Vladimir Azarin
We approximate a chain recurrent dynamical system by periodic dynamical systems. This is similar to the well known Bohr theorem on approximation of almost periodic functions by periodic functions.
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In these expository notes we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.
Furstenberg, using tools from topological dynamics, defined the notion of a central subset of positive integers, and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-Čech compactification, this combinatorial theorem has been generalized and extended to the Central Sets Theorem. The algebraic techniques also discovered many sets, which are not central, that satisfy the conclusion of the Central Sets Theorem. We call such sets C sets. Since C sets are defined combinatorially, it is natural to ask if this notion admits a dynamical characterizatio…
We establish a dynamical version of Kuratowski-Mycielski Theorem on the existence of "large" invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results on this topic and also obtain some new results.
On sets of pointwise recurrence and dynamically thick sets
本文研究了点态复发集和动态厚集的组合特征，并引入了动态分段 syndetic 性，扩展了相关领域的研究成果。
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.
We consider a homeomorphism on a totally disconnected, compact metric space and define a binary relation on the family of clopen subsets. We will show that the comparability of any clopen sets with respect to the relation is equivalent to the unique ergodicity of the homeomorphism.
For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach to this result is under the settings of topological dynamical systems, which is also applicable to infinite dimensional $C^1$ dynamical systems.
In this article, we deal with several notions in dynamical systems. Firstly, we prove that both closure function and orbital function are idempotent on set-valued dynamical systems. And we show that the compact limit set of a connected set is also connected. Furthermore, we prove that the $Ω$-limit set of a compact set is quasi-attracting.
The modern qualitative theory of dynamical systems is thoroughly intertwined with the fairly young science of topology. Strange and even bizarre constructions of topology are found sooner or later in dynamics of discrete or continuous dynamical systems. In the present paper we show that the wild Fox-Artin arc naturally emerges in dynamics as an invariant manifold of a fixed point and as a heteroclinic intersection.
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 …
It is known by the Conley's theorem that the chain recurrent set $CR(ϕ)$ of a deterministic flow $ϕ$ on a compact metric space is the complement of the union of sets $B(A)-A$, where $A$ varies over the collection of attractors and $B(A)$ is the basin of attraction of $A$. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called \emph{absorbing condition}. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove th…
Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems
本文研究不具有物理测度的一维映射，证明存在一个测度单纯形，其中每个测度的吸引盆具有完全Hausdorff维数。
In this text, we prove then that any minimal effective dynamical system on a Cantor set $\mathcal{A}^{\mathbb{N}}$ can be simulated by a minimal $\mathbb{Z}^3$-SFT, in a sense that we explicit here. This notion is a generalization of simulation by factor and sub-action defined by Hochman.
In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including $A$-coupled expanding systems. We prove that Li-Yorke pairs of $A$- coupled-expanding system under some conditions have full hausdorff dimension on the invariant set. we generalize the result of [8] on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and having a self-similar invariant set to the case of dynamical system conjugated to some kind of subshifts. And Hausdorff dimension of "chaotic invariant set" for some ki…
We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathb…
For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice.
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…
Meyer sets, Pisot numbers, and self-similarity in symbolic dynamical systems
本文从算术与动力学视角探讨准晶与自相似性的联系，重点阐述Pisot数在定义简单模型集及作为Meyer集伸缩因子中的作用，并介绍Yves Meyer关于Pisot或Salem数保持Meyer集的刻画。
Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.
In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of Li-Yorke, asymptotic and distal pairs in constant--length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li--Yorke pairs, none of which is recurrent.
