scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

scholarly journals Estimates of topological entropy of continuous maps with applications

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Xiao-Song Yang ◽  
Xiaoming Bai
Keyword(s):  
Topological Entropy ◽  
Simple Proof ◽  
Metric Space ◽  
Simple Theory ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Continuous Maps

We present a simple theory on topological entropy of the continuous maps defined on a compact metric space, and establish some inequalities of topological entropy. As an application of the results of this paper, we give a new simple proof of chaos in the so-calledN-buffer switched flow networks.

CHAOTIC MAP WITH THE PROPERTY OF RECURRENCE

2007 ◽  
Vol 21 (15) ◽  
pp. 2711-2721 ◽  
Author(s):  
LIDONG WANG ◽  
ZHENYAN CHU ◽  
XIAODONG DUAN
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Chaotic Map ◽  
Compact Metric Space ◽  
Continuous Map

In this paper, we consider a continuous map f: X→X, where X is a compact metric space, and discuss the existence of a chaotic set of f specially (as X=[0,1]). We prove that f has a positively topological entropy if and only if it has an uncountably chaotic set in which each point is recurrent and is not weakly periodic.

On Generic Properties of Nonautonomous Dynamical Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1850102 ◽  
Author(s):  
Francisco Balibrea ◽  
Jaroslav Smítal ◽  
Marta Štefánková
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Compact Space ◽  
Metric Space ◽  
Generic Properties ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonautonomous Dynamical Systems ◽  
Uniformly Convergent ◽  
Generic System

We consider nonautonomous dynamical systems consisting of sequences of continuous surjective maps of a compact metric space [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] denote the space of systems [Formula: see text], which are uniformly convergent, or equicontinuous, or pointwise convergent (to a continuous map), respectively. We show that for [Formula: see text], the generic system in any of the spaces has infinite topological entropy, while, if [Formula: see text] is the Cantor set, the generic system in any of the spaces has zero topological entropy. This shows, among others, that the general results of the above type for arbitrary compact space [Formula: see text] are difficult if not impossible.

Some properties of positive entropy maps

2013 ◽  
Vol 34 (3) ◽  
pp. 765-776 ◽  
Author(s):  
A. ARBIETO ◽  
C. A. MORALES
Keyword(s):  
Invariant Measure ◽  
Metric Space ◽  
Metric Spaces ◽  
Acta Math ◽  
Positive Entropy ◽  
Canonical Coordinates ◽  
Compact Metric Space ◽  
Compact Metric Spaces ◽  
Positive Topological Entropy ◽  
Continuous Map

AbstractWe prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

On the polynomial entropy for morse gradient systems

2019 ◽  
Vol 69 (3) ◽  
pp. 611-624
Author(s):  
Jelena Katić ◽  
Milan Perić
Keyword(s):  
Metric Space ◽  
Singular Set ◽  
Compact Metric Space ◽  
Easy Method ◽  
Gradient Systems ◽  
Continuous Map ◽  
Negative Gradient

Abstract We adapt the construction from [HAUSEUX, L.—LE ROUX, F.: Polynomial entropy of Brouwer homeomorphisms, arXiv:1712.01502 (2017)] to obtain an easy method for computing the polynomial entropy for a continuous map of a compact metric space with finitely many non-wandering points. We compute the maximal cardinality of a singular set of Morse negative gradient systems and apply this method to compute the polynomial entropy for Morse gradient systems on surfaces.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

Chain recurrence and attraction in non-compact spaces

1991 ◽  
Vol 11 (4) ◽  
pp. 709-729 ◽  
Author(s):  
Mike Hurley
Keyword(s):  
Metric Space ◽  
Basins Of Attraction ◽  
The Other ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Fundamental Interest ◽  
Compact Spaces ◽  
Continuous Map ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractIn the study of a dynamical systemf:X→Xgenerated by a continuous mapfon a compact metric spaceX, thechain recurrent setis an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ off. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

Horseshoes and the Conley index spectrum

2000 ◽  
Vol 20 (2) ◽  
pp. 365-377 ◽  
Author(s):  
MARIA C. CARBINATTO ◽  
JAROSLAW KWAPISZ ◽  
KONSTANTIN MISCHAIKOW
Keyword(s):  
Metric Space ◽  
Conley Index ◽  
Spectral Information ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Full Shift ◽  
Shift Dynamics ◽  
Isolating Neighborhood

Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semi-conjugacy onto the full shift dynamics on two symbols.

scholarly journals On the computability of rotation sets and their entropies

2018 ◽  
Vol 40 (2) ◽  
pp. 367-401 ◽  
Author(s):  
MICHAEL A. BURR ◽  
MARTIN SCHMOLL ◽  
CHRISTIAN WOLF
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
Topological Entropy ◽  
Metric Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Subshift Of Finite Type ◽  
Continuous Potential ◽  
Rotation Set ◽  
Entropy Functions

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

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