本文研究了双曲面上多尖点旋绕过程的多标度分析，建立了其Hausdorff维度谱的条件变分原理，并证明了该维度谱的解析性。
It is known that all uniformly expanding or hyperbolic dynamics have no phase transition with respect to Hölder continuous potentials. In \cite{BC21}, is proved that for all transitive $C^{1+α}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transf…
In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.
In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $ϕ$ is an Hölder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we de…
We prove that the system resulting of coupling the standard map with a fast hyperbolic system is robustly non-uniformly hyperbolic.
We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus for a set of (large) parameters of positive Lebesgue measure.
In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.
We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.
For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non uniformly hyperbolic dynamical systems developed by L.S.Young.
We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.
We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally we show that these equilibrium states vary continuously in the weak$^\ast$ topology within such systems.
This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.
We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.
We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a u-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched cocycles.
Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$
本文研究了$\mathbb{P}^k$上内射映射的平衡态的广义分形分析，通过体积维度理论定义了局部维度谱，并证明其与温度函数形成Legendre变换对。
We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small topological degree cases not considered in the previous article. This proves the existence of a $\mathcal C^1$ open set of non-uniformly hyperbolic systems, that intersects essentially every homotopy classes in $\mathbb T^2$, where the Lyapunov exponents vary continuously.
In this article we prove that for a $C^{1+α}$ diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular points that are not uniformly hyperbolic is larger than or equal to the metric entropy of the hyperbolic ergodic measure. In the process of proof, we give an abstract general mechanism to study topological entropy of irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property an…
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions …
Measures of maximal entropy for non-uniformly hyperbolic maps
该研究探讨了非均匀超双曲映射中最大熵测度的唯一性和有限性，通过同宿类分析和多种不同模型的应用，证明了相关测度的有限性或唯一性。
We give an essentially equivalent formulation of the backward contracting property, defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least two which are at most finitely renormalizable and have only hyperbolic periodic points, as well as for all smooth interval maps with non-flat critical points.
In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials.
This is a lightning introduction to some modern techniques used in the study of the statistical properties of hyperbolic dynamical systems. The emphasis is not in presenting a comprehensive theory but rather in fleshing out the main ideas in the simplest and fastest possible manner so that the reader can quickly get the intuition necessary to easily read the more technical (and more complete) accounts of the theory.
We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.
We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,μ)$ is a non-uniformly hyperbolic system and $A:M \to GL(d,\mathbb{R}) $ is an $α$-Hölder continuous map satisfying $ A(f^{n-1}(p))\ldots A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ and $n\in \mathbb{N}$, there exists a measurable map $P:M\to GL(d,\mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $μ$-almost every $x\in M$. Moreover, we prove that whenever the measure $μ$ has local product structure the transfer map $P$ is $α$-H…
In this note, we obtain verifiable sufficient conditions for the extreme value distribution for a certain class of skew product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of $Φ(p) = -\log(d(p, p_0))$ that is of the first type. In particular, we establish the Type I distribution for $S^1$ extensions of piecewise $C^2$ uniformly expanding maps of the interval, non-uniformly expanding maps of th…
We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated to horseshoes with Hausdorff dimension $<22/21$ of conservative surface diffeomorphisms.
We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity…
We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.
We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We also consider the more general situation where the allowed products of matrices satisfy a Markovian rule.
We consider a partially hyperbolic C1-diffeomorphism f on a smooth compact manifold M with a uniformly compact f-invariant center foliation. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space of the center foliation is a torus and f induces a hyperbolic automorphism on it, in particular, f is centrally transitive. We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set …
Discontinuity example for the Lyapunov exponents on the boundary of the uniformly hyperbolic set
研究通过构造一个局部常数的协同映射，展示Lyapunov指数在C^0拓扑更细的拓扑下出现不连续点，且该映射位于一致超双曲集边界，通过C_{δ- log}拓扑逼近证明Lyapunov指数可消失。
Lyapunov exponents for uniformly hyperbolic random matrix products
本文研究了由马尔可夫移位驱动的随机矩阵乘积的最大学李雅普诺夫指数，证明了在项目性均匀超双曲条件下，该指数可表示为无限矩阵的显式表达式，并提出高效的多项式时间近似算法。
In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.
