Lyapunov spectrum for rational maps

李雅普诺夫指数熵连续性对于非平坦一维映射

Entropy Continuity of Lyapunov Exponents for Non-flat 1-dimensional Maps

该研究证明了非平坦一维映射中李雅普诺夫指数的熵连续性，并展示了其在熵上的统一可积性。

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.

Expansiveness, Lyapunov exponents and entropy for set valued maps

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for se…

关于Markov-Rényi映射的快速Lyapunov谱

On fast Lyapunov spectra for Markov-Rényi maps

本文提出几何方法研究Markov-Rényi映射的快速Lyapunov谱，证明其为分段常数且可能在无穷远不连续，推广了Gauss映射的既有结果。

Observable Lyapunov irregular sets for planar piecewise expanding maps

For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_σ$ $(0<σ<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_σ$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).

Lyapunov exponents for random perturbations of coupled standard maps

In this paper, we give a quantitative estimate for the sum of the first $N$ Lyapunov exponents for random perturbations of a natural class $2N$-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both `weak' and `strong' coupling regimes.

Uniform hyperbolicity for random maps with positive Lyapunov exponents

We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.

On the pointwise Lyapunov exponent of holomorphic maps

We prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.

The Lyapunov exponent of holomorphic maps

For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.

Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of "stable" and "unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

周期轨道Lyapunov指数的正态分布

Normal distribution of Lyapunov exponents of periodic orbits for expanding circle maps

研究显示，扩张圆映射中周期点的Lyapunov指数分布近似正态，误差随周期增长而减小，验证了中心极限定理在有限周期轨道中的应用。

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…

Counting the Lyapunov inflections in piecewise linear systems

Following the pioneering work of Iommi-Kiwi and Jenkinson-Pollicott-Vytnova, we continue to study the inflection points of the Lyapunov spectrum in this work. We prove that for any 3-branch piecewise linear expanding map on an interval, the number of its Lyapunov inflections is bounded above by 2. Then we continue to show that, there is a 4-branch piecewise linear expanding map, such that its Lyapunov spectrum has exactly 4 inflection points. These results give an answer to a question by Jenkinson-Pollicott-Vytnova on the least number of branches needed to observe 4 inflections in the Lyapunov…

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.

Lyapunov spectrum for Hénon-like maps at the first bifurcation

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the conti…

正Lyapunov指数与随机守恒动力学中的可积性

Lyapunov exponents and rigidity in random billiards and standard maps

研究随机保守动力学中正Lyapunov指数与系统可积性的关系，证明在特定条件下Lyapunov指数消失的判据。

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.

作为可数Markov区间映射的Newton-Raphson方法的Lyapunov谱

The Lyapunov spectrum as the Newton-Raphson method for countable Markov interval maps

本文研究具有无限多分支的MRL区间映射，证明其Lyapunov谱可表示为拓扑压的Legendre变换，并与应用于该拓扑压的Newton-Raphson方法直接相关，推广了已有结论。

无限可重整 Lorenz 映射的 Lyapunov 指数与随机稳定性

Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps

本文研究无限可重整压缩 Lorenz 映射，证明在有界几何或先验界条件下其临界值满足慢回归，并据此得到零逐点 Lyapunov 指数与随机稳定性。

C^r 一维映射的 Lyapunov 指数连续性

Continuity of Lyapunov exponents for C^r one-dimensional maps

本文研究区间或圆周 C^r 一维映射，证明大熵条件下 Lyapunov 指数的熵连续性，并推出熵的上半连续性与几何势的一致可积性。

部分I：α-复发与等价Lyapunov映射

Partial section I: $α$-recurrence and equivariant Lyapunov maps

本文研究了流在紧致流形上的α-等价Lyapunov映射的存在性及其与α相关的复发集分类。

具有一个拓扑吸引子的映射的 α-极限集与 Lyapunov 函数

α-limit sets and Lyapunov function for maps with one topological attractor

研究具有单一拓扑吸引子的连续映射，通过分层 $A$-$R$ 对分解刻画各点 α-极限集，并基于弱 Morse 分解构造有界 Lyapunov 函数以描述除贫集外的轨道行为。

关于自守测度、李雅普诺夫指数与有理映射的不稳定性

On automorphic measures, Lyapunov exponents and instability of rational maps

引入复特征值的自守测度，证明其存在导致有理映射的不稳定性，并建立遍历与组合条件确保此类测度存在。

Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps

We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.

