Statistical properties of one-dimensional maps under weak hyperbolicity assumptions

Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents

We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.

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.

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。

关于实二次映射非游荡集双曲性的注记

A note on the hyperbolicity of the non-wandering sets of real quadratic maps

本文针对实二次映射在参数接近-2时的非游荡集，给出两种证明其双曲性的方法，虽结果非新，但提供了更易获取的证明途径。

Hyperbolicity and Quasi-hyperbolicity in Polynomial Diffeomorphisms of ${\Bbb C}^2$

We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.

具有某些双曲性的一维映射的Birkhoff谱

Birkhoff spectra for one-dimensional maps with some hyperbolicity

研究了一维光滑动力系统的多重分形分析。对于由抽象动力系统建模的拓扑混合$C^2$映射，利用双曲测度的局部维数刻画了由连续函数的Birkhoff平均得到的水平集的豪斯多夫维数.对不变概率测度的遍历盆也给出了相应的刻画。并证明了拟正则点集的补具有全豪斯多夫维数。

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.

Hyperbolicity for conservative twist maps of the 2-dimensional annulus

These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus and focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones. We will also give some open questions. This course is the second part of a minicourse that was begun by R. Potrie. S…

广义临界准圆映射的重正化超几何性

Hyperbolicity of renormalization of critical quasicircle maps

本文研究了临界准圆映射的重正化超几何性，扩展了Yampolsky的结果，通过冠状重正化算子证明了其超几何固定点性质。

Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller's theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly …

$\u2164^2$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。

The maximal entropy measure detects non-uniform hyperbolicity

We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure. Using the same tools in the proof of these results we give an extension of a result of Carleson, Jones and Yoccoz, that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.

实二次族中的双曲性是稠密的

Hyperbolicity is dense in the real quadratic family

它表明，非双曲实二次多项式的拓扑和拟对称共轭类是相同的。利用拟共形刚性，每类映射在二次映射族中只有一个代表，证明了双曲映射是稠密的。

Renormalization and Siegel disks for complex Hénon maps

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.

Breakdown of hyperbolicity for quasiperiodic attracting invariant circles in a family of three-dimensional Henon-like maps

We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasipe…

Statistical properties for compositions of standard maps with increasing coefficent

The Chirikov standard map family is a one-parameter family of volume-preserving maps exhibiting hyperbolicity on a `large' but noninvariant subset of phase space. Based on this predominant hyperbolicity and numerical experiments, it is anticipated that the standard map has positive metric entropy for many parameter values. However, rigorous analysis is notoriously difficult, and it remains an open question whether the standard map has positive metric entropy for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing parameter. Whe…

Dimensional entropies and semi-uniform hyperbolicity

We describe dimensional entropies introduced in a previous work list some of their properties and give some new proofs. These entropies allowed the definition of entropy-expanding maps. We introduce a new notion of entropy-hyperbolicity for diffeomorphisms. We indicate some simple sufficient conditions (some of them new) for these properties. We conclude by some work in progress and more questions.

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão and Vargas. We also prove density of (real) hyperboli…

超越一致双曲性的刚性测度：环面上Cartan作用的不变测度

Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori

证明了Z^k（k>1）在（k+1）维环面上的每一个光滑作用都保持绝对连续测度.这是第一个已知的结果关于阿贝尔群的同伦同态存在的一个不变的几何结构是从同伦数据。

C^r单峰映射重整化的整体双曲性

Global hyperbolicity of renormalization for C^r unimodal mappings

本文推广了M. Lyubich关于类二次芽到具有二次临界点的C^r单峰映射空间的重正化的整体双曲性的最新结果。我们表明，在这个空间中的重整化算子的有界型极限集有一个不变的双曲结构提供rge 2+ α与α接近1。作为Lyubich结果和我们结果之间的中间步骤，我们证明了重整化算子在实解析映射的巴拿赫空间中是双曲的。我们构造了局部稳定流形，证明了它们构成了一个连续的层，其叶是C^1余维1，是周围空间的Banach子流形，并且对于某个β>0，其完整性是C^{1+β}。我们还证明了全局稳定集也是C^1浸入（余维1）子流形，只要rge 3+ alpha且alpha接近1。作为推论，我们推导出在一般的单参数C^r单峰映射族中，对应于有界组合型无限重正化映射的参数集是一个豪斯多夫维数小于1的康托尔集.

Existence and Regularity of Invariant Graphs for Cocycles in Bundles: partial hyperbolicity case

We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and non-locally-compact bundles. The regularity includes (uniform) $ C^0 $ continuity, Hölder continuity and smoothness. To illustrate the power of our results and methods, a number of applications to both well-posed and ill-posed semilinear differential equations and abstract infinite-dimensional dynamical systems are given. These applications include the existen…

On non-uniformly hyperbolicity assumptions in one-dimensional dynamics

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.

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent $α>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely …

Geometric pressure for multimodal maps of the interval

This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps f of a finite union of compact intervals in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transiti…

Statistical hyperbolicity for harmonic measure

We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moment with respect to the Teichmuller metric, and whose supports generate non-elementary subgroups. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.

