Pintu Debnath, Sayan Goswami
There is a long history of studying Ramsey theory using the algebraic structure of Stone-Cech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic large sets. In this article we will study elementary characterization of essential F sets. And also study the mixing properties arises from essential F sets.
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We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set.
For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is positive. In particular, we give a straightforward proof that any mildly mixing transformation T has a dense algebra C such that T is lightly mixing on C. Also, we provide a hierarchy for the properties of lightly mixing, sweeping out and uniform sweeping out for dense collections, and dense algebras of sets. As a result, stronger mixing realizations are given fo…
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.
We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $ε$-over-recurrence and show that given $ε> 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $ε$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigi…
For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable examples. We also discuss rigid rank one transformations of finite measure that mix along a given set, and some analogues for group actions.
We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing.
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…
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 $m \in A$.
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.
We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant $δ$-scrambled set for some $δ>0$ if and only if it has a fixed point and not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.
For any dimension $d\geq 3$ we construct $C^{1}$-open subsets of the space of $C^{3}$ vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure.
Slow polynomial mixing, dynamical Borel-Cantelli lemma and Hausdorff dimension of dynamical diophantine sets
本文针对缓慢的多项式混合系统，建立了动力学Borel-Cantelli引理与特定动力学丢番图集Hausdorff维数的最优性结果。
In this paper, some characterizations about transitivity, mildly mixing property, $\mathbf{a}$-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\mathbf{a}$-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\mathbf{a}$-transitive, equicontinuous,…
The ring of polynomial over a finite field $F_q[x]$ has received much attention, both from a combinatorial viewpoint as in regards to its action on measurable dynamical systems. In the case of $(\mathbb{Z},+)$ we know that the ideal generated by any nonzero element is an IP$^*$-set. In the present article we first establish that the analogous result is true for $F_q[x]$. We further use this result to establish some mixing properties of the action of $(F_q[x],+)$. We shall also discuss on Khintchine's recurrence for the action of $(F_q[x]\setminus\{0\},\cdot)$.
It is well-known that, for sufficiently mixing dynamical systems, the number of visits to balls and cylinders of vanishing measure is approximately Poisson compound distributed in the Kac scaling. Here we extend this kind of results when the target set is an arbitrary set with vanishing measure in the case of $ϕ$-mixing systems. The error of approximation in total variation is derived using Stein-Chen method. An important part of the paper is dedicated to examples to illustrate the assumptions, as well as applications to temporal synchronisation of $g$-measures
We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniforml…
If we perturb a completely integrable Hamiltonian system with two degrees of freedom, the perturbed flow might display, on every energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincaré section of the flow. Each one of these laminations carry a unique invariant probability measure for the flow on which mixing is impossible in this low dimensional frame. We prove that if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling functio…
Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of th…
Exponential mixing for Gibbs measures on self-conformal sets and applications
本文证明了满足OSC的 $C^{1+\alpha}$ 共形IFS生成的自共形集上Gibbs测度的指数混合性，并据此获得了递归集与收缩目标集的尖锐渐近计数，同时将Mattila刚性定理推广至无分离条件的自共形集。
We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.
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 prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables) then the time $τ(x,S_{r})$ needed for a typical point $x$ to enter for the first time a set $S_{r}=\{x:f(x)\leq r\}$ which is a sublevel of a Lipschitz funcion $f$ scales as $\frac{1}{μ(S_{r})}$ i.e. \begin{equation*} \underset{r\to 0}{\lim }\frac{\log τ(x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log μ(S_{r})}{\log (r)}. \end{equation*} This generalizes a previous result obtained for balls. We will also consider relations with the return time distributio…
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…
We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if $\mathcal{C}$ has zero entropy. When the entropy of $\mathcal{C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly str…
We consider a $ϕ$-mixing shift $T$ on a sequence space $Ω$ and study the number of returns $\{ T^kω\in U\}$ to a union $U$ of cylinders of length $n$ until the first return $\{ T^kω\in V\}$ to another union $V$ of cylinder sets of length $m$. It turns out that if probabilities of the sets $U$ and $V$ are small and of the same order then the above number of returns has approximately geometric distribution. Under appropriate conditions, we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymp…
A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite t…
We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly …
We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion…
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。
Dynamical compartments in stirred tank reactors and Markov state modeling for mixing quantification: a transfer operator approach
本文提出利用转移算子方法识别搅拌釜反应器中的相干流结构，通过计算停留时间和混合时间分析混合行为，并利用马尔可夫状态模型描述反应器内腔体的宏观传输动力学。
The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, are currently the subject of intense study. In the autonomous setting, the use of transfer operators (Perron-Frobenius operators) to identify invariant and almost-invariant sets has been particularly successful. In the nonautonomous (time-dependent) setting, \emph{coherent sets}, a time-parameterised family of minimally dispersive sets, are a natural extension of almost-invariant sets. The present w…
One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for…
We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mixing.
The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do…
Chaotic behaviour on invariant sets of linear operators
该研究探讨带不变集的拓扑向量空间上算子的多种混沌与混合性质，建立了不变集上的相关性质与其闭张成空间上类似性质的关联，并给出示例说明结果。
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…
Using the combinatorial properties of subsets of integers, a classification of metric dynamical systems was given in [V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure-preserving systems, Colloquium Mathematicum, 110(1), (2008), 117-150]. As a result, some new families emerged. Here, their counterparts in topological dynamics has been considered. The differences will be discussed and new classes of systems in topological dynamics will be introduced.
We consider $ψ$-mixing dynamical systems $(\mathcal{X},T,B,μ)$ and we find conditions on families of sets $\{\mathcal{U}_n\subset \mathcal{X}:n\in\mathbb{N}\}$ so that $μ(\mathcal{U}_n)τ_n$ tends in law to an exponential random variable, where $τ_n$ is the entry time to $\mathcal{U}_n.$
In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.
In the paper we study what sets can be obtained as $α$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $α$-limit sets are $ω$-limit sets and for all but finitely many points $x$, we can obtain every $ω$-limits set as the $α$-limit set of a backward trajectory starting in $x$. For zero entropy maps, every $α$-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.
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