Simple Lyapunov spectrum for certain linear cocycles over partially hyperbolic maps

Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarc'h-Raugi, Gol'dsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main …

由扩张圆周自同态强迫的圆周映射的负 Lyapunov 指数

Negative Lyapunov exponent of circle maps forced by expanding circle endomorphisms

本文研究受扩张圆周自同态强迫的斜积圆周映射，构造 $C^1$ 开集证明纤维 Lyapunov 指数几乎处处为负，并给出依赖于映射选择性特征的一致上界，从而得到局部同步。

基于李雅普诺夫指数的霍普夫纤维化有理自映射吸引域计算算法

Algorithms for computing basins of attraction associated with a rational self-map of the Hopf fibration based on Lyapunov exponents

本研究构建复有理函数迭代理论以规避分母为零引发的问题，提出基于李雅普诺夫指数的方法计算霍普夫纤维化自同态的吸引周期与吸引域，算法无需预先确定不动点或吸引周期。

Quantitative approximations of the Lyapunov exponent of a rational function over valued fields

We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly non-archimedean absolute value, in terms of the multipliers of periodic points of the rational function. This quantifies both our former convergence result over general fields and the one-dimensional version of Berteloot--Dupont--Molino's one over archimedean fields.

一个具有有理系数且无局部多项式李雅普诺夫函数的全局渐近稳定的多项式向量场

A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

本文提出一个二维七次多项式向量场，具有有理系数，全局渐近稳定但无局部解析李雅普诺夫函数。

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 …

Lyapunov exponents and bifurcation current for polynomial-like maps

We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.

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…

稳定 Lyapunov 指数的刚性与 Anosov 映射的可积性

Rigidity of stable Lyapunov exponents and integrability for Anosov maps

本文研究了环面上不可约 Anosov 映射的稳定 Lyapunov 指数刚性与可积性，证明了一维及特定高维稳定丛下可积性与 Lyapunov 指数相等的等价关系，并得出拓扑共轭在稳定丛上光滑的结论。

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

We study the effects of IID random perturbations of amplitude $ε> 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $ε> 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the …

平均场极限下全局耦合映射的可观测李雅普诺夫指数

Observable Lyapunov Exponents for Globally Coupled Maps in the Mean-Field Limit

本文提出可观测李雅普诺夫指数，用于区分大型耦合系统的微观混沌与宏观稳定，证明其由自洽传递算子线性化决定，并应用于弱耦合扩张映射，表明宏观扰动衰减。

Flexibility of exponents for expanding maps on a circle

We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to log 2 which, in addition, is less than or equal to the Lyapunov exponent of g with respect to the measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree 2.

The Lyapunov spectrum is not always concave

We characterize one-dimensional compact repellers having nonconcave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra.

关于S-单峰映射的李雅普诺夫指数的存在性

Existence of the Lyapunov exponent for $S$-unimodal maps

研究S-单峰映射李雅普诺夫指数，证明非平坦临界点的该映射对几乎每个勒贝格点李雅普诺夫指数存在且为常数λ_T，给出其为0的充要条件，还得出相关特殊映射的结论。

Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach

This works investigates the Lyapunov-Oseledets spectrum of transfer operator cocycles associated to one-dimensional random paired tent maps depending on a parameter $ε$, quantifying the strength of the \emph{leakage} between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent $λ_2^ε$ within an error of order $ε^2|\log ε|$. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that $λ_1^ε=0$ and $λ_2^ε$ are simple, and are the only exceptional Lyapunov …

Continuity of Lyapunov exponents in the C0 topology

We prove that the Bochi-Mañé theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.

Rational maps with rational multipliers

In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.

Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems

We explore the concept of metastability in random dynamical systems, focussing on connections between random Perron-Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron-Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

Rationality of the zeta function for Ruelle-expanding maps

We will prove that the zeta function for Ruelle-expanding maps is rational.

On Rational Maps with Two Critical Points

This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled ``On Bicritical Rational Maps'' in September 1997; revised and retitled April 1999)

Rational Misiurewicz maps are rare

We show that the set of Misiurewicz maps has Lebesgue measure zero in the parameter space of rational maps for any fixed degree greater than or equal to 2.

李雅普诺夫谱刚性与随机阿诺索夫微分同胚的同时线性化

Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms

本文研究了随机阿诺索夫微分同胚的李雅普诺夫谱刚性，证明了在特定条件下系统可同时线性化，并探讨了正矩阵随机游走的正李雅普诺夫指数刚性。

Multifractal analysis of Lyapunov exponent for the backward continued fraction map

In this note we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then it becomes equal to zero.

Bounded hyperbolic components of bicritical rational maps

We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.

Puiseux series dynamics of Quadratic rational maps

We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.

On decomposable rational maps

If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.

Real bounds and Lyapunov exponents

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfy the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.

二次有理映射周期曲线中横截性的渐近性

Asymptotics of Transversality in Periodic Curves of Quadratic Rational Maps

本文计算了具有周期标记临界点的二次有理映射模空间的欧拉特征数，从而研究其横截性的渐近行为。

On rational maps with buried critical points

In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.