When are all closed subsets recurrent?

2016 ◽  
Vol 37 (7) ◽  
pp. 2223-2254 ◽  
Author(s):  
JIE LI ◽  
PIOTR OPROCHA ◽  
XIANGDONG YE ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Periodic Points ◽  
Topological Dynamical System ◽  
Open Questions ◽  
Weakly Mixing ◽  
Uniformly Rigid ◽  
Compact Subsets

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

scholarly journals On weak mixing, minimality and weak disjointness of all iterates

2011 ◽  
Vol 32 (5) ◽  
pp. 1661-1672 ◽  
Author(s):  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Dynamical System ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Open Questions ◽  
Weakly Mixing ◽  
Recurrence Theorem ◽  
Topological Weak Mixing ◽  
Mixing Property ◽  
Transitive System

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→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 [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

scholarly journals Topological Dynamics of Zadeh’s Extension on Upper Semi-Continuous Fuzzy Sets

2017 ◽  
Vol 27 (10) ◽  
pp. 1750165 ◽  
Author(s):  
Xinxing Wu ◽  
Xianfeng Ding ◽  
Tianxiu Lu ◽  
Jianjun Wang
Keyword(s):  
Dynamical System ◽  
Fuzzy Sets ◽  
Topological Dynamics ◽  
Property A ◽  
Weakly Mixing ◽  
Uniformly Rigid ◽  
Mixing Property

In this paper, some characterizations are obtained on the transitivity, mildly mixing property, a-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh’s extensions restricted on some invariant closed subsets of all upper semi-continuous fuzzy sets in the level-wise metric. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and a-transitive, equicontinuous, uniformly rigid) if and only if the corresponding Zadeh’s extension is transitive (resp., mildly mixing, a-transitive, equicontinuous, uniformly rigid).

Chaos of Transformations Induced Onto the Space of Probability Measures

2016 ◽  
Vol 26 (13) ◽  
pp. 1650227 ◽  
Author(s):  
Xinxing Wu
Keyword(s):  
Dynamical System ◽  
Weak Topology ◽  
Probability Measures ◽  
Weakly Mixing ◽  
Borel Probability ◽  
Uniformly Rigid

For a dynamical system [Formula: see text], let [Formula: see text] be its induced dynamical system on the space of Borel probability measures with weak*-topology. It is proved that [Formula: see text] is [Formula: see text]-transitive (resp., exact, uniformly rigid) if and only if [Formula: see text] is weakly mixing and [Formula: see text]-transitive (resp., exact, uniformly rigid), where [Formula: see text] is an [Formula: see text]-vector of integers. Moreover, some analogous results are obtained for the hyperspace.

scholarly journals Thickly Syndetical Sensitivity of Topological Dynamical System

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Heng Liu ◽  
Li Liao ◽  
Lidong Wang
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Topological Dynamical System ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Devaney Chaos ◽  
Weakly Mixing

Consider the surjective continuous mapf:X→X, whereXis a compact metric space. In this paper we give several stronger versions of sensitivity, such as thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, and strong sensitivity. We establish the following. (1) If(X,f)is minimal and sensitive, then(X,f)is syndetically sensitive. (2) Weak mixing implies thick sensitivity. (3) If(X,f)is minimal and weakly mixing, then it is thickly syndetically sensitive. (4) If(X,f)is a nonminimalM-system, then it is thickly syndetically sensitive. Devaney chaos implies thickly periodic sensitivity. (5) We give a syndetically sensitive system which is not thickly sensitive. (6) We give thickly syndetically sensitive examples but not cofinitely sensitive ones.

scholarly journals Periodic points and transitivity on dendrites

2016 ◽  
Vol 37 (7) ◽  
pp. 2017-2033 ◽  
Author(s):  
GERARDO ACOSTA ◽  
RODRIGO HERNÁNDEZ-GUTIÉRREZ ◽  
ISSAM NAGHMOUCHI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
Dense Orbit ◽  
Topological Graphs ◽  
Periodic Decomposition ◽  
Dense Set ◽  
Compact Subsets

We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It is also possible that a dynamical system is transitive but there is a unique periodic point which, in fact, is the unique fixed point. We also prove that on almost meshed continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all non-empty compact subsets.

scholarly journals On $ n $-tuplewise IP-sensitivity and thick sensitivity

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jian Li ◽  
Yini Yang
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Necessary Conditions ◽  
Minimal System ◽  
Topological Dynamical System ◽  
Minimal Points ◽  
Sufficient And Necessary Conditions ◽  
Weakly Mixing ◽  
Factor Maps

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id="M3">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M4">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id="M7">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id="M8">\begin{document}$ U $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id="M10">\begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i&lt;j\le n}d(T^k x_i,T^k x_j)&gt;\delta\Bigr\} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id="M11">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id="M12">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id="M17">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id="M18">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP<inline-formula><tex-math id="M19">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.</p>

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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