R. Markarian, M. J. Pacifico, J. Vieitez
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.
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On the ergodicity of anti-symmetric skew products with singularities and its applications
本文提出一种新方法证明具有端点奇异性的区间交换变换斜积的遍历性，适用于非对称协环，改进了传统对数奇异性的限制，并应用于局部哈密顿流谱分解的误差项等分布研究。
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.
Quantitative infinite mixing for non-compact skew products
本文研究纤维为实直线的非紧致斜积系统，在可及性假设下，对基上吉布斯测度与纤维上勒贝格测度之积这一无限不变测度，证明了一类稠密可观测量满足定量克里克伯格混合。
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.
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。
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 consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^d)$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uni…
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。
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。
Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish u…
We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows of cyclic covers.
We study global-local mixing for a family of accessible skew products with an exponentially mixing base and non-compact fibers, preserving an infinite measure. For a dense set of almost periodic global observables, we prove rapid mixing; and for a dense set of global observables vanishing at infinity, we prove polynomial mixing. More generally, we relate the speed of mixing to the "low frequency behaviour" of the spectral measure associated to our global observables. Our strategy relies on a careful choice of the spaces of observables and on the study of a family of twisted transfer operators.
At the end of 1960-ths Yu.S.Ilyashenko stated the problem: is it true that for any one-dimensional holomorphic foliation with singularities on a Stein manifold leaves intersecting a transversal disc can be uniformized so that the uniformization function would depend holomorphically on the transversal parameter? In the present paper we construct a two-dimensional counterexample. This together with the previous result of Yu.S.Ilyashenko (Lemma 1) implies existence of a counterexample given by a foliation on affine (projective) algebraic surface by level curves of a rational function with singula…
We consider reparametrizations of Heisenberg nilflows. We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial time-changes within a dense subspace of smooth time-changes are mixing. Equivalently, in the language of special flows, we consider special flows over linear skew-shift extensions of irrational rotations of the circle. Without assuming any Diophantine condition on the frequency, we define a dense class of smooth roof functions for which the corresponding special flows are mixing. Mixing is produced by a mechanism known as stretching of Birkhoff sums. The complement o…
Step skew products with interval fibres and a subshift as a base are considered. It is proved that if the fibre maps are continuous, piecewise monotone, expanding and surjective and the subshift has the specification property and a periodic orbit such that the composition of the fibre maps along this orbit is mixing, then the corresponding step skew product has the specification property.
On Transitivities for Skew Products
本文研究了非紧纤维斜积的传递性性质，探讨了普遍性条件和超循环准则与拓扑传递、弱混合和混合动力学性质的关系。
We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density 1. These results strengthen Parry's result that such systems are uniquely ergodic. Extending the work of Furstenberg, we prove an analogous theorem for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper u…
In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the aver…
We prove that for any compact connected Lie group G and a typical interval exchange transformation T, not isomorphic to a rotation, the skew product of T with a typical G-valued function, constant on the intervals, is weakly mixing.
We consider skew extensions of expanding maps by compact Lie groups. For a class of natural invariant measures, we prove an explicit lower bound on the rate of (exponential) mixing involving topological pressure. Proof uses representation theory and some elliptic pde estimates.
Ergodicity of explicit logarithmic cocycles over IETs
本文证明了区间交换变换上具有对数奇异性的余循环所构造的斜积扩张的遍历性，并给出了亏格二中极小局部哈密顿流的显式遍历扩张例子。
We study skew-products of the form (x,ω)\mapsto (Tx, ω+ϕ(x)) where T is a nonuniformly expanding map on a space X, preserving a (possibly singular) probability measure \tildeμ, and ϕ:X\to S^1 is a C^1 function. Under mild assumptions on \tildeμand ϕ, we prove that such a map is exponentially mixing, and satisfies the central and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi.
