scholarly journals Orbital shadowing, internal chain transitivity and -limit sets

2016 ◽  
Vol 38 (1) ◽  
pp. 143-154 ◽  
Author(s):  
CHRIS GOOD ◽  
JONATHAN MEDDAUGH
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
Compact Metric Space ◽  
Limit Sets ◽  
Continuous Map ◽  
Orbital Shadowing ◽  
Limit Shadowing ◽  
Chain Transitivity

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let $\unicode[STIX]{x1D714}_{f}$ be the collection of $\unicode[STIX]{x1D714}$-limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\unicode[STIX]{x1D714}_{f}$ in the Hausdorff metric coincides with $\mathit{ICT}(f)$. In this paper, we prove that $\unicode[STIX]{x1D714}_{f}=\mathit{ICT}(f)$ if and only if $f$ satisfies Pilyugin’s notion of orbital limit shadowing. We also characterize those maps for which $\overline{\unicode[STIX]{x1D714}_{f}}=\mathit{ICT}(f)$ in terms of a variation of orbital shadowing.

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 multi-sensitivity with respect to a vector

2018 ◽  
Vol 32 (15) ◽  
pp. 1850166 ◽  
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Continuous Map ◽  
Periodic Sets ◽  
Vector Formula ◽  
Compact Subsets

Consider the surjective continuous map [Formula: see text]: [Formula: see text] defined on a compact metric space X. Let [Formula: see text] be the space of all non-empty compact subsets of X equipped with the Hausdorff metric and define [Formula: see text]: [Formula: see text] by [Formula: see text] for any [Formula: see text]. In this paper, we introduce several stronger versions of sensitivities, such as multi-sensitivity with respect to a vector, [Formula: see text]-sensitivity, strong multi-sensitivity. We obtain some basic properties of the concepts of these sensitivities and discuss the relationships with other sensitivities for continuous self-map on [0,[Formula: see text]1]. Some sufficient conditions for a dynamical system to be [Formula: see text]-sensitive are presented. Also, it is shown that the strong multi-sensitivity of f implies that [Formula: see text] is [Formula: see text]-sensitive. In turn, the [Formula: see text]-sensitivity of [Formula: see text] implies that [Formula: see text] is [Formula: see text]-sensitive. In particular, it is proved that if [Formula: see text] is a multi-transitive map with dense periodic sets, then f is [Formula: see text]-sensitive. Finally, we give a multi-sensitive example which is not [Formula: see text]-sensitive.

EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE OF PULLBACK ATTRACTORS ON PARAMETERS

2004 ◽  
Vol 04 (03) ◽  
pp. 373-384 ◽  
Author(s):  
DESHENG LI ◽  
P. E. KLOEDEN
Keyword(s):  
Metric Space ◽  
Continuous Dependence ◽  
Hausdorff Metric ◽  
Regularity Conditions ◽  
Pullback Attractors ◽  
Compact Metric Space ◽  
Nonautonomous Dynamical Systems ◽  
The Family ◽  
Nonempty Compact ◽  
Autonomous Dynamical System

The equi-attraction properties of uniform pullback attractors [Formula: see text] of nonautonomous dynamical systems (θ,ϕλ) with a parameter λ∈Λ, where Λ is a compact metric space, are investigated; here θ is an autonomous dynamical system on a compact metric space P which drives the cocycle ϕλon a complete metric state space X. In particular, under appropriate regularity conditions, it is shown that the equi-attraction of the family [Formula: see text] uniformly in p∈P is equivalent to the continuity of the setvalued mappings [Formula: see text] in λ with respect to the Hausdorff metric on the nonempty compact subsets of X.

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.

On the set of expansive measures

2018 ◽  
Vol 20 (07) ◽  
pp. 1750086 ◽  
Author(s):  
Keonhee Lee ◽  
C. A. Morales ◽  
Bomi Shin
Keyword(s):  
Metric Space ◽  
Weak Topology ◽  
Hausdorff Metric ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Whole Space ◽  
Isolated Points ◽  
Borel Probability ◽  
Heteroclinic Points

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

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.

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.

Chaos to Multiple Mappings

2017 ◽  
Vol 27 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Lidong Wang ◽  
Yingcui Zhao ◽  
Yuelin Gao ◽  
Heng Liu
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
Strong Sense ◽  
Sufficient Condition ◽  
Compact Metric Space ◽  
Distributional Chaos ◽  
Nonwandering Set

Let [Formula: see text] be a compact metric space and [Formula: see text] be an [Formula: see text]-tuple of continuous selfmaps on [Formula: see text]. This paper investigates Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence from a set-valued view. On the basis of this research, we draw the main conclusions as follows: (i) If [Formula: see text] has a distributionally chaotic pair, especially [Formula: see text] is distributionally chaotic, the strongly nonwandering set [Formula: see text] contains at least two points. (ii) We give a sufficient condition for [Formula: see text] to be distributionally chaotic in a sequence and chaotic in the strong sense of Li–Yorke. Finally, an example to verify (ii) is given.

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