In this paper, we consider local and uniform invariance preserving steplength thresholds on a set when a discretization method is applied to a linear or nonlinear dynamical system. For the forward or backward Euler method, the existence of local and uniform invariance preserving steplength thresholds is proved when the invariant sets are polyhedra, ellipsoids, or Lorenz cones. Further, we also quantify the steplength thresholds of the backward Euler methods on these sets for linear dynamical systems. Finally, we present our main results on the existence of uniform invariance preserving steplen…
We study the transport properties of nonautonomous chaotic dynamical systems over a finite time duration. We are particularly interested in those regions that remain coherent and relatively non-dispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detects maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratosph…
In this paper, we set up a general correspondence between the algebra properties of $\bN$ and the sets defined by dynamical properties. In particular, we obtain a dynamical characterization of C-sets, where C-sets are the sets satisfying the strong Central Sets Theorem. As an application, we show that Rado systems are solvable in C-sets.
Local stable and unstable sets for random dynamical systems
针对正熵双侧连续丛随机动力系统，在缺乏光滑性时用局部稳定/不稳定集替代不变流形，证明纤维熵由不稳定集上的作用决定，并建立其Hausdorff维数下界。
Interpolation sets for dynamical systems
本文研究动力系统中的插值集问题，给出了极小、混合、严格遍历、零熵等多类拓扑动力系统插值集的组合刻画，还探讨了弱插值集并解答了相关公开问题。
Let $f: M \to M$ be a $C^{1+α}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $μ$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic on the support set $W$ of $μ$ and $W$ is locally maximal. For any subset $A\subset W$ with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the $A$-exceptional set and the limit $A$-exceptional set.
Functional Limit Theorems for Dynamical Systems with Correlated Maximal Sets
本文研究动力系统中重尾平稳过程的聚类行为，将 pilling 过程从单点极大情形推广到同一轨道上多点取得最大值的相关极大集情形，并在分段线性扩张的一维和二维动力学下给出显式表达式。
In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $ω(x)$ is equal to an interval), 2) for which $ω(c)$ is a Cantor set, and 3) for which $ω(x)=ω(c)$ for Lebesgue almost all x. So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor.
Recently, Horváth, Song, and Terlaky [\emph{A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation}] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem are utilized…
We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.
It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $β$ and then relate the properties of $β$ to ones of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.
A dynamical algorithm to compute hyperbolic Julia sets in polynomial time
本文提出一种动态算法，用于在多项式时间内计算复多项式的双曲Julia集，并确立其下可计算性。
In this paper, the study of the global orbit pattern (gop) formed by all the periodic orbits of discrete dynamical systems on a finite set $X$ allows us to describe precisely the behaviour of such systems. We can predict by means of closed formulas, the number of gop of the set of all the function from $X$ to itself. We also explore, using the brute force of computers, some subsets of locally rigid functions on $X$, for which interesting patterns of periodic orbits are found.
Dynamical covering sets in self-similar sets
本文研究自相似集上动态覆盖集的大小，通过压力函数刻画收缩速率下的覆盖条件与Hausdorff维数，结合经典维数理论与鞅方法。
Two Results for the Omega Limit Sets of Dynamical Systems
该研究针对动力系统的Ω极限集展开，提出可采用满足比李雅普诺夫定理、拉萨尔定理常规假设更宽松条件的函数来估计Ω极限集，给出了两个相关结果。
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…
Dichotomy laws for the Hausdorff measure of shrinking target sets in $β$-dynamical systems
本文研究β-动力系统中收缩目标集的Hausdorff测度，证明其测度为零或满取决于特定级数的收敛与发散，从而给出完备的测度论刻画。
In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete systems, instead of the traditional Lyapunov method, our novel proofs are based on the S-lemma, which enables us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets. Finally, …
On Sets of Periodic Orbit Lengths in Finitely Presented Dynamical Systems
本文研究有限生成动力系统中周期点长度的分类，利用Artin-Mazur齐斯函数的对数导数和Skolem-Mahler-Lech定理，扩展了移位有限类型的已知分类。
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 present paper we mainly focus on non-recurrent dynamical orbits with empty syndetic center and show that twelve different statistical structures over mixing expanding maps or transitive Anosov diffeomorphisms all have dynamical complexity in the sense of strongly distributional chaos.
