Vitaly Bergelson, Alexander Gorodnik
The first part of the paper contains a survey of weakly mixing group actions of general group, and the second part discusses a special example of a weakly mixing action -- the SL(2,Z)-action on the torus T^2.
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This paper is concerned with relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems. It is shown that weakly mixing implies topologically weakly mixing and sensitivity for measurable systems with a fully supported measure; and topological weakly mixing implies sensitivity for general dynamical systems. However, the inverse conclusions are not true and some counterexamples are given. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weaken.
We show existence of smooth, weakly mixing reparametrizations of some linear flows on $\mathbb{T}^2$ for which all orbits sampled at prime times are dense.
The notion of $Δ$-weakly mixing subsets is introduced for countable torsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the existence of $Δ$-weakly mixing subsets, and while there exists a finitely generated torsion-free discrete solvable group action which has positive topological entropy but without any $Δ$-weakly mixing subsets.
We show that in a typical polygon the billiard map as well as its associated subshift obtained by coding orbits by the sequence of sides they visit are topologically weakly mixing.
Analytic weakly mixing diffeomorphisms on odd dimensional spheres
本文提出共轭逼近方案，构造奇数维球面上关于体积测度弱混合的实解析微分同胚。
The notion of $Δ$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $Δ$-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of $Δ$-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. Moreover, Our results generalize several well known results and also answer several open questions.
Anti-classification results for weakly mixing diffeomorphisms
该研究将遍历理论反分类结果推广至弱混合系统，证明其同构与Kakutani等价关系非Borel集，表明无法通过可计算不变量对这类系统分类，还针对特定微分同胚验证了结论。
We prove that a class of infinite measure preserving transformations, satisfying a "strong" weak mixing condition, generates all rigidity sequences of all conservative ergodic invertible measure preserving transformations defined on a Lebesgue $σ$-finite measure space.
The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an introductory section recalling some definitions and classic results, we establish some necessary conditions for this to happen, and in the final section we show, using Ratner's theory, that the enveloping semigroup of the `time one map' of a classical horocycle flow is weakly mixing.
The Return Map of the Cross Section of Horizontally Short Lattice Surfaces is Weakly Mixing
本文证明水平短平移曲面空间上不稳定水平环流回归映射的弱混合性，将Cheung-Quas的方形环面结果推广至所有格曲面，方法基于弱混合判据与Siegel-Veech变换的定量界。
Let $(X, T)$ be a weakly mixing minimal system, $p_1, \cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,\cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $x\in X_0$ $$\{ (T^{p_1(n)}x, \cdots, T^{p_d(n)}x): n\in \mathbb{Z}\}$$ is dense in $X^d$.
We construct a rank one infinite measure preserving transformation $T$ such that for all sequences of nonzero integers $\{k_{1},..., k_{r}\}$, $T^{k_{1}}\times...\times T^{k_{r}}$ is ergodic.
We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.
In this paper, we propose a mild condition, named Condition $(**)$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $σ$-algebra is a characteristic $σ$-algebra for the averages along a collection satisfying Condition $(**)$. We introduce the notion of $Δ$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $Δ$-weakly mixing subsets along a collection of "good" sequences. As a consequence, we show that positive topological entropy implies multi-variant Li-Yorke c…
We study the behavior of maximal multiplicities $mm (R^n)$ for the powers of a weakly mixing automorphism $R$. For some special infinite set $A$ we show the existence of a weakly mixing rank-one automorphism $R$ such that $mm (R^n)=n$ and $mm(R^{n+1}) =1$ for all $n\in A$. Moreover, the cardinality $cardm(R^n)$ of the set of spectral multiplicities for $R^n$ is not bounded. We have $cardm(R^{n+1})=1$ and $cardm(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: $mm(R^{n}) =n$ for $n=1,2,3,..., 2009, 2010$ but $mm(T…
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.
We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation. We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting.
Pointwise Ergodic Theorems for Higher Levels of Mixing
本文在弱混合和强混合系统中强化了Birkhoff ergodic定理，证明了弱混合系统的点wise定理比Wiener-Wintner定理更强，并分别刻画了弱混合和强混合系统。
In this paper it is shown that every non-periodic ergodic system has two topologically weakly mixing, fully supported models: one is non-minimal but has a dense set of minimal points; and the other one is proximal. Also for independent interests, for a given Kakutani-Rokhlin tower with relatively prime column heights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower with same property, which can be used in Weiss's proof of the Jewett-Krieger's theorem and the proofs of our theorems. Applications of the results are given.
For a commutative non-autonomous dynamical system we show that topological transitivity of the non-autonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in non-autonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic non-autonomous system on intervals, it is proved that weakly mixing implies Devaney chaos. Given a periodic non-autonomous system, it is shown…
We construct an increasing sequence of natural numbers $(m_n)_{n=1}^{+\infty}$ with the property that $(m_n þ[1])_{n\geq 1}$ is dense in $\T$ for any $þ\in \R\setminus \Q$, and a continuous measure on the circle $μ$ such that $\lim_{n\to +\infty}\int_{\T}\|m_nθ\|dμ(θ)=0$. Moreover, for every fixed $k\in \N$, the set $\{n\in \N:\,k\nmid m_n \}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.