The non-hyperbolicity of irrational invariant curves for twist maps and all that follows

The key result of this article is key lemma: if a Jordan curve $γ$ is invariant by a given C 1+$α$ -diffeomorphism f of a surface and if $γ$ carries an ergodic hyperbolic probability $μ$, then $μ$ is supported on a periodic orbit. From this Lemma we deduce three new results for the C 1+$α$ symplectic twist maps f of the annulus: 1. if $γ$ is a loop at the boundary of an instability zone such that f |$γ$ has an irrational rotation number, then the convergence of any orbit to $γ$ is slower than exponential; 2. if $μ$ is an invariant probability that is supported in an invariant curve $γ$ with an…

一维映射族中非一致双曲性发生的可计算条件

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

基于可计算初始条件，提出并证明了Jakobson-Benedicks-Carleson型定理，给出非一致双曲性（随机动力学）发生的显式可计算下界，并应用于二次族，证明绝对连续不变概率测度对应的参数集测度至少为10^{-5000}。

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。

Inducing schemes for multi-dimensional piecewise expanding maps

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity.

联合典型周期优化：具有稳定双曲性的系统

Joint typical periodic optimization: systems with stable hyperbolicity

本文研究了具有稳定双曲性的系统，提出了一个更广泛的联合扰动框架，并为几种重要的自然家庭建立了新的联合典型周期优化结果。

Asymptotic expansion of smooth interval maps

We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various different ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unicritical maps. This also solves a conjecture of Luzzatto in dimension 1. Combined with a result of Nowicki and Przytycki, these considerations im…

指数映射和二次多项式的分支轨迹：局部连通性、纤维平凡性和双曲性稠密性

Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

研究了二次（和单临界）多项式和指数映射的分支轨迹。我们概述了一个证明，指数分岔轨迹是连接的，这是一个模拟Douady和哈伯德的著名定理，（边界）的Mandelbrot集是连接的。对于这些参数空间，一个主猜测是双曲动力系统是稠密的。对于二次多项式，这将遵循著名的更强猜想，即分支轨迹（或等价的曼德尔布罗特集）是局部连通的。事实证明，形式上稍微弱一点的陈述是足够的，即分叉轨迹中的每个点都是参数射线的着陆点。对于指数映射，分支轨迹不是局部连通的。我们描述了一个不同的猜想（平凡的纤维），自然概括的作用，局部连通二次或单临界多项式。

A survey on MLC, Rigidity and related topics

The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic family $\{z^2+c\}_{c\in\mathbb{C}}$. We describe recent advances on MLC and the relations between MLC, the Density of Hyperbolicity Conjecture, the Rigidity Conjecture, the No Invariant Line Fields Conjecture, and the Triviality of Fibers Conjecture. We treat families of unicritical polynomials and rational maps as well as the exponential family and familie…

Large deviation principles of one-dimensional maps for Hölder continuous potentials

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

Julia sets of complex Hénon maps

There are two natural definitions of the Julia set for complex Hénon maps: the sets $J$ and $J^\star$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set $J^\star$, under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in the paper [For06], but the proof is incomplete. Our proof closely follows ideas from [For06], deviating at two points where substantial dissipativity is used. We show that $J = J^\star$ al…

多维非一致双曲性、鲁棒指数混合与吸引盆问题

Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem

本文证明双曲主导遍历cu-Gibbs态的三类吸引盆重合，解决一大类非一致双曲系统的吸引盆问题；还得到具多维非一致双曲主导分裂系统的鲁棒指数混合物理测度实例，两项结果源于扩展Gibbs-Markov-Young结构构造。

复动力学中的非一致双曲性

Non-uniform hyperbolicity in complex dynamics

我们研究了满足可和性条件的有理函数，这是一类沿临界轨道展开的弱条件。假设它们的适当的版本，我们得到许多很好的性质：存在一个唯一的，遍历的，和非原子共形测度的指数等于朱利亚集的豪斯多夫维数。存在关于这个共形测度的绝对连续的不变测度。朱利亚集的Minkowski维数严格小于2。要么朱利亚集是整个球面，要么动力学是不稳定的。对于这样的多项式和Blaschke积，朱利亚集是共形可去的。不变Fatou分支的边界的连通分支是局部连通的。最后，我们得到了Jakobson-Benedicks-Carleson定理的一个共形类似定理，并证明了Julia集的豪斯多夫维数对Mandelbrot集上几乎所有点关于调和测度的外连续性.部分结果推广到了区间上的多峰映射。

Topological invariance of the Collet-Eckmann condition for one-dimensional maps

This paper is devoted to study the topological invariance of several non-uniform hyperbolicity conditions of one-dimensional maps. In contrast with the case of maps with only one critical point, it is known that for maps with several critical points the Collet-Eckmann condition is not in itself invariance under topological conjugacy. We show that the Collet-Eckmann condition together with any of several slow recurrence conditions is invariant under topological conjugacy. This extends and gives a new proof of a result by Luzzatto and Wang that also applies to the complex setting.

