Construction of measures with dilation

Approximating Lyapunov exponents and stationary measures

We give a new proof of E. Le Page's theorem on the Holder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first Lyapunov exponent, as well as the stationary measure, for such random cocycles.

平衡矩阵对的Lyapunov最大化测度

Lyapunov maximizing measures for balanced pairs of matrices

本文研究了平衡矩阵对的Lyapunov最大化测度的存在性和Sturmian性质。

Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as: - the Lyapunov exponents of minimizing measures; -the weak KAM solutions. In particular, we deduce that the support of every minimizing measure all of whose Lyapunov exponents are zero is C1-regular almost everywhere.

Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures

In this paper, we consider a monic, centred, hyperbolic polynomial of degree $d \ge 2$, restricted on its Julia set and compute its Lyapunov exponents with respect to certain weighted Lyubich's measures. In particular, we show a certain well-behavedness of some coefficients of the Lyapunov exponents, that quantifies the non-well-behavedness in a system.

三维流形上$C^1$星向量场的双曲测度的Lyapunov指数

The Lyapunov Exponents of Hyperbolic Measures for $C^1$ Star Vector Fields on Three-dimensional Manifolds

该研究针对三维流形上的$C^1$星向量场，证明了非奇点支集的遍历双曲不变测度及其Lyapunov指数均可由周期测度及其Lyapunov指数逼近，推进了动力系统相关理论。

Lyapunov optimizing measures and periodic measures for $C^2$ expanding maps

We prove that there exists an open and dense subset $\mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.

$C^1$向量场具有主导分解的超几何测度的Lyapunov指数

The Lyapunov Exponents of Hyperbolic Measures for $C^1$ Vector Fields with Dominated Splitting

本文研究了$C^1$向量场中具有主导分解的超几何测度的Lyapunov指数，证明其可被周期测度近似。

Expansive Measures versus Lyapunov exponents

In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive is expanding. Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can not be accumulated by positively measure expansive diffeomorphis have a dominated sppliting.

Lyapunov exponents for random maps

It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper, we show that under any physical noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite. This result is a Lyapunov exponent version of Araújo's theorem on the existence and finitude of time averages. Furthermore, we numerically compute the Lyapunov exponents for a surface flow with an attracting heterocl…

Measures with positive Lyapunov exponent and conformal measures in rational dynamics

Ergodic properties of rational maps are studied, generalising the work of F.\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure.

纤维束上循环的具有简单Lyapunov谱的测度的共存性

Coexistence of measures with simple Lyapunov spectrum for fiber-bunched cocycles

该研究针对可逆双曲传递移位上的Hölder连续纤维束上循环，在Grassmann流形强不可约条件下，证明了具有全支撑与乘积结构的遍历测度的简单Lyapunov谱具有共存性。

$\mathcal{C}^2$ 曲面微分同胚中具有大 Lyapunov 指数的测度同宿类的有限性

Finitness of measured homoclinic classes with large Lyapunov exponents for $\mathcal{C}^2$ surface diffeomorphisms

本文证明了 $\mathcal{C}^2$ 曲面微分同胚中具有大 Lyapunov 指数的测度同宿类的有限性，并由此导出系统熵较大时最大熵遍历测度集合的有限性。

Robust vanishing of all Lyapunov exponents for iterated function systems

Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$-open sets of IFS's admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS's on the flag manifold.

Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism $L$ of $\mathbb T^2$. It is known that the positive Lyapunov exponent of $f$ with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of $L$, which, in addition, is less than or equal to the Lyapunov exponent of $f$ with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

Maximal measure and entropic continuity of Lyapunov exponents for $C^r$ surface diffeomorphisms with large entropy

We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.

Uniform hyperbolic approximations of measures with non zero Lyapunov exponents

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Omega_n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence mu_n of f-invariant ergodic probability measures with supp (mu_n) in Omega_n we have mu_n -> mu in the weak-* topology.

