Uniformly perfect analytic and conformal attractor sets

一类非一致双曲吸引子的关联衰减

Decay of Correlations for some Non-Uniformly Hyperbolic Attractors

针对光滑实现问题，研究非一致双曲吸引子动力系统的关联衰减，通过对一致双曲微分同胚施加减速程序构造系统，证明其关联存在多项式上下界，结果适用于Smale-Williams螺线及其小扰动。

On the uniform perfectness of the boundary of multiply connected wandering domains

We investigate in which cases the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.

The well-posedness and the regularity of global attractor for a couple stress fluid through porous layer with the local thermal non-equilibrium effect

In the article, we aim to investigate the well-posedness of solution and the regularity of the global attractor for the couple stress fluid in saturated porous media with the local thermal non-equilibrium effect. To be more specific, we firstly show the existence and uniqueness of global weak solution to the model by making use of the standard Galerkin method. Second, relying on verifying the uniformly compact condition required, we prove the existence of the global attractor of the model in the space where the weak solution resides. Finally, we improve the regularity of global attractor by un…

Hereditarily Non Uniformly Perfect Sets

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an example of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

McKean--Vlasov 随机反应扩散方程的平均原理与拉回吸引子收敛性

Averaging Principle and Pullback Attractor Convergence for McKean--Vlasov Stochastic Reaction--Diffusion Equations

该研究针对环面上带快速振荡系数的McKean--Vlasov随机反应扩散方程建立三个平均原理，证明解的均方收敛、有界整体解的均方距离一致消失及拉回吸引子上半连续收敛到全局吸引子，并给出相关应用模型。

在均匀扰动的非光滑区域中具有诺伊曼边界条件的反应扩散方程全局吸引子的一致\(L^{\infty}\)有界性

Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains

研究一族在非光滑区域上具诺伊曼边界条件的半线性抛物方程，通过特定条件建立适定性与吸引子存在性，利用多种方法证明吸引子族\(L^\infty\)一致有界，且在区域体积收敛时关于强\(H^1\)拓扑在\(\mu = 0\)处上半连续。

Decay of correlations, quantitative recurrence and logarithm law for contracting Lorenz attractors

In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.

Exponential decay of correlations for nonuniformly hyperbolic flows with a C^{1+α} stable foliation, including the classical Lorenz attractor

We prove exponential decay of correlations for a class of $C^{1+α}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.

On uniformly disconnected Julia sets

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $\mathbb{S}^n$, for $n\geq 2$. Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if $E$ is a compact, uniformly perfect and uniformly disconnec…

Hereditarily non Uniformly Perfect non-Autonomous Julia Sets

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the…

On cut sets of attractors of iterated function systems

In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.

On the volume of singular-hyperbolic sets

An attractor $Λ$ for a 3-vector field $X$ is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that $C^{1+α}$ singular-hyperbolic attractors, for some $α>0$, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if $Λ$ is a singular-hyperbolic attractor for $X$ then either it has zero volume or $X$ is an Anosov flow. We also present examples of $C^1$ singular-hyp…

Multidimensional Rovella-like attractors

We present a multidimensional flow exhibiting a Rovella-like attractor: a transitive invariant set with a non-Lorenz-like singularity accumulated by regular orbits and a multidimensional non-uniformly expanding invariant direction. Moreover, this attractor has a physical measure with full support but persists along certain0909.1033 submanifolds of the space of vector fields. As in the 3-dimensional Rovella-like attractor, this example is not robust. The construction introduces a class of multidimensional dynamics, whose suspension provides the Rovella-like attractor, which are partially hyperb…

Rotation intervals and entropy on attracting annular continua

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

一致双曲流的良好诱导方案及其在关联指数衰减中的应用

Good inducing schemes for uniformly hyperbolic flows, and applications to exponential decay of correlations

该研究针对$C^{1+α}$流的Axiom A吸引子，构造了诱导集为光滑嵌入不稳定圆盘的指数返回可数马尔可夫扩张，规避了马尔可夫划分边界不规则问题，为SRB测度关联指数衰减问题提供了初等方法。

厚薄非自治朱利亚集

Thick-Thin non-Autonomous Julia Sets

本文构造了一个非自治多项式序列，其朱利亚集既非一致完全也非遗传非一致完全，并证明其可分解为一致完全集的可数并集与HNUP子集。

非线性平面迭代函数系统（IFS）的表面排斥子与吸引子的维数

Dimensions of surface repellers and attractors of non-linear planar IFSs

该研究证明了$C^r$通用表面排斥子的维数公式，推广了满足强分离条件的平面非线性迭代函数系统吸引子的维数结果，并弱化了Weierstrass型函数图像二分结果的正则性要求。

SRB measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one $SRB$ measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including …

洛伦兹吸引子的饱和集的拓扑熵

Topological entropy of saturated sets for Lorenz attractors

该研究针对连续流的饱和集拓扑熵，在奇异双曲动力学框架下，证明了几何洛伦兹吸引子相关的拓扑熵公式，发现其与一致双曲情形存在差异，表明离散时间饱和熵理论无法直接推广到奇异流。

Properties of the maximal entropy measure and geometry of Hénon attractors

We consider an abundant class of non-uniformly hyperbolic $C^2$-Hénon like diffeomorphisms called strongly regular and which corresponds to Benedicks-Carleson parameters. We prove the existence of $m>0$ such that for any such diffeomorphism $f$, every invariant probability measure of $f$ has a Lyapunov exponent greater than $m$, answering a question of L. Carleson. Moreover, we show the existence and uniqueness of a measure of maximal entropy, this answers a question of M. Lyubich and Y. Pesin. We also prove that the maximal entropy measure is equi-distributed on the periodic points and is fin…

条件性随机稳定性与均匀超几何集上的平衡态

Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets

该研究通过条件性随机稳定性分析，揭示了均匀超几何集上平衡态在噪声趋于零时的收敛性质，并扩展至Axiom A微分同胚的瞬态混沌场景。

稠密分支树作为类 Hénon 和类 Lozi 吸引子的模型

Densely branching trees as models for Hénon-like and Lozi-like attractors

本文证明类 Hénon 与类 Lozi 吸引子可由具有稠密分支点的度量树自然扩张建模，从而统一刻画其拓扑与动力学，并排除更简单的一维模型。

Large deviations bound for semiflows over a non-uniformly expanding base

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from t…

Parabolic carpets

We introduce and study a family of non-conformal and non-uniformly contracting iterated function systems. We refer to the attractors of such systems as parabolic carpets. Roughly speaking they may be thought of as nonlinear analogues of self-affine carpets which are allowed to have parabolic fixed points. We compute the $L^q$-spectrum of a class of weak Gibbs measures supported on parabolic carpets as well as the box dimensions of the carpet itself.