Complex box bounds for real maps

In this paper we prove complex bounds, also referred to as a priori bounds, for real analytic (and even C3) interval maps. This means that we associate to such a map a complex box mapping (which provides a kind of Markov structure), together with universal geometric bounds on the shape of the domains. Such bounds show that the first return maps to these domains are well-controlled, and consequently form one of the corner stones in many recent results on one-dimensional dynamics: renormalisation theory, rigidity results, density of hyperbolicity, local connectivity.

Riemann 曲面上的对应关系与非一致双曲性

Correspondences on Riemann surfaces and non-uniform hyperbolicity

本文证明 Riemann 曲面上的对应关系具有弱双曲性，并应用于有理映射：对具有 4 个后临界点的映射，存在有限的不变曲线或图同伦类集合，所有曲线或图在迭代提升下被吸引，从而提供拓扑标准形。

Expansivity properties and rigidity for non-recurrent exponential maps

We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.

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.

Renormalization of almost commuting pairs

In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demontsrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses an attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.

Dynamics of post-critically finite maps in higher dimension

We study the dynamics of post-critically finite endomorphisms of P^k(C). We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a mild regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a weak transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins, thus partially extending to any dimension a result of Fornaess-Sibony and Ron…

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.

SL（2，R）上圈一致双曲性与零李雅普诺夫指数的二分性

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles

考虑由保测度动力系统$T:X \to X$和映射$A:X \to \mathit{SL}(2,\mathbb{R})$诱导的线性上循环$(T,A)$.当映射$A$保持固定时，我们讨论了$(T,A)$的上李雅普诺夫指数对动态$T$的依赖性.我们引入明确的条件上的上循环，允许扰动的动态，在弱和一致的拓扑结构，使指数下降任意接近零。在弱拓扑中，如果$X$是紧致连通流形，则对于$C^r$（$r \ge 1$）开稠密映射集$A$，要么对每个$T$，$(T,A)$是一致双曲的，要么对一般可测$T$，$(T,A)$的李雅普诺夫指数为零.对于连续的情形，我们得到：如果$X$的维数大于2，则对于$C^r$（$r \ge 1$）类属映射$A$，存在一个保体积同胚$T$的剩余集，对于该剩余集，要么$(T,A)$是一致双曲的，要么$(T,A)$的李雅普诺夫指数为零.

Chaotically driven sigmoidal maps

We consider skew product dynamical systems $f:Θ\times\mathbb{R}\toΘ\times\mathbb{R}, f(θ,y)=(Tθ,f_θ(y))$ with a (generalized) baker transformation $T$ at the base and uniformly bounded increasing $C^3$ fibre maps $f_θ$ with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for $f$ we prove that the presence of these fibres restricts considerably the possible structures of invariant measures - both topologically and measure theoretically, and that this finally allows to provide a "thermodynamic formula" for the Hausdorff …

Large entropy implies existence of a maximal entropy measure for interval maps

We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that $h_{\rm top}(f)>2\log\|f'\|_{\infty}/r$ possesses measures with maximal entropy.

Equilibrium measures at temperature zero for Hénon-like maps at the first bifurcation

We develop a thermodynamic formalism 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. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to+\infty$ minimizes the unstable Lyapunov exponent. …

Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

For strongly dissipative Hénon maps at the first bifurcation where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e., prove the existence and uniqueness of an invariant probability measure which maximizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $t\in\mathbb R$ is in a certain large interval and $J^u$ is the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

满足非增长条件的有理映射的一个连接引理

A connecting lemma for rational maps satisfying a no growth condition

本文引入并研究了复有理映射的一个不涉及增长条件的非一致双曲性条件。我们称之为后向收缩。我们证明了这个条件比Collet-Eckmann条件弱，比指数为1的可和性条件弱。我们的主要结果是向后收缩有理映射的一个连接引理，大致上说，我们可以扰动一个有理映射，使朱利亚集的每个临界轨道与一个不在临界点上积累的轨道连接起来。这个结果的证明是基于Thurston算法和拟共形映射的一些刚性性质。证明了后向收缩有理映射的朱利亚集的勒贝格测度为零，当它不是整个黎曼球面时.本文的基本工具是具有向后迭代的马尔可夫性的集合，这些向后迭代是真实一维动力学中美好区间的全纯模拟。

Golden mean Siegel disk universality and renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like ma…

几乎每一个实二次映射不是正则的就是随机的

Almost every real quadratic map is either regular or stochastic

我们证明了所有可能的实组合类型的重整化算子的一致双曲性。我们由此得到实二次族$P_c: x\mapsto x^2+c$中的无穷可重正化参数值集有零测度。这就产生了标题中的陈述（其中"正则“意味着有一个吸引循环，"随机”意味着有一个绝对连续的不变测度）。最后给出了MLC问题的一个应用。

Statistical stability for multidimensional piecewise expanding maps

We present sufficient conditions for the (strong) statistical stability of some classes of multidimensional piecewise expanding maps. As a consequence we get that a certain natural two-dimensional extension of the classical one-dimensional family of tent maps is statistically stable.