ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence [Formula: see text] of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence [Formula: see text] either forms an equicontinuous family of maps or is uniformly convergent. We also show that for any continuous maps f and g from a compact topological space into itself the topological entropies h(f ◦ g) and h(g ◦ f) are equal.

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Alder's topological entropy of generalized discrete dynamical system

2012 International Conference on Systems and Informatics (ICSAI2012) ◽

10.1109/icsai.2012.6223078 ◽

2012 ◽

Cited By ~ 1

Author(s):

Xiaoqian Song ◽

Liangwei Wang

Keyword(s):

Dynamical System ◽

Topological Entropy ◽

Discrete Dynamical System

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Global asymptotic stability of inhomogeneous iterates

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202011316 ◽

2002 ◽

Vol 29 (3) ◽

pp. 133-142 ◽

Cited By ~ 5

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.

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ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY

Fractals ◽

10.1142/s0218348x10004956 ◽

2010 ◽

Vol 18 (03) ◽

pp. 363-370 ◽

Cited By ~ 2

Author(s):

DONGKUI MA ◽

MIN WU

Keyword(s):

Variational Principle ◽

Hausdorff Dimension ◽

Topological Space ◽

Topological Entropy ◽

Continuous Map ◽

Compact Topological Space ◽

Metric Function

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

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ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499001206 ◽

1999 ◽

Vol 09 (09) ◽

pp. 1719-1729 ◽

Cited By ~ 5

Author(s):

LLUÍS ALSEDÀ ◽

MOIRA CHAS ◽

JAROSLAV SMÍTAL

Keyword(s):

Dynamical Systems ◽

Topological Entropy ◽

Discrete Dynamical Systems ◽

Limit Sets ◽

Interval Maps ◽

Positive Topological Entropy ◽

Countable Ordinal ◽

Continuous Maps

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

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ABSTRACT ω-LIMIT SETS

Journal of Symbolic Logic ◽

10.1017/jsl.2018.11 ◽

2018 ◽

Vol 83 (2) ◽

pp. 477-495 ◽

Cited By ~ 3

Author(s):

WILL BRIAN

Keyword(s):

Dynamical System ◽

Hausdorff Space ◽

Complete Characterization ◽

Topological Dynamics ◽

Limit Set ◽

Limit Sets ◽

Shift Map ◽

Image Position ◽

Continuous Surjection

AbstractThe shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

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On the origin and development of some notions of entropy

Topological Algebra and its Applications ◽

10.1515/taa-2015-0006 ◽

2015 ◽

Vol 3 (1) ◽

Cited By ~ 1

Author(s):

Francisco Balibrea

Keyword(s):

Applied Sciences ◽

Dynamical Systems ◽

Topological Entropy ◽

Discrete Dynamical Systems ◽

Point Of View ◽

Topological Dynamics ◽

Theoretical Point ◽

Compact Metric Space ◽

Continuous Maps ◽

And Control

AbstractDiscrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper we are dealing mainly with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn such notions have to evolve to other recent situations where it is necessary to give some extended versions of them adapted to the new problems.

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Kinematics in Biology: Symbolic Dynamics Approach

Mathematics ◽

10.3390/math8030339 ◽

2020 ◽

Vol 8 (3) ◽

pp. 339

Author(s):

Carlos Correia Ramos

Keyword(s):

Dynamical System ◽

Topological Entropy ◽

Empirical Data ◽

Symbolic Dynamics ◽

Discrete Dynamical System ◽

Building Blocks ◽

Geometrical Method ◽

Two Parameters ◽

Complex Trajectories

Motion in biology is studied through a descriptive geometrical method. We consider a deterministic discrete dynamical system used to simulate and classify a variety of types of movements which can be seen as templates and building blocks of more complex trajectories. The dynamical system is determined by the iteration of a bimodal interval map dependent on two parameters, up to scaling, generalizing a previous work. The characterization of the trajectories uses the classifying tools from symbolic dynamics—kneading sequences, topological entropy and growth number. We consider also the isentropic trajectories, trajectories with constant topological entropy, which are related with the possible existence of a constant drift. We introduce the concepts of pure and mixed bimodal trajectories which give much more flexibility to the model, maintaining it simple. We discuss several procedures that may allow the use of the model to characterize empirical data.

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Entropy on noncompact sets

Topological Algebra and its Applications ◽

10.1515/taa-2019-0003 ◽

2019 ◽

Vol 7 (1) ◽

pp. 29-37

Author(s):

Jose S. Cánovas

Keyword(s):

Topological Entropy ◽

Topological Spaces ◽

Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

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On topological quotient maps preserved by pullbacks or products

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100045850 ◽

1970 ◽

Vol 67 (3) ◽

pp. 553-558 ◽

Cited By ~ 72

Author(s):

B. J. Day ◽

G. M. Kelly

Keyword(s):

Topological Space ◽

Topological Spaces ◽

Quotient Maps ◽

Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

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