We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\om,x)=(\te\om,T_\om x)$ together with a $T$-invariant measure, whose base map $\te$ satisfies certain topological and mixing conditions and the maps $T_\om$ on the fibers are certain non-singular distance expanding maps. Our results hold true when $\te$ is either a sufficiently fast mixing Markov shift or a (non-uniform) Young tower with at least one periodic point and polynomial tails. %In fact, our conditions will be satisfied %when $\om$ is the whole orbit t…
We investigate the decay rates of correlations for nonuniformly hyperbolic systems with or without singularities, on piecewise Hölder observables. By constructing a new scheme of coupling methods using the probability renewal theory, we obtain the optimal bounds for decay rates of correlations for a large class of such observables. We also establish the alpha-mixing property for time series generated by these systems, which leads to a vast ranges of limiting theorems. Our results apply to rather general hyperbolic systems with singularities, including Bunimovich flower billiards, semidispersin…
Irregular sets for a class of skew product transformations
针对由唯一遍历同胚系统驱动且纤维满足Anosov与拓扑混合性质的斜积变换，本文证明若某纤维上连续可观测量的不规则集非空，则$\boldsymbol{\text{P}}$-几乎处处纤维上的该集合必非空、剩余且具全纤维拓扑熵，关键技术为纤维规格性质的轨道粘合。
We provide an explicit rigorous derivation of a diffusion limit - a stochastic differential equation with additive noise - from a deterministic skew-product flow. This flow is assumed to exhibit time-scale separation and has the form of a slowly evolving system driven by a fast chaotic flow. Under mild assumptions on the fast flow, we prove convergence to a stochastic differential equation as the time-scale separation grows. In contrast to existing work, we do not require the flow to have good mixing properties. As a consequence, our results incorporate a large class of fast flows, including t…
We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property (a property originally proved by Ratner for the horocycle flow), i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has impli…
We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of any genus are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations.
We consider a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a class of (piecewise) partially hyperbolic skew products whose behavior on the preserved fibration is dominated by the expansion of the base map. In particular we apply the results to power law mixing toral extensions. It turns out that in this case, the dependence of the physical measure on small deterministic perturbations, in a suitable anis…
Locally conformal expanding Actions: Markov Partition and Thermodynamic of the induced skew product
本文研究了由有限个 $C^{1+α}$ 共形局部微分同胚生成的拓扑混合局部共形半群作用，提出了满足有限像和有限环性质的可数 Markov 分割，并基于诱导映射等价于可数型子移位的塔构造，证明了可提升测度的存在性及诱导斜积的热力学形式。
We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,μ)$, where $μ$ is a Gibbs measure, an aperiodic Hölder continuous cocycle $ϕ:M\to \mathbb R$ with zero mean and a zero-entropy flow $(K_t,N,ν)$. We then study the skew product $$ T_ϕ(x,y)=(Tx,K_{ϕ(x)}y), $$ acting on $(M\times N,μ\times ν)$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_ϕ$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a s…
We discuss dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robust topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.
Doubly Intermittent Full Branch Maps with Critical Points and Singularities
研究一类具有两个中性不动点及临界点或奇异点的一维全分支映射，在温和假设下证明唯一混合绝对连续不变测度存在，并分析其相关衰减与极限定理。
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。
We study the problem of lifting various mixing properties from a base automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where $\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$, $\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$ and $\tfs$ acts on the product space $(X\times Y,\cb\ot\cc,μ\otν)$ by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever $T$ is ergodic (mildly mixing, mixing) but $\tfs$ is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor $\ca\subset\cc$ of $\cs$ the cor…
In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew prod…
Statistical properties for mixing Markov chains with applications to dynamical systems
本文为弱混合 Markov 系统建立了抽象的指数大偏差估计和中心极限定理，并应用于混合随机-拟周期动力系统等斜积动力学，适用于无谱隙性质的系统。
Let $T=(T_t^f)_{t\in \mathbb{R}}$ be a special flow built over an IET $T : T \to T$ of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T. We show that $T$ satisfies so-called switchable Ratner's property. A consequence of this fact is that such flows are mildly mixing. Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.