We give an example of a weakly mixing vector field $b\in L^\infty([0,1],\text{BV}(\mathbb{T}^2))$ which is not strongly mixing. The example is based on a work of Chacon who constructed a weakly mixing automorphism which is not strongly mixing on $([0,1],\mathcal{B}([0,1]),|\cdot|)$, where $\mathcal{B}([0,1])$ are the Borel subsets of $[0,1]$ and $|\cdot|$ is the one-dimensional Lebesgue measure.
For $n\geq 1$ we consider the class JP($n$) of dynamical systems whose every ergodic joining with a Cartesian product of $k$ weakly mixing automorphisms ($k\geq n$) can be represented as the independent extension of a joining of the system with only $n$ coordinate factors. For $n\geq 2$ we show that, whenever the maximal spectral type of a weakly mixing automorphism $T$ is singular with respect to the convolution of any $n$ continuous measures, i.e. $T$ has the so-called convolution singularity property of order $n$, then $T$ belongs to JP($n-1$). To provide examples of such automorphisms, we …
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.
We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{\infty}$-diffeomorphisms which preserve both a smooth volume $ν$ and a measurable Riemannian metric is dense in $\mathcal{A}_α \left(M\right)= \overline{ \left\{h \circ S_α \circ h^{-1} : h \in \text{Diff}^{\infty}\left(M, ν\right) \right\}}^{C^{\infty}}$ for every Liouvillean number $α$. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly c…
In this note, we construct torsion-free countable, amenable, weakly mixing groups, which answer a question of V. Bergelson. Some results related to verbal subgroups and crystallographic groups are also presented.
Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts
本文研究一阶子位移的词复杂度，构造了容许弱混合测度且复杂度渐近接近1.5q的子位移，并证明非里程表一阶变换对应的子位移满足limsup p(q)-1.5q=∞，且该结果最优。
We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.
Interval Translation Maps with Weakly Mixing Attractors
针对具有 Cantor 吸引子的双参数区间平移映射族，本文利用类似 Rauzy 归纳的参数空间动力系统 $G$，判定其线性递归性或弱混合性。
We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if $A \subset \mathbb N$ is a weakly mixing set and $p(x) \in \mathbb Z[x]$ a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers $N$ can be represented as $N = n_1 + n_2$, where $p(n_1) + m,\ p(n_2) + m \in A$ for some $m \in A$.
We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k} tends to infinity, or such that n_{k} divides n_{k+1} for each k, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, fo…
The article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map $f\times f^2 \times...\times f^m$, where $f\colon X\ra X$ is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is $Δ$-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Colloq. Math. 120 (2010), no. 1, 127--138]. The first one is a multi-transitive non weakly mixing system, and the second…
Weak mixing in rational billiards
本文完全刻画了在几乎每个方向上弱混合的有理多边形与平移曲面，证明了Gutkin的长期猜想，并指出此类曲面不容许到圆的仿射因子映射。
Shadowing, average shadowing and transitive properties of multiple mappings
该研究从集值视角定义多重映射的多类动力学性质，证明单映射的对应性质不必然传递给其构成的多重映射，给出多重映射满足传递等性质的充分条件，明确跟踪类性质的迭代不变性及相关性质间的推导关系。
A view towards mixing in holomorphic correspondences
本文研究全纯对应在紧连通复流形上的动力学混合性质，连接到Londhe提出的全纯对应 ergodic理论，并通过例子对比与映射的情况，研究两个全纯对应乘积以刻画弱混合性。
Polynomial Mixing for a Weakly Damped Stochastic Nonlinear Schrödinger Equation
本文利用耦合方法，证明了1D有界区域上具加性噪声的弱阻尼随机非线性Schrödinger方程在聚焦与散焦情形下的多项式混合，从而确立了不变测度的唯一性。
We exhibit rationally ergodic, weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.
In this article we will see some properties that guarantee that a product of an ergodic non-singular action and a probability preserving ergodic action is also an ergodic action. We will start by proving 'The multiplier theorem' for locally compact abelian groups. Then we will show that for certain locally compact Polish groups (Moore groups, and minimally weakly mixing groups), a non-singular G action is weakly mixing if and only if any finite dimensional G-invariant subspace of L_\infty is trivial. Finally, we will show that the Gaussian action associated to the infinite dimensional irreduci…
Hoehn and Mouron [Ergod. Th. \& Dynam. Sys. (2014) \textbf{34}, 1897--1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn-Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Topology (2001) \textbf{40}, 551--569].