Equality of the dynamical sets of two commuting transcendental entire functions
本文研究了两个可交换的超越整函数f和g的动力学性质，其中g的形式为af^p + b，其中a,b∈C，p∈N，且a≠0,1。研究发现f和g的逃逸集、填充Julia集和蹦床集都相同。作为直接结果，我们得出f和g的Julia集相同。该定理扩展了Poon和Yang1998年的结果。此外，遵循Wang和Yang的研究，我们考虑了一个非常数多项式Q和可交换的整个函数f和g，满足关系Q(g)=aQ(f)+b，其中a≠0,1，b∈C。在更一般的情况下，我们证明了f和g的逃逸集、填充Julia集和蹦床集相等。
We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational experiments that this approach can successfully converge to approximations of (maximal) invariant sets of arbitrary topology, dimension and stability as, e.g., saddle type invariant sets with complicated dynamics. We further propose to extend this approach by adding a Lennard-Jones type potential term to the objective function which yields more evenly distribu…
Let K denote a compact invariant set for a strongly monotone semiflow in an ordered Banach space E, satisfying standard smoothness and compactness assumptions. Suppose the semiflow restricted to K is chain transitive. The main result is that either K is unordered, or else K is contained in totally ordered, compact arc of stationary points; and the latter cannot occur if the semiflow is real analytic and dissipative. As an application, entropy is 0 when E = R^3 . Analogous results are proved for maps. The main tools are results of Mierczynski [27 ] and Terescak [37 ]
Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,\infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,\infty),+)$, C-sets near zer…
It has been recently shown that, under appropriate hypotheses, the $ω$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the action of a finitely generated free group or monoid. We give general definitions for several types of limit sets and analogous notions of internal transitivity. We then demonstrate that these limit sets are completely characterized by internal transitivity properties in shifts of finite type and general dynamical systems exhibiting a form of the shadowing property.
Partial semigroup partial dynamical systems and Partial Central Sets
本文在部分半群框架下提出大集合新概念，引入部分半群部分动力系统（PSPDS），并证明中心集的拓扑动力学刻画与代数刻画等价。
We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.
In this paper, tools to study forward invariance properties with robustness to dis- turbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of dif- ferential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without dis- turbances), including solutions that have persistent flows and jumps, that are Zeno, and th…
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 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…
The term "hybrid system" refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical system on a compact space. This allows one to study its limit sets. We examine the Morse decompositions of hybrid systems, find a sufficient condition for the existence of a non-trivial Morse decomposition, and study the Morse sets of such a decomposition. Finally, we consider the case in which the Markovian perturbations are small, showing that trajectories…
Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-Čech Compactification of discrete semigroup. They found that central sets are the members of the minimal idempotents of $βS$, the Stone-Čech Compactification of a semigroup $\left(S,\cdot\right)$. We know that any closed subsemigroup of $βS$ is generated by a filter. We call a set $A$ to be a $\mathcal{F}$-central set if it is a member of a minimal idempotent of a closed subsemigroup of …
Shadowing With Small Diameter Sets Of Bounded Cardinality
本文研究动力系统中伪轨道可被有界基数小直径集跟踪的性质，证明混合sofic子移位、带指定性质的满射动力系统只需基数≤2的集合即可满足该性质，且准有限型子移位也具备该性质。
Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form $x+τy$ for $x,y \in \mathbb{R}$, and $τ^2 = 1$ but $τ\neq \pm 1$, are the natural number system in which to encode geometric properties of the Minkowski space $\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set parameterizes connectedness of hyperbolic Julia sets. We give a wall-and-chamber decom…
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh-Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than $1/2$. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.
In this article, the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Especially, we use topological pressure to describe the relative multifractal spectrum of ergodic averages and give a positive answer to the conjecture posed by L. Olsen (J. Math. Pures Appl. {\bf 82} (2003)).
We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J. C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra.
Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.
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