Invariant Curves for Degenerate Hyperbolic Maps of the Plane

Notes on $s$ and $u$-states for cocycles over partially hyperbolic maps

In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.

Multifractal analysis in non-uniformly hyperbolic interval maps

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.

A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities

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.

某些与Anosov同伦的映射的极大熵测度的双曲性

Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov

证明了具有紧致二维中心叶理的环面部分双曲系统遍历极大熵测度的双曲性。

Semi-hyperbolic maps are rare

We prove in this paper that the set of semi-hyperbolic rational maps has Lebesgue measure zero in the space of rational maps of the Riemann sphere for a fixed degree d at least 2. It generalises an earlier result by J. Graczyk and the author who proved the same thing in degree 2.

Existence of SRB measures for hyperbolic maps with weak regularity

We prove that a $C^1$ hyperbolic map whose differential is regular enough has an SRB measure. The precise regularity condition is weaker than H{ö}lder and was mentionned by various authors through the developement of expanding and uniformly hyperbolic dynamics.

Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts

We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class $\mathcal B$), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1.

On hyperbolic rational maps with finitely connected Fatou sets

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence…

Statistical properties of one-dimensional maps under weak hyperbolicity assumptions

For a real or complex one-dimensional map satisfying a weak hyperbolicity assumption, we study the existence and statistical properties of physical measures, with respect to geometric reference measures. We also study geometric properties of these measures.

Continuity of topological entropy for perturbations of time-one maps of hyperbolic flows

We consider a $C^1$ neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle.

Persistent Markov partitions and hyperbolic components of rational maps

Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.

非均匀超双曲映射的最大熵测度

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。

Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

We consider semigroup actions on the unit interval generated by strictly increasing $C^r$-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and the Schwarzian…

Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties

Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results but gives also new insights on the statistical properties of these systems.

DA映射中心叶层上非双曲遍历测度的分解

Disintegrations of non-hyperbolic ergodic measures along the center foliation of DA maps

该研究针对3维环面上同伦于Anosov的部分双曲微分同胚，证明其非双曲遍历测度存在与每个中心叶交点至多两个的满测度子集，推进了非双曲遍历测度沿中心叶层分解的研究。

Hyperbolic Components

Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called the hyperbolic locus. The various connected components of this hyperbolic locus are called hyperbolic components, and those hyperbolic components with compact closure (or equivalently those contained in the "connectedness locus") are called bounded hyperbolic components. It is shown that each bounded hyperbolic component is a topological cell containing a uni…

Statistics of multipliers for hyperbolic rational maps

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.

双曲吸引子与扩张映射族中SRB熵泛函的行为

Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps

本文综述均匀双曲系统族开分支上SRB熵泛函的研究现状，探讨其极值、梯度流及边界附近行为，涉及Anosov系统等多类双曲系统。

Coexistence of critical orbit types in sub-hyperbolic polynomial maps

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex numbers {\bf C} to itself which have degree two or more in each copy. As a consequence of these results we prove a transitivity relation between hyperbolic components in parameter space which was conjectured by Milnor.

Hyperbolic components in spaces of polynomial maps

We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of this type, the post-critically bounded maps form a compact subset $\cl^{S}$ called the connectedness locus, and the hyperbolic maps in $\cl^{S}$ form an open set $\hl^{S}$ called the hyperbolic connectedness locus. The various connected components $H_α\subset \hl^{S}$ are called hyperbolic components. It is shown that each hyperbolic component is a topologica…

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.

Equilibrium states of interval maps for hyperbolic potentials

We study the thermodynamic formalism of sufficiently regular interval maps for Holder continuous potentials. We show that for a hyperbolic potential there is a unique equilibrium state, and that this measure is exponentially mixing. Moreover, we show the absence of phase transitions: The pressure function is real analytic at such a potential.

Equivalent characterizations of hyperbolic Hölder potential for interval maps

Consider a topologically exact $C^3$ interval map without non-flat critical points. Following the works we did in \cite{LiRiv12two}, we give two equivalent characterizations of hyperbolic Hölder continuous potential in terms of the Lyapunov exponents and the measure-theoretic entropies of equilibrium states for those potentials.

Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities

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.

具有多重断点的重整化双曲性

Hyperbolicity of renormalization for maps with multiple breaks

本文在断点小且旋转数有界类型的条件下，为多重断点圆周映射构造双曲马蹄，并证明其刚性类为解析子流形。

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 …

Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps

We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with infinite derivative (like the first return maps of Lorenz like flows). We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error, with respect to the Wasserstein distance. We apply this to the rigorous computation of the dimension of the measure. We present a rigorous implementation of the algorithms using interval arithmetics, and the result of the computation …

First hyperbolic times for intermittent maps with unbounded derivative

We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last thirty years. The statistics of these maps are controlled by the order of tangency that a certain "cut function" makes with the boundary of the square. Using a large deviations result of Melbourne and Nicol we obtain…

Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.

简单二维映射中双曲分裂刚性的数值研究

A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

本文通过数值模拟研究简单二维解析映射中的双曲分裂刚性，证明其双曲结构非光滑，为光滑动力系统中产生非光滑相空间结构的全局机制提供了例证。

Statistical properties of uniformly hyperbolic maps and transfer operators' spectrum

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.

