Some properties of chain recurrent sets in a nonautonomous discrete dynamical system

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Dhaval Thakkar ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Shadowing Property

AbstractIn this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

scholarly journals Martelli Chaotic Properties of a Generalized Form of Zadeh’s Extension Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yaoyao Lan ◽  
Chunlai Mu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Discrete Dynamical System ◽  
Extension Principle ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Compact Subsets

LetXdenote a compact metric space and letf : X→Xbe a continuous map. It is known that a discrete dynamical system (X,f) naturally induces its fuzzified counterpart, that is, a discrete dynamical system on the space of fuzzy compact subsets ofX. In 2011, a new generalized form of Zadeh’s extension principle, so-calledg-fuzzification, had been introduced by Kupka 2011. In this paper, we study the relations between Martelli’s chaotic properties of the original andg-fuzzified system. More specifically, we study the transitivity, sensitivity, and stability of the orbits in system (X,f) and its connections with the same ones in itsg-fuzzified system.

scholarly journals Functional envelope of a non-autonomous discrete system

2017 ◽  
Vol 4 (1) ◽  
pp. 98-107
Author(s):  
Ali Barzanouni
Keyword(s):  
Metric Space ◽  
Discrete System ◽  
Compact Metric Space ◽  
Continuous Maps ◽  
Functional Envelope

Abstract Let (X, F = {fn}n =0∞) be a non-autonomous discrete system by a compact metric space X and continuous maps fn : X → X, n = 0, 1, ....We introduce functional envelope (S(X), G = {Gn}n =0∞), of (X, F = {fn}n =0∞), where S(X) is the space of all continuous self maps of X and the map Gn : S(X) → S(X) is defined by Gn(ϕ) = Fn ∘ ϕ, Fn = fn ∘ fn-1 ∘ . . . ∘ f1 ∘ f0. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope.

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.

scholarly journals A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Risong Li ◽  
Xiaoliang Zhou
Keyword(s):  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Stable Map ◽  
Asymptotic Average Shadowing

We prove that if a continuous, Lyapunov stable mapffrom a compact metric spaceXinto itself is topologically transitive and has the asymptotic average shadowing property, thenXis consisting of one point. As an application, we prove that the identity mapiX:X→Xdoes not have the asymptotic average shadowing property, whereXis a compact metric space with at least two points.

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.

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$\text{Rot}(\unicode[STIX]{x1D6F7})$.

scholarly journals Global asymptotic stability of inhomogeneous iterates

2002 ◽  
Vol 29 (3) ◽  
pp. 133-142 ◽  
Author(s):  
Yong-Zhuo Chen
Keyword(s):  
Dynamical System ◽  
Asymptotic Stability ◽  
Metric Space ◽  
Complete Metric Space ◽  
Discrete Dynamical System ◽  
Closed Convex Cone ◽  
Nonlinear Difference Equations ◽  
Finite Dimensional ◽  
Uniformly Convergent ◽  
The Stability

Let(M,d)be a finite-dimensional complete metric space, and{Tn}a sequence of uniformly convergent operators onM. We study the non-autonomous discrete dynamical systemxn+1=Tnxnand the globally asymptotic stability of the inhomogeneous iterates of{Tn}. Then we apply the results to investigate the stability of equilibrium ofTwhen it satisfies certain type of sublinear conditions with respect to the partial order defined by a closed convex cone. The examples of application to nonlinear difference equations are also given.

Directional entropy of ℤ+k-actions

2015 ◽  
Vol 16 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Jinlian Zhang ◽  
Wenda Zhang
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Compact Metric Space ◽  
Finite Graphs ◽  
Nonautonomous Dynamical System

In this paper, topological and measure-theoretic directional entropies are investigated for [Formula: see text]-actions. Let [Formula: see text] be a [Formula: see text]-action on a compact metric space. For each ray [Formula: see text] in [Formula: see text] we introduce a notion of positive expansivity for [Formula: see text] along [Formula: see text]. We apply the technique of “coding” which was given by Boyle and Lind in [1] to show that these directional entropies are both continuous at positively expansive directions. We relate the directional entropies of a [Formula: see text]-action at a ray [Formula: see text] to the entropies of a nonautonomous dynamical system which induced by the compositions of a sequence of maps along [Formula: see text]. And hence the variational principle relating topological and measure-theoretic directional entropies is given at positively expansive directions. Applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the [Formula: see text]-subshift actions on [Formula: see text], [Formula: see text]-actions on finite graphs and certain smooth [Formula: see text]-actions on Riemannian manifolds.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

The limit shadowing property and Li–Yorke’s chaos

2016 ◽  
Vol 09 (01) ◽  
pp. 1650007
Author(s):  
Manseob Lee
Keyword(s):  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Limit Shadowing

Let [Formula: see text] be a compact metric space, and let [Formula: see text] be a homeomorphism. We show that if [Formula: see text] has the limit shadowing property then [Formula: see text] is chaotic in the sense of Li–Yorke. Moreover, [Formula: see text] is dense Li–Yorke chaos.

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