Hao Wu
Let $f$ be a Hénon-Sibony map (regular polynomial automorphism) of $\mathbb{C}^k$ and let $μ$ be the equilibrium measure of $f$. In this paper we prove that $μ$ is exponentially mixing for plurisubharmonic test functions.
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Mixing and CLT for Hénon-Sibony maps: plurisubharmonic observables
本文证明复Hénon映射及更一般的Hénon-Sibony映射的最大熵测度对多次调和可观测函数具有指数混合性和中心极限定理。
The bifurcation measure is exponentially mixing
本文证明紧凯勒流形上亚纯映射序列的一般混合定理，并由此推出带适当多标记点的有理映射族的分岔测度具有指数混合性。
Let $f$ be a holomorphic automorphism of a compact Kähler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $μ$ be its equilibrium measure. In this paper, we prove that $μ$ is exponentially mixing for all d.s.h.\ test functions.
Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also holds for automorphisms on compact Kaehler manifolds of higher dimension under a natural condition on their dynamical degrees.
Every complex H{é}non map is exponentially mixing of all orders and satisfies the CLT
本文研究复 Hénon 映射的统计性质，证明其最大熵测度对 Hölder 可观测量呈所有阶指数混合，并由此推出中心极限定理。
Exponential mixing of all orders on Kähler manifolds: (quasi-)plurisubharmonic observables
本文证明紧致Kähler流形上具有简单上同调作用的全纯自同构的最大熵测度对所有d.s.h.可观测函数具有全阶指数混合性，并由此推导出中心极限定理。
Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds
本文研究上同调作用简单的紧 Kähler 流形自同构，证明其唯一最大熵测度对 Hölder 可观测量具有所有阶指数混合性，并推出 CLT，结果覆盖所有正熵紧 Kähler 曲面自同构。
Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$
针对$\boldsymbol{\text{mathbb{P}}^k}$上一般双有理映射的最大熵测度，推广了Hénon映射下的所有阶指数混合及Hölder可观测量CLT结论，通过引入合适的检验函数空间解决了不定集导致的Hölder映射迭代不稳定问题。
We show that a topologically mixing $C^\infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.
We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and regularity of solutions of the cohomological equation for such automorphisms. Our method is based on the quantitative equidistribution results for polynomial maps combined with Diophantine estimates.
Let $f$ be a $C^{1+α}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $μ$. We show that if $f:(M,μ)\to (M,μ)$ is exponentially mixing then it is Bernoulli.
It is known that locally constant toral extensions of hyperbolic systems can never mix at an exponential rate. In this note we exhibit some examples of non-abelian locally constant compact extensions of the shift map which are exponentially mixing for Holder-$L^2$ observables. The proof rests on a result of Bourgain-Gamburd and a decoupling argument.
A Smooth Autonomous Flow with Exponential Mixing on the Three-Torus
该研究在三维环面上构造了具指数混合性质的光滑自治散度为零的向量场，解答了相关公开问题，通过外科手术从Anosov悬挂流得到目标向量场，还证明了保持指数混合的转移定理。
Non-monotonic velocity profiles are an inherent feature of mixing flows obeying non-slip boundary conditions. There are, however, few known models of laminar mixing which incorporate this feature and have proven mixing properties. Here we present such a model, alternating between two non-monotonic shear flows which act in orthogonal (i.e. perpendicular) directions. Each shear is defined by an independent variable, giving a two-dimensional parameter space within which we prove the mixing property over open subsets. Within these mixing windows, we use results from the billiards literature to est…
Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing
该研究证明有限视界的非周期线性重复 Lorentz 气体对任意 Hölder 可观测量均无指数或拉伸指数混合，并给出依赖 α 的多项式混合速度界。
Let $f: [0,1]\times [0,1] \setminus {1/2} \to [0,1]\times [0,1]$ be the $C^\infty$ endomorphism given by $$f(x,y)=(2x- [2x], y+ c/|x-1/2|- [y+ c/|x-1/2|]),$$ where $c$ is a positive real number. We prove that $f$ is topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue measure. Furthermore we prove that the speed of mixing is exponential.
Exponential decay of random correlations for random Anosov systems mixing on fibers
本文借助Birkhoff锥方法，证明了满足Anosov性与纤维拓扑混合的外力驱动曲面Anosov系统，其对应唯一随机SRB测度的Hölder可观测量淬灭随机关联具有指数衰减的统计特性。
Stretched-exponential mixing for surface semiflows and Anosov flows
本文证明曲面悬挂半流在屋顶函数 Lipschitz 且非局部常上同调时关于 SRB 测度拉伸指数混合，并推广至双曲斜积、吸引子及余维一拓扑混合 Anosov 流。
Mixing Rates for Analytic Observables: Toral Endomorphisms and Anosov Blaschke Deformations
该研究分析解析观测值的混合率，构造两类Anosov Blaschke形变，证明真正非线性系统存在双指数混合，且尖锐指数不连续，还为环面自同构及高阶作用建立双指数界。
We consider the Teichmuller flow on the unit cotangent bundle of the moduli space of compact Riemann surfaces with punctures. We show that it is exponentially mixing for the Ratner class of observables. More generally, this result holds for the restriction of the Teichmuller flow to an arbitrary connected component of stratum. This result generalizes [AGY] which considered the case of strata of squares.