Good Banach spaces for piecewise hyperbolic maps via interpolation

We introduce a weak transversality condition for piecewise C^{1+α} and piecewise hyperbolic maps which admit a C^{1+α} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, g…

弱超双曲收缩洛伦兹映射的随机稳定性

Stochastic stability for weakly hyperbolic contracting Lorenz maps

研究满足指数1可和条件的随机扰动收缩洛伦兹映射的扩张性质，证明在强意义下随机稳定性，即平稳测度密度收敛到未扰动映射物理测度密度的L1范数。

Equilibrium states of weakly hyperbolic one-dimensional maps for Holder potentials

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, "hyperbolic". We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Holder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Holder continuous functions. Another consequence is that every Holder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.

Transversality in the setting of hyperbolic and parabolic maps

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcat…

Structure of partially hyperbolic Hénon maps

We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…

Physical measures at the boundary of hyperbolic maps

We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a "small" subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastical stability. The physical measures are obtained as zero-noise limits which are shown to satisfy the Entropy Formula.

双曲复Hénon映射的可计算性性质

Computability properties of hyperbolic complex Hénon maps

本文建立了复Hénon映射（双曲多项式微分同胚）的Julia集可计算性、双曲性半可判定性以及参数空间中双曲性轨迹的下可计算性。

The growth rate inequality for Thurston maps with non hyperbolic orbifolds

Let $f: S^2 \to S^2$ be a continuous map of degree $d$, $|d|>1$, and let $N_nf$ denote the number of fixed points of $f^n$. We show that if $f$ is a Thurston map with non hyperbolic orbifold, then either the growth rate inequality $\limsup \frac{1}{n} \log N_nf\geq \log |d|$ holds for $f$ or $f$ has exactly two critical points which are fixed and totally invariant.

Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric unfolding. The normal forms are used to analyse bifurcations of $n$-periodic orbits. In particular, for $n\ge6$ we find regions of parameters where the normal form has "meandering'' invariant curves.

Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.

Hyperbolicity of the invariant sets for the real polynomial maps

It is well known that for $a>4$, the dynamical behaviors of the logistic map $f_a(x)=ax(1-x)$ on the maximal invariant compact set are "simple" which could be clearly explained by the theories of hyperbolic dynamics and symbolic dynamics. Is it possible that similar phenomena could be observed in general real polynomial maps? In this paper, we study this problem by investigating the real polynomial map $f_a(x)=ag(x)$, where $a$ is a parameter, and $g$ is a real-coefficient polynomial, which has at least two different real zeros or only one real zero.

Parabolic Flows Renormalized by Partially Hyperbolic Maps

We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and…

保面积映射与Reeb流的正双曲周期轨道

Positive hyperbolic periodic orbits for area-preserving maps and Reeb flows

该研究针对保面积映射与Reeb流，证明闭三维接触流形上强非退化Reeb向量场若有至少3个简单周期轨道则存在无穷多简单正双曲周期轨道，核心结合曲面结果与破书分解定理完成证明。

$\u2164^2$上映射的非一致双曲性

Non-uniform hyperbolicity of maps on $\mathbb{T}^2$

该研究在二维环面线性自同态的两类同伦类中构造出非一致双曲映射，还得到带稳定遍历性的结果，部分回应了已有文献中的相关问题。

Local limit theorems in relatively hyperbolic groups II : the non-spectrally degenerate case

This is the second of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this second paper, we restrict our attention to non-spectrally degenerate random walks and we prove precise asymptotics of the probability $p_n(e, e)$ of going back to the origin at time $n$. We combine techniques adapted from thermodynamic formalism with the rough estimates of the Green function given by the first paper to show that $p_n(e, e) \sim CR^{-n} n^{-3/2}$ , where $R$ is the spectral radius of the random walk. This generalizes results of W. Woess for free products and r…

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\bf R}^{2n}$

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $Σ\subset{\bf R}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $Σ$ when $n$ is even. When $n$ is odd, there exist at least $n$ closed characteristics with odd Maslov-type indices on $Σ$ and at least $(n-1)$ of them are non-hyperbolic. Here we call a compact star-shaped hypersurface $Σ\subset {\bf R}^{2n}$ {\rm index perfect} if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed charac…

Topological characterization of the hyperbolic maps in the Sine family

The purpose of this paper is to establish a topological characterization of all the hyperbolic maps in the Sine family $\{λ\sin(z) \:\big{|}\:λ\ne 0\}$ which have super-attracting cycles.

Thurston type Theorem for sub-hyperbolic rational maps

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem.

Hyperbolic components of McMullen maps

In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the boundary of the unbounded hyperbolic component. This is a dynamical analogue of McMullen's theorem that cusps are dense on the Bers' boundary of Teichmüller space.

圆盘上的保面积微分同胚与正双曲轨道

Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits

本文证明了开圆盘上保面积非退化微分同胚在具有至少2个内部周期点时必有无穷多个正双曲周期点，并将其应用于切透镜空间与切3维球面，细化了Hofer等人的结果。

Semi-hyperbolic rational maps and size of Fatou components

Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${ń}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${ń}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.

Symbolic dynamics for nonuniformly hyperbolic maps with singularities in high dimension

We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.