We investigate a renewal scheme for non-uniformly hyperbolic semiflows that closely resembles the renewal scheme developed in the discrete time case, in order to obtain sharp estimates for the correlation function. Also, the involved observables are supported on a flow-box of unbounded length. The present abstract setting does not require the use of Markov structure. However, the classes of examples covered here are rather restrictive. In these examples, it is easier to exploit the full force of the method and get optimal results for observables supported on finite length flow-boxes.
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $A$ over $f$ is Holder continuous and takes values in $GL(d,R)$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d,R)$-valued cocycle $A$ we prove that the Lyapunov exponents of $A$ with respect to $μ$ can be approximated by the Lyapunov exponents of $A$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-di…
We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient d…
We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measure…
In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recu…
Hausdorff Dimension and Flexibility for Attractors of Non-Uniformly Hyperbolic Flows
该研究将几何洛伦兹吸引子的维数理论推广到更一般的非均匀双曲吸引子，证明此类吸引子含维数大于2的双曲马蹄，建立中间维数性质，得到相关动力学谱的鲁棒非空内部结果。
Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums
本文结合密度、熵和多重分形分析研究遍历测度结构，证明了传递Anosov系统中遍历测度的稠密$G_δ$性质，建立了“多马蹄”熵稠密性质的一般框架，并推广至非双曲系统与多种定量谱分析。
Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows
本文为三维正速度流构造符号动力学，对每个χ>0编码所有χ-双曲不变测度的全测度集，包括高熵遍历测度和鞍型周期轨道，并应用于编码同宿类。
Non-uniformly hyperbolic endomorphisms
该研究展示了存在大量非均匀超双曲保面积双射映射，通过变形线性映射获得显式例子，并证明其在C^1拓扑下具有连续变化的积分Lyapunov指数
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…
For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to an extreme value distribution, and obtain estimates on the rate of convergence. We show that our results are applicable to a range of examples, and include new results for Lorenz maps, certain partially hyperbolic systems, and non-uniformly expanding systems with sub-exponential decay of correlations. For applications where analytic results are not readily av…
Decay of Correlations for some Non-Uniformly Hyperbolic Attractors
针对光滑实现问题，研究非一致双曲吸引子动力系统的关联衰减，通过对一致双曲微分同胚施加减速程序构造系统，证明其关联存在多项式上下界，结果适用于Smale-Williams螺线及其小扰动。
This article is an expanded version of some notes for my talk at the ``Ergodic Theory and Dynamical Systems Workshop'' (from March 22 to March 25, 2012) held at the Department of Mathematics of the University of North Carolina at Chapel Hill. In the aforementioned talk, it was discussed some recent results on the fractal geometry of certain objects -- \emph{non-uniformly hyperbolic horseshoes} -- constructed by Jacob Palis and Jean-Christophe Yoccoz in their recent \emph{tour-de-force} work around \emph{heteroclinic bifurcations of surface diffeomorphisms}. The goal of the present article is t…
In the 1960s and 1970s a large part of the theory of dynamical systems concerned the case of uniformly hyperbolic or Axiom A dynamical system and abstract ergodic theory of smooth dynamical systems. However since around 1980 an emphasize has been on concrete examples of one-dimensional dynamical systems with abundance of chaotic behavior (Collet &Eckmann and Jakobson). New proofs of Jakobson's one-dimensional results were given by Benedicks and Carleson \cite{BC85} and were considerably extended to apply to the case of Hénon maps by the same authors \cite{BC91}. Since then there has been a con…
In this article we introduce a gluing orbit property, weaker than specification, for both maps and flows. We prove that flows with the $C^1$-robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property. We also prove a level-1 large deviations principle and a level-2 large deviations lower bound for for semiflows with the gluing orbit property. As a consequence we establish a level-1 large deviations principle for hyperbolic flows and every continuous observable, and also a level-2 large deviations lower bound. Finally, sinc…
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