一个孤立的 Lyapunov 指数

An isolated Lyapunov exponent

本文构造了一个膨胀基础动力学上的连续线性 cocycle，证明了存在非周期轨道支撑的特殊测度使其 Lyapunov 指数远离零，从而违反了周期逼近性质。

Abundance of observable Lyapunov irregular sets

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke [OY2008], whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoc…

同宿类的李雅普诺夫谱

Lyapunov spectrum of homoclinic classes

研究同宿类C^1泛型微分同胚的测度的李雅普诺夫谱，证明其内部非空，且内部任意向量对应于同宿类上完全支撑的测度的谱。

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

A minimum principle for Lyapunov exponents and a higher-dimensional version of a Theorem of Mane'

We consider compact invariant sets Λfor C^{1} maps in arbitrary dimension. We prove that if Λcontains no critical points then there exists an invariant probability measure with a Lyapunov exponent λwhich is the minimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λis uniformly expanding if every invariant probability measure supported on Λis hyperbolic repelling. This generalizes a well known theorem of Mane' to the higher-dimensional setting.

Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting

We consider locally minimizing measures for the conservative twist maps of the $d$-dimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic such measures (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and the unstable Oseledet's bundles gives an upper bound of the sum of the positive Lyapunov exponents and a lower bound of the smallest positive Lyapunov exponent. Some more precise results are proved too.

Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies

We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

$SL(2,\mathbb{R})$典型上循环的Lyapunov最大化测度的熵

Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles

针对满足挤压、扭转及Mather集上映射一一条件的一步$SL(2,\mathbb{R})$典型上循环，论文证明其Lyapunov最大化测度熵为零。

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

In \cite{Ch91a} it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this article we study the set of points with infinite Lyapunov exponents. Using the cell structure developed in \cite{BSC90,Ku} we construct an ergodic invariant probability measure with infinite topological entropy supported on this set. Since the topological entropy is infinite this is a measure of maximal entropy. From the construction it is clear that there many such measures can coexist on a single component of topological transitivity. We also construct an ergodic inv…

Return time statistics for invariant measures for interval maps with positive Lyapunov exponent

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around a.e. point. We also show a `polynomial Gibbs property' for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hoelder potentials.

Nonexistence of Lyapunov Exponents for Matrix Cocycles

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_δ$ set…

Lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions

The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry-Mather theory was developed by Garibaldi \& Thieullen that provides the existence of some minimizing measures. Using the two Green bundles G- and G+ that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: -the angle between G- and G+ along the suppor…

On the periodic approximation of Lyapunov exponents for semi-invertible cocycles

We prove that, for semi-invertible linear cocycles, Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.

Lyapunov Exponents of Linear Cocycles Over Markov Shifts

The Lyapunov exponents of GL(2)-cocycles over Markov shifts depend continuously on the underlying data, that is, on the matrix coefficients and the Markov measure transition probabilities.

遍历斜积中零李雅普诺夫指数

Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems

研究遍历斜积的性质，证明其存在非超bolic测度，并展示这些测度的支撑全集和弱*极限。

Continuity properties of Lyapunov exponents for surface diffeomorphisms

We study the entropy and Lyapunov exponents of invariant measures $μ$ for smooth surface diffeomorphisms $f$, as functions of $(f,μ)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion f…

On self-affine measures with equal Hausdorff and Lyapunov dimensions

Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{φ_λ(x) = A_λx + v_λ\}_{λ\inΛ}$ and a probability vector $p=(p_λ)_λ>0$. Assume the strong separation condition holds. Let $γ_{1}\ge...\geγ_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_λ\}_{λ\inΛ}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_λ\}_{λ\inΛ}$. We show that if $γ_{m+1}>γ_{m}=...=γ_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $μ_{F}$ satisfies $\dim_{H}μ_{F}+D>(m+1)(d-m)…

Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $μ$ is an arbitrary $f$-invariant measure with positive Lyapunov exponents on the boundary of $A$, then $μ$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessability of every "good" $q$ i.e. such $q$ for which "small neighbourhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem …

On density of positive Lyapunov exponents for $C^1$ symplectic diffeomorphisms

Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $ω$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \[ \left\{ f\in \mathcal{S}ym^1_ω(M)\,| \begin{array}{ll} &\mbox{The largest Lyapunov exponent }λ_1(f,\,x)>0\\ &\mbox{ for a positive measure set } \end{array} \right\} \] is dense in $\mathcal{S}ym^1_ω(M)$. \end{abstract} \end{center}

Lyapunov spectrum for exceptional rational maps

We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain $σ$-finite conformal measures. This allows to extend previous results to exceptional rational maps.