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…
On the variational principle for a class of skew product transformations
本文针对满足特定性质的唯一遍历同胚驱动的斜积变换，利用纤维指定性质证明两类变分原理，验证其具有Gundlach-Kifer指定性质，得到Hölder势的唯一平衡态并给出相关系统实例。
We consider completely irrational nilflows on any nilmanifold of step at least $2$. We show that there exists a dense set of smooth time-changes such that any time-change in this class which is not measurably trivial gives rise to a mixing nilflow. This in particular reproves and generalizes to any nilflow (of step at least $2$) the main result proved in [AFU] for the special class of Heisenberg (step $2$) nilflows, and later generalized in [Rav2] to a class of nilflows of arbitrary step which are isomorphic to suspensions of higher-dimensional linear toral skew-shifts.
Khintchin conjecture and related topics
本文研究了Khintchin类函数与序列的性质，探讨了其在紧阿贝尔群上极小映射中的应用，并提出了相关开放问题。
We study the spectral gap for transfer operators of the skew product $F: \mathbb{T}^d\times \mathbb{T}^\ell\to \mathbb{T}^d\times \mathbb{T}^\ell$ given by $F(x,y)=(Tx, y+τ(x) \pmod{ \mathbb{Z}^\ell})$, where $T: \mathbb{T}^d\to \mathbb{T}^d$ is a $C^\infty$ uniformly expanding endomorphism, and the fiber map $τ: \mathbb{T}^d\to \mathbb{R}^\ell$ is a $C^\infty$ map. We construct a Hilbert space $\mathcal{W}^{-s}$ for any $s<0$, which contains all the Hölder functions of Hölder exponents $|s|$ on $ \mathbb{T}^d\times \mathbb{T}^\ell$. Applying the method of semiclassical analysis, we obtain the…
Limit theorems for inhomogeneous $ϕ$-mixing Markov chains
该研究证明非均匀φ-混合马尔可夫链的极限定理，验证其在两类条件下成立，应用于随机环境中斜积等，还得到最优CLT速率等成果。
Discrete-time dynamics, step-skew products, and pipe-flows
本文通过三种动态系统近似研究了广义ergodic动力学的规律，证明了这些系统的路径空间分布可以任意接近，重新确认了无法区分确定性或随机过程以及连续或离散时间序列的原则。
We prove a structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows with a compactly generated Abelian acting group (e.g. $\Z^d$-flows and $\R^d$-flows). The main result states that every such extension apart from a coboundary can be represented by a perturbation of a so-called Rokhlin skew product. We obtain as a corollary that the topological ergodic decomposition of the skew product extension into prolongations is continuous and compact with respect to the Fell topology on the hyperspace. The right translation acts minimally on this d…
Dynamics of Ostrowski skew-product: I. Limit laws and Hausdorff dimensions
本研究利用转移算子研究Ostrowski斜积动力系统，证明其存在具分段全纯密度的绝对连续不变测度与指数混合性，推导了算术随机变量的中心极限定理，并得到有界数字集Hausdorff维数的Bowen–Ruelle型谱隐式估计。
The measure of maximal entropy for random skew products on compact complex surfaces
研究紧复曲面上自同构群随机斜积的最大纤维熵测度，证明了在适当条件下该测度唯一存在，并给出了其正则构造与混合性。
Lower bounds on mixing rates for a class of mixing flows on surfaces
研究具有不对称对数奇点的局部哈密顿流，证明相关函数衰减不能一致快于log t的幂次，并给出下界估计，补充了上界结果。
Ergodic properties of infinite extension of symmetric interval exchange transformations
本文研究了对称区间交换变换的无限扩张的遍历性质，证明了特定余循环的斜积在遍历基下是遍历的，并在唯一遍历性假设下推广到更一般的函数，最后通过弱混合情形得到无限遍历指数。
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.
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 via invariant foliations and relatively Anosov homeomorphisms
该研究构造了一类闭流形上的保体积同胚，实现了Hölder可观测量的相关性指数衰减，肯定了Dolgopyat和Pesin的相关猜想，还给出了无穷多个无对应微分同胚的此类流形实例。
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 $d$-…
We consider skew products $$T_ϕ:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!ϕ(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\frak g$, $F_1:X\to X$ the time-one map of a measure-preserving flow, and $ϕ\in C^1(X,G)$ a cocycle. Then, we define the degree of $ϕ$ as a suitable function $P_ϕM_ϕ:X\to\frak g$, we show that it transforms in a natural way under Lie group homomorphisms and under the relation of $C^1$-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irre…
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