Approximately transitive dynamical systems and simple spectrum

Answering two open problems on Banks theorem for non-autonomous dynamical systems

This paper shows that (1) there exists a topologically transitive NADS having two disjoint invariant periodic orbits with dense periodic points, which is finitely generated but not periodic; (2) there exists a topologically transitive non-finitely generated NADS having two disjoint invariant periodic orbits with dense periodic point, which is not sensitive. This answers positively the Open Problems 4.1 and 4.2 posed in [8].

Dynamical compactness and sensitivity

To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system $(X,T)$ given by a compact metric space $X$ and a continuous surjective self-map $T:X \to X$. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is…

Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.

Genericity in Topological Dynamics

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. O…

非自治离散动力系统中若干更强的传递性形式

Several stronger forms of transitivity in non-autonomous discrete dynamical systems

本文在非自治离散动力系统中引入并研究了多种更强的传递性概念，证明了其迭代不变性，给出了充分必要条件，并通过反例揭示了与自治系统的差异，最后建立了等价性条件。

传递同胚的涌现秩序谱

Emergent order spectrum for transitive homeomorphisms

本文研究了传递同胚的动力系统中，通过极限顺序诱导的秩序谱的丰富性，证明了在特定条件下，该谱能包含所有可数无限散列的顺序类型及有理数的顺序类型。

CR-动力系统中的传递性

Transitivity in CR-Dynamical Systems

本文研究了CR-动力系统，引入了新型传递点和传递性，并开发了传递树工具以确定不同类型传递点间的关系。

Ergodic Optimization Restricted On Certain Subsets Of Invariant Measures

In this article, we pay attention to transitive dynamical systems having the shadowing property and the entropy functions are upper semicontinuous. As for these dynamical systems, when we consider ergodic optimization restricted on the subset of invariant measures whose metric entropy are equal or greater than a given constant, we prove that for generic real continuous functions the ergodic optimization measure is unique, ergodic, full support and have metric entropy equal to the given constant. Similar results also hold for suspension flows over transitive subshift of finite type, Cr (r\geq 2…

Transitivity and Mixing Properties of Set-Valued Dynamical Systems

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two topological properties for set-valued functions and generalize some results from single-valued case to set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

On multi-transitivity with respect to a vector

A topological dynamical system $(X,f)$ is said to be multi-transitive if for every $n\in\mathbb{N}$ the system $(X^{n}, f\times f^{2}\times \dotsb\times f^{n})$ is transitive. We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets, answering a question proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661--1672]. We also show that multi-transitive systems are Li-Yorke chaotic.

Another invariant for AT actions

We construct a collection of numerical invariants for approximately transitive (AT) actions (of $\Z$). We use them (sometimes supplemented by other invariants to show that members of various one-parameter families of AT actions are mutually non-isomorphic.

Topological dynamics of Zadeh's extension on the space of upper semi-continuous fuzzy sets

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,…

Conjugacy in the Cantor Set Automorphism Group

We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically defined subsets of $C_s(X)$, e. g. the sets of topologically transitive maps, chain transitive maps, chain mixing maps, etc. In each case, we consider whether there exist elements with a dense conjugacy class and if so, what the generic elements look like.

Torus actions and automorphism groups of singularities of integrable dynamical systems

We show that in the neighborhood of each ``finite type'' singular orbit of a real analytic integrable dynamical system (hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this orbit. We also show that the local automorphism group of the system near such an orbit is

Reconstructing a minimal topological dynamical system from a set of return times

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric …

Full Groups and Orbit Equivalence in Cantor Dynamics

In this note we consider dynamical systems $(X,G)$ on a Cantor set $X$ satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems $(X_1,G_1)$ and $(X_2,G_2)$ are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known Dye's theorem established originally for ergodic measure-preserving actions.

超越规范的动力系统复发光谱

The recurrence spectrum for dynamical systems beyond specification

本文提出（W'）-规范概念，证明了特定子移位的复发集具有满豪斯多夫维度，并扩展到广泛类别的分段扩张区间映射。

Finite type coarse expanding conformal dynamics

We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group act…

The systems with almost Banach mean equicontinuity for abelian group actions

In this paper, we give the concept of Banach-mean equicontinuity and prove that three concepts, Bnanach-, Weyl- and Besicovitch-mean equicontinuity of a dynamic system with abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system with abelain group action is zero. As an application with our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an abelian groups is zero.

Some Properties of Recurrent Sets for Endomorphisms of Topological Groups

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, it is obtained that some dynamical properties induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property.

关于局部热力学形式的一些方面

On some aspects of local thermodynamical formalism

本文提出调整局部拓扑熵以考虑轨道初始暂态，通过Lyapunov指数计算translocal熵，并推广了部分变化原理。

与多项式相关的一些动力学性质

Some dynamical properties related to polynomials

本文证明了整数中厚合成集在多项式映射下的高维厚合成性质，并构造了一个传递、强混合且非极小的拓扑动力系统，展示了特定回归点集不稠密的动力学特征。

Dynamical properties of some adic systems with arbitrary orderings

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli-Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every $k$ the subshift …

The Non-Archimedean Theory of Discrete Systems

In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior of the system w.r.t. variety of word transformations performed by the system: We call a system completely transitive if, given arbitrary pair $a,b$ of finite words that have equal lengths, the system $\mathfrak A$, while evolution during (discrete) time, at a certain moment transforms $a$ into $b$. To every system $\mathfrak A$, we put into a correspondence a family $\mathcal F_{\mathfrak A}$ of continuous maps of a suitable non-Archim…

Quotients of $\mathbb{N}^*$, $ω$-limit sets, and chain transitivity

$\mathbb{N}^* = β\mathbb{N} \setminus \mathbb{N}$ has a canonical dynamical structure provided by the shift map, the unique continuous extension to $β\mathbb{N}$ of the map $n \mapsto n+1$ on $\mathbb{N}$. Here we investigate the question of what dynamical systems can be written as quotients of $\mathbb{N}^*$. We prove that a dynamical system is a quotient of $\mathbb{N}^*$ if and only if it is isomorphic to the $ω$-limit set of some point in some larger system. This provides a full external characterization of the quotients of $\mathbb{N}^*$. We also prove, assuming MA$_{σ\text{-centered}}(κ)…

多标度熵谱的完全实现

Complete realization of multifractal entropy spectra and pressure functions

本文研究了连续势函数在广泛动力系统上的多标度熵谱的完全实现问题，证明了对于任意H>0和在紧区间上连续凹函数，每个具有拓扑熵H的动力系统都存在一个连续势函数，其多标度熵谱恰好为该函数，并且该谱可以由任意多个非同调势函数实现。此外，文章还研究了熵谱在Hausdorff图度量下的势依赖性，证明了非平凡混合有限型移位系统的熵谱下半连续性和上半连续性的密集失败。最后，通过多标度熵谱与压力函数之间的Legendre变换对偶性，推导了压力函数在全体实数上定义的压力灵活性定理，从而回答了Kucherenko和Quas提出的问题。