Upper bound on the regularity of the Lyapunov exponent for random products of matrices

We prove that if $μ$ is a finitely supported measure on $\text{SL}_2(\mathbb{R})$ with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not $α$-Hölder around $μ$ for any $α$ exceeding the Shannon entropy of $μ$ over the Lyapunov exponent of $μ$.

Positive Lyapunov exponents for cocycles over time one anosov maps

We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.

Continuity of Lyapunov Exponents for Random 2D Matrices

The Lyapunov exponents of locally constant GL(2;C)-cocycles over Bernoulli shifts depend continuously on the cocycle and on the invariant probability. The Oseledets decomposition also depends continuously on the cocycle, in measure.

On the Lyapunov numbers

We introduce and study the Lyapunov numbers -- quantitative measures of the sensitivity of a dynamical system $(X,f)$ given by a compact metric space $X$ and a continuous map $f:X \to X$. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.

Regularity of foliations and Lyapunov exponents of partially hyperbolic dynamics

In this work we study relations between regularity of invariant foliations and Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of desintegration of Lebesgue measure which can be considered as a criterium for rigidity of Lyapunov exponents.

Iterated function systems of affine expanding and contracting maps on the unit interval

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existe…

Lyapunov `Non-typical' Points of Matrix Cocycles and Topological Entropy

It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman's Sub-additional Ergodic Theorem) that the set of `non-typical' points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any H$\ddot{o}$der continuous cocycles over hyperbolic systems, in this article we show that either all ergodic measures have same Maximal Lyapunov exponents or the set of Lyapunov `non-typical' points have full topological entropy and packing topological entropy. Moreover, we give an estimate of B…

Periodic approximation of exceptional Lyapunov exponents for semi-invertible operator cocycles

We prove that for semi-invertible and Hölder continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.

Uniqueness Of Ergodic Optimization Of Top Lyapunov Exponent For Typical Matrix Cocycles

In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov expoent for typical matrix cocyles. By using the results we obtain, we prove that in any non-uniquely ergodic minimal dynamical system, the Lyapunov-irregular points are typical for typical matrix cocyles.

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^1$ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Continuity of Lyapunov exponents is equivalent to continuity of Oseledets subspaces

We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.

具有非零李雅普诺夫指数的微分同胚的涌现性

Emergence for diffeomorphisms with nonzero Lyapunov exponents

研究保持双曲测度的 $C^{1+α}$ 微分同胚的高点态涌现性点集，基于不稳定Hausdorff维数给出其维数下界，并证明SRB测度情形下该点集具有满Hausdorff维数。

测地流中通过Lyapunov指数水平集的曲率刚性

Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows

本文通过不变测度研究Lyapunov指数水平集，证明零Lyapunov指数水平集全测度蕴含流形平坦、水平集全测度蕴含截面曲率常数，并给出无共轭点流形上阻碍光滑共轭的新判据，统一证明了已有刚性结果。

Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems

Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $μ$ with positive entropy, and let $\mathcal{A}$ be a Hölder continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mat…

Mono-atomic disintegration and Lyapunov exponents for derived from Anosov diffeomorphisms

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.

Ergodic optimization of Birkhoff averages and Lyapunov exponents

Ergodic optimization is the study of extremal values of asymptotic dynamical quantities such as Birkhoff averages or Lyapunov exponents, and of the orbits or invariant measures that attain them. We discuss some results and problems.

Lyapunov-maximising measures for pairs of weighted shift operators

Motivated by recent investigations of ergodic optimisation for matrix cocycles, we study the measures of maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that for generic pairs of weighted shift operators the Lyapunov-maximising measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximising measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that in contrast to the matrix case, the Lyapunov-maximising measures of pairs of bounded operators are in general no…

Lyapunov指数在一点Lyapunov谱情况下的正则性：半单情况

Regularity of Lyapunov exponents at one-point Lyapunov spectra: the semisimple case

本文研究了Lyapunov指数作为紧支撑概率测度空间上的函数的正则性，证明了在半单概率测度具有单点Lyapunov谱时，Lyapunov指数在Wasserstein距离下点wise log-Hölder连续。