Chaos in a topologically transitive system

2005 ◽  
Vol 48 (7) ◽  
pp. 929-939 ◽  
Author(s):  
Xiong Jincheng
Keyword(s):  
Topologically Transitive ◽  
Transitive System

scholarly journals Sufficient conditions under which a transitive system is chaotic

2009 ◽  
Vol 30 (5) ◽  
pp. 1277-1310 ◽  
Author(s):  
E. AKIN ◽  
E. GLASNER ◽  
W. HUANG ◽  
S. SHAO ◽  
X. YE
Keyword(s):  
Dynamical System ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Minimal System ◽  
Topologically Transitive ◽  
Transitive System

AbstractLet (X,T) be a topologically transitive dynamical system. We show that if there is a subsystem (Y,T) of (X,T) such that (X×Y,T×T) is transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We then show that many of the known sufficient conditions in the literature, as well as a few new results, are corollaries of this statement. In fact, the kind of chaotic behavior we deduce in these results is a much stronger variant of Li–Yorke chaos which we call uniform chaos. For minimal systems we show, among other results, that uniform chaos is preserved by extensions and that a minimal system which is not uniformly chaotic is PI.

scholarly journals Properties of invariant measures in dynamical systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2257-2294 ◽  
Author(s):  
JIAN LI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Entropy Function ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Ergodic Measures ◽  
One To One ◽  
Transitive System

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

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).

scholarly journals Characterization of topologically transitive attractors for analytic plane flows

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Martin Shoptrajanov ◽  
Nikita Shekutkovski
Keyword(s):  
Necessary Conditions ◽  
Topological Characterization ◽  
Plane Flow ◽  
Limit Sets ◽  
Topologically Transitive ◽  
Analytic Plane

We give necessary conditions for a set to be topologically transitive attractor of an analytic plane flow using topological characterization of ?-limit sets and the concept of upper semi-continuity of multi valued maps.

scholarly journals An answer to Furstenberg’s problem on topological disjointness

2019 ◽  
Vol 40 (9) ◽  
pp. 2467-2481 ◽  
Author(s):  
WEN HUANG ◽  
SONG SHAO ◽  
XIANGDONG YE
Keyword(s):  
Open Subset ◽  
Dense Subset ◽  
Minimal System ◽  
Open Neighbourhood ◽  
Countable Dense Subset ◽  
Weakly Mixing ◽  
Transitive System

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

Remarks on Livšic' theory for nonabelian cocycles

1999 ◽  
Vol 19 (3) ◽  
pp. 703-721 ◽  
Author(s):  
KLAUS SCHMIDT
Keyword(s):  
Dynamical System ◽  
Homomorphic Image ◽  
Periodic Points ◽  
Skew Product ◽  
Topologically Transitive ◽  
Polish Group ◽  
Bounded Distortion

Let $(X,\phi)$ be a hyperbolic dynamical system and let $(G,\delta)$ be a Polish group. Motivated by Nicol and Pollicott, and then by Parry we study conditions under which two Hölder maps $f,g: X\longrightarrow G$ are Hölder cohomologous.In the context of Nicol and Pollicott we show that if $f$ and $g$ are measurably cohomologous and the distortion of the metric $\delta $ by the cocycles defined by $f$ and $g$ is bounded in an appropriate sense, then $f$ and $g$ are Hölder cohomologous.Two further results extend the main theorems recently presented by Parry. Under the hypothesis of bounded distortion we show that, if $f$ and $g$ give equal weight to all periodic points of $\phi $, then $f$ and $g$ are Hölder cohomologous. If the metric $\delta $ is bi-invariant, and if the skew-product $\phi _f$ defined by $f$ is topologically transitive, then conjugacy of weights implies that $g$ is Hölder conjugate to $\alpha \cdot f$ for some isometric automorphism $\alpha $ of $G$. The weaker condition that $g$-weights of periodic points are close to the identity whenever their $f$-weights are close to the identity implies that $g$ is continuously cohomologous to a homomorphic image of $f$.

Topological Transitivity on the Torus

1994 ◽  
Vol 37 (4) ◽  
pp. 549-551 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Vector Field ◽  
Finite Number ◽  
Fixed Points ◽  
Analytic Vector ◽  
Topological Transitivity ◽  
Topologically Transitive ◽  
Transitive Flow ◽  
Real Analytic ◽  
Analytic Vector Field

AbstractT. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.

scholarly journals When is a dynamical system mean sensitive?

2017 ◽  
Vol 39 (06) ◽  
pp. 1608-1636 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS ◽  
JIE LI ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
The Other ◽  
Positive Entropy ◽  
Topological Dynamical System ◽  
Other Hand ◽  
Open Question ◽  
Uniquely Ergodic ◽  
Mean Sensitivity ◽  
Transitive System

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

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