Countable Markov shifts with exponential mixing
本文针对可数马尔可夫移位上的势，给出其强正递归性与相关性指数衰减等价的一组条件，并通过量化移位在边界处的行为完成证明，揭示了两者的等价关系。
The aim of this paper is to establish exponential mixing of frame flow for the measure of maximal entropy on a convex cocompact hyperbolic manifold. Consequences include results on the decay of matrix coefficients and on effective equidistribution of holonomies. The main technical point is a spectral bound on certain twisted transfer operators, which we obtain by building on Dolgopyat's framework. This extends and strengthens earlier work of Dolgopyat, Stoyanov, Pollicott, and others.
Strong Positive recurrence for potential and exponential mixing of equilibrium states of surface diffeomorphisms
本文研究了具有正熵的表面微分同胚的大类势能的强正回复性，并建立了平衡态的指数衰减相关函数和有效内在遍历性。
Quantitative exponential mixing for the randomized Chirikov standard map
本文研究了随机Chirikov标准映射在二维环面上的混合性质，通过建立不可压缩随机动力系统准则，证明了在激励强度足够大时，随机映射具有几乎必然的定量指数混合性，并给出了更温和的参数条件以获得定性指数混合性和增强的耗散性。
Consider the skew product $F:\mathbb{T}^2 \to \mathbb{T}^2$, $F(x,y)= (f(x),y+τ(x))$, where $f:\mathbb{T}^1\to \mathbb{T}^1$ is a piecewise $\mathscr{C}^{1+α}$ expanding map on a countable partition and $τ:\mathbb{T}^1 \to \mathbb{R}$ is piecewise $\mathscr{C}^1$. It is shown that if $τ$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $τ$ and $f$, then $F$ is mixing at a stretched-exponential rate.
Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.
Expanding on average diffeomorphisms of surfaces: exponential mixing
本文证明了闭曲面上保体积微分同胚的平均扩张元组相关的 Bernoulli 随机动力系统具有指数混合性质。
We consider the skew product $F: (x,u) \mapsto (f(x), u + τ(x))$, where the base map $f : \mathbb{T}^{1} \to \mathbb{T}^{1}$ is piecewise $\mathcal{C}^{2}$, covering and uniformly expanding, and the fibre map $τ: \mathbb{T}^{1} \to \mathbb{R}$ is piecewise $\mathcal{C}^{2}$. We show the dichotomy that either this system mixes exponentially or $τ$ is cohomologous (via a Lipschitz function) to a piecewise constant.
We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…
We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the pha…
Loss of Exponential Mixing in a Non-Monotonic Toral Map
该研究针对正交帐篷形剪切复合的二维环面保测映射，借助混沌台球成果证明其强混合性与赫尔德可观测量的多项式关联衰减，指出其为遍历边界分段线性映射的典型，存在减慢混合的零测集且扰动下可生椭圆岛。
Exponential mixing of measures of maximal entropy for certain skew products
本文研究了某些斜积中最大熵测度的指数混合性，通过纤维熵与李雅普诺夫指数的连续性关系，证明了满足强正回复性质的斜积具有有限数量的最大熵测度，且这些测度具有良好的统计性质。
Exponential mixing of frame flows for three dimensional manifolds of quarter-pinched negative curvature
研究了三维四分针尖负曲率流形的框架流指数混合性，通过构造Young塔模型和加强Dolgopyat估计，扩展到Anosov流的torus扩展。
Let $M$ be a closed $3$-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.
Let $Γ$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $Γ\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow an…
On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the …
Exponential mixing of the Teichmüller flow on affine invariant manifolds
本文针对有限型非例外可定向曲面，利用符号编码方法重新证明了 Avila 和 Gouezel 的结果：Teichmüller 流关于任意遍历 SL(2,R)-不变 Borel 概率测度具有指数混合性。
We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geometric consequence is that the $\SL(2,\R)$ action in the moduli space has a spectral gap.