Weakly Mixing Polygonal Billiards
本文证明了存在非有理多边形的剩余集，使得台球流关于Liouville测度弱混合，其核心方法是通过Baire范畴论证及平移曲面方向对流的遍历性与特征值性质。
Let $(X,T)$ be a topological dynamical system and $\mathcal{F}$ be a Furstenberg family (a collection of subsets of $\mathbb{Z}_+$ with hereditary upward property). A point $x\in X$ is called an $\mathcal{F}$-transitive one if $\{n\in\mathbb{Z}_+:\, T^n x\in U\}\in\F$ for every nonempty open subset $U$ of $X$; the system $(X,T)$ is called $\F$-point transitive if there exists some $\mathcal{F}$-transitive point. In this paper, we aim to classify transitive systems by $\mathcal{F}$-point transitivity. Among other things, it is shown that $(X,T)$ is a weakly mixing E-system (resp.\@ weakly mixin…
We provide a simple criterion for a non-singular and conservative Bernouilli action to have a weakly mixing Maharam extension. As an application, we show that every countable amenable group admits a stable type III_1 Bernoulli action, answering a recent question by Vaes and Wahl.
Given a switched system, we introduce weakly mixing sets of type 1, 2 and Xiong chaotic sets of type 1, 2 with respect to a given set and show that they are equivalent respectively.
This work contains the following results: the trajectory fullness of the homoclinic groups, their connections with factors, K-property, weak multiple mixing; the ergodicity of the weakly homoclinic group for Gauss and Poisson actions; the triviality of the weakly homoclinic group for classes of rank-1 actions. Some open problems are discussed.
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.
Size of exceptional sets in weakly mixing systems
本文研究了Chacon变换及一类称为限制紧映射的切割-堆叠变换的例外集，构造了统一的例外集J，并证明了其在对数尺度下的有效性，同时探讨了弱混合模型中的例外集障碍问题。
BCZ map is weakly mixing
本文证明了BCZ映射具有弱混合性，利用其源于测地线流对horocycle流重整化的自相似性，回答了2006年综述中关于该映射的最后一个遗留问题。
Mixing (of all orders) rank-one actions $T$ of Heisenberg group $H_3(\Bbb R)$ are constructed. The restriction of $T$ to the center of $H_3(\Bbb R)$ is simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions of $H_3(\Bbb R)$ are also constructed. A rigid weakly mixing rank-one action $T$ is constructed such that the restriction of $T$ to the center of $H_3(\Bbb R)$ is not isomorphic to its inverse.
Local spectral estimates and quantitative weak mixing for substitution $\mathbb{Z}$-actions
本文研究本原、非周期且替换矩阵有理不可约的弱混合替换 $\mathbb{Z}$-作用，证明无单位圆特征值时谱测度一致对数 Hölder 正则并给出幂-对数弱混合速率界，同时刻画 Salem 替换情形下代数谱参数的非一致 Hölder 正则性。
We study weak mixing of all orders for weakly mixing measure preserving dynamical systems, where the dynamics is given by the action of an abelian second countable topological group which has an invariant measure under the group operation. One of the main technical tools we use is a van der Corput lemma for Hilbert space valued functions on a second countable topological group.
Let $\mathcal X$ be an infinite locally compact separable metric space with metric $ρ$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $β= \sup \big\{ ρ(x, y): \{x, y \} \subset \mathcal X \big\}$. In this note, we show (Theorem 4) that, for any countably infinite set $\{x_1, x_2, \cdots\}$ of points in $\mathcal X$ with compact orbit closures $\overline{O_f(x_i)}$'s, there exist an infinite set $\mathcal M$ of positive integers and countably infinitely many pairwise disjoint Cantor sets ${\mathcal S}^{(1)}, {\mathcal S}^{(2)}, \cdots$ of totally tran…
It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable. If $G$ has the Haagerup property then (and only then) these actions can be chosen of 0-type. If $G$ is amenable and unimodular then $G$ has weakly mixing Bernoulli actions of any possible Krieger's type.
Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nilfactor.
Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equi…
For a weakly mixing bounded rank-one construction the disjointness of its powers is proved. For non-rigid constructions we get minimal self-joinings. Examples of non-mixing rank one actions with explicit weak closure are proposed.
Weak mixing for dynamical systems with ${\bm{\mathscr M}}_{\bmα}$-shadowing
本文研究具有$\mathscr{M}_{\alpha}$-跟踪性质的动力系统的弱混合性，证明了$M$-系统在$\alpha\in[0,1)$时弱混合，$E$-系统在$\alpha\in(0,1)$时弱混合，并构造反例说明参数范围最优。
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.
Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},μ)$ which is faithful but not essentially free. In other words, $G$ admits a weakly mixing nontrivial faithful IRS. We also prove that every non-elementary hyperbolic group admits a characteristic random subgroup with the same properties.
We give conditions for the exactness of Rokhlin endomorphisms, apply these to random walks on locally compact, second countable topological groups and obtain that the action on the Poisson boundary of an adapted random walk on such a group is weakly mixing.
Properties of Mixing BV vector fields
研究混合BV向量场的密度性质，通过子方块置换型BV向量场的密度及适当扰动方法，证明遍历、弱混合、强混合等不同混合性态的散度为零向量场在对应集合中的拓扑分类性质，相关逼近结果具独立价值。
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