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.

scholarly journals New results on topological dynamics of antitriangular maps

2001 ◽  
Vol 2 (1) ◽  
pp. 51 ◽  
Author(s):  
Francisco Balibrea ◽  
J.S. Cánovas ◽  
A. Linero
Keyword(s):  
Metric Space ◽  
Topological Dynamics ◽  
Compact Metric Space ◽  
Special Analysis ◽  
Continuous Maps

<p>We present some results concerning the topological dynamics of antitriangular maps, F:X<sup>2</sup>→ X<sup>2 </sup>with the formvF(x,y)=(g(y),f(x)), where (X,d) is a compact metric space and f,g : X→ X are continuous maps. We make an special analysis in the case of X = [0,1].</p>

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.

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 Iterated function systems: transitivity and minimality

2019 ◽  
Vol 38 (3) ◽  
pp. 97-109 ◽  
Author(s):  
Hadi Parham ◽  
F. H. Ghane ◽  
A. Ehsani
Keyword(s):  
Metric Space ◽  
Chaotic Dynamics ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Finite Family ◽  
Function System ◽  
Topological Transitivity ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Continuous Maps

In this paper, we study the chaotic dynamics of iterated function systems (IFSs) generated by a finite family of maps on a compact metric space. In particular, we restrict ourselves to topological transitivity, fiberwise transitivity, minimality and total minimality of IFSs. First, we pay special attention to the relation between topological transitivity and fiberwise transitivity. Then we generalize the concept of periodic decompositions of continuous maps, introduced by John Banks [1], to iterated function systems. We will focus on the existence of periodic decompositions for topologically transitive IFSs. Finally, we show that each minimal abelian iterated function system generated by a finite family of homeomorphisms on a connected compact metric space X is totally minimal.

On Preimage Entropy of a Sequential Mapping

2008 ◽  
Vol 15 (04) ◽  
pp. 345-357 ◽  
Author(s):  
Wen-Chiao Cheng
Keyword(s):  
Metric Space ◽  
Compact Metric Space ◽  
Fixed Sequence ◽  
Continuous Maps

Biś [2] introduced the notion of entropy-like invariants as an extension of the concept of pointwise preimage entropy. This paper follows Biś concept and extends the behaviour of preimage entropy to the case of a fixed sequence of maps. Therefore, some results are obtained, and we also show that for any continuous maps S and T from a compact metric space onto itself, the maps S ◦ T and T ◦ S have the same pointwise preimage entropy.

Chaos to multiple mappings from a set-valued view

2020 ◽  
Vol 34 (11) ◽  
pp. 2050108
Author(s):  
Hongbo Zeng ◽  
Lidong Wang ◽  
Tao Sun
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Topological Conjugacy ◽  
Compact Metric Space ◽  
Distributional Chaos ◽  
Continuous Maps ◽  
Weakly Mixing

Let [Formula: see text] be a compact metric space and [Formula: see text] be an [Formula: see text]-tuple of continuous maps from [Formula: see text] to itself. In this paper, we investigate the multiple mappings dynamical system [Formula: see text] with Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence properties from a set-valued view. On the basis of this research, we draw main conclusions as follows: (i) two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li–Yorke chaos or distributional chaos. (ii) Hausdorff metric Li–Yorke [Formula: see text]-chaos is equivalent to Hausdorff metric distributional [Formula: see text]-chaos in a sequence. (iii) By giving two examples, we show that there is non-mutual implication between that the multiple mappings [Formula: see text] is Hausdorff metric Li–Yorke chaos and that each element [Formula: see text] [Formula: see text] in [Formula: see text] is Li–Yorke chaos. (iv) For the multiple mappings, weakly mixing implies the Hausdorff metric strongly Li–Yorke chaos and Hausdorff metric distributional chaos in a sequence.

Shadowing property and topological ergodicity for iterated function systems

2019 ◽  
Vol 33 (23) ◽  
pp. 1950272
Author(s):  
Yingcui Zhao ◽  
Lidong Wang ◽  
Fengchun Lei
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Skew Product ◽  
Semi Group ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Iterated Function ◽  
Continuous Maps

Let [Formula: see text] be a compact metric space and [Formula: see text] be two continuous maps on [Formula: see text]. The iterated function system [Formula: see text] is the action of the semi-group generated by [Formula: see text] on [Formula: see text]. In this paper, we introduce the definitions of shadowing property, average shadowing property and topological ergodicity for [Formula: see text] and give some examples. Then we show that (1) if [Formula: see text] has the shadowing property then so do [Formula: see text] and [Formula: see text]; (2) [Formula: see text] has the shadowing property if and only if the step skew product corresponding to [Formula: see text] has the shadowing property. At last, we prove a Lyapunov stable iterated function system having the average shadowing property is topologically ergodic.

scholarly journals Devaney Chaos in Nonautonomous Discrete Systems

2016 ◽  
Vol 26 (11) ◽  
pp. 1650190 ◽  
Author(s):  
Hao Zhu ◽  
Yuming Shi ◽  
Hua Shao
Keyword(s):  
Metric Space ◽  
Chaotic Behavior ◽  
Convergent Sequence ◽  
Discrete Systems ◽  
Periodic Points ◽  
The Other ◽  
Compact Metric Space ◽  
Devaney Chaos ◽  
Continuous Maps ◽  
Uniformly Convergent

This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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