Exponential Mixing for Hyperbolic Flows on Non-Compact Spaces
研究非紧相空间上的双曲流，证明对足够光滑的观测函数，其SRB测度下的相关函数指数衰减，通过三诱导方案构建均匀双曲Poincaré映射，应用在模曲面测地流的指数混合性证明中。
We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mi…
Strong positive recurrence and exponential mixing for diffeomorphisms
本研究引入闭流形上微分同胚的强正递归（SPR）性质，证明其可通过具谱隙的可数状态马尔可夫移位编码，推导最大熵测度的多种遍历性质，验证正熵C∞曲面微分同胚及更多低正则、高维情形满足SPR。
Exponential mixing for singular skew-products
本文研究了非一致扩张映射与分段光滑映射构成的奇异斜积系统，证明了在允许纤维映射奇异行为的温和假设下，该映射相对于唯一的 SRB 测度呈指数混合，扩展了已有结论。
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。
The Bernoulli property for $PSL_2\mathbb{R}$ skew products over exponentially mixing base
本文证明当 $PSL_2\mathbb{R}$ 斜积的余循环具有非零 Lyapunov 指数时，该斜积具有 Bernoulli 性质，且无需指数混合条件，提供了新方法。
Exponential mixing for Hamiltonian shear flow
研究环面$\bbT^2$上交替两哈密顿剪切的平流方程，通过随机交替向量场注入随机性，证明其具正最大李雅普诺夫指数与指数混合，并将框架应用于随机时间皮埃尔安贝尔模型及类奇里科夫标准映射模型。
We study the class of open continuous-time mechanical particle systems introduced in the paper by Khanin and Yarmola [Ergodic Properties of Random Billiards Driven by Thermostats. Commun. Math. Phys. 320, no. 1, 121-147 (2013)]. Using the discrete-time results from that paper we demonstrate rigorously that, in continuous time, a unique steady state exists and is sub-exponentially mixing. Moreover, all initial distributions converge to the steady state and, for a large class of initial distributions, convergence to the steady state is sub-exponential. The main obstacle to exponential convergenc…
Exponential scalar mixing for the 2D Navier-Stokes equations with degenerate stochastic forcing
本文针对仅含有限个受迫模态的退化随机二维 Navier-Stokes 方程，融合渐近强 Feller 框架与混合理论，证明其输运的被动标量呈指数混合。
Exponential mixing for heterochaos baker maps and the Dyck system
本文研究非 Markov 分段仿射异混沌面包师变换，证明其所有阶混合，并对中心主要扩张或收缩情形得到 Hölder 函数的指数混合，进而用于 Dyck 系统。
Exponential mixing of frame flows for convex cocompact hyperbolic manifolds
本文证明了凸余紧双曲流形标架流的指数混合，通过基于Dolgopyat方法建立转移算子的谱界，并给出了矩阵系数渐近公式与闭测地线全纯指数等分布等应用。
Exponential mixing for the randomly forced NLS equation
本文研究空间局域阻尼和随机噪声作用下受随机外力的非线性薛定谔方程不变测度的指数混合，利用指数渐近紧性与控制性质建立其遍历性，拓展了相关色散方程统计行为的系列研究。
There exists a $C^2$-open and $C^1$-dense subset of vector fields exhibiting singular-hyperbolic attracting sets (with codimension-two stable bundle), in any $d$-dimensional compact manifold ($d\ge3$), which mix exponentiallu with respect to any physical/SRB invariant probability measure. More precisely, we show that given any connected singular-hyperbolic attracting set for a $C^2$-vector field $X$, there exists a $C^1$-close multiple of $X$ of class $C^2$, generating a topologically equivalent flow, which is robustly exponentially mixing with respect to any physical measure for all vector fi…
Exponential mixing of constrained random dynamical systems via controllability conditions
本文提出确定性可控性条件以证明随机受迫约束动力系统的指数混合性质，并将其应用于弹塑性非光滑模型，获得新的遍历性结果，展示了控制理论工具处理约束系统正则性问题的价值。
We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set at a certain time, for almost all random orbits, decays in a (stretched) exponential fashion, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications, we obtain almost sure stretched exponential decay of random correlations for Viana maps, as for a class of non-uniformly expanding local …
Exponential mixing for nonlinear Schrödinger equations perturbed by bounded degenerate noise
研究非线性薛定谔方程在有界退化噪声扰动下的指数混合性，通过引入线性化系统的渐近紧致性增强耦合方法，结合全局稳定性、非线性平滑性和几何控制，建立新的指数混合准则。
We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $C^2$ open subset of vector fields in any $d$-…
Exponential mixing for random nonlinear wave equations: weak dissipation and localized control
本文提出随机动力系统指数混合的新判据，应用于具退化阻尼和局部化噪声的非线性波动方程，解决弱耗散与随机性交互的挑战。
Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows
本文研究了三维周期盒中随机化ABC流的拉格朗日性质，证明了其正的最大Lyapunov指数与几何遍历性，并据此论证了该流场对被动标量是指数混合场，且为被动向量提供了普适理想运动学发电机实例。
We prove exponential mixing for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations (Theorem 3)
We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.
For a large class of quickly mixing dynamical systems, we prove that the error in the almost sure approximation with a Brownian motion is of order O((log n)^a) with a $\ge$ 2. Specifically, we consider nonuniformly expanding maps with exponential and stretched exponential decay of correlations, with one-dimensional H{ö}lder continuous observables.
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