On the Entropy Points and Shadowing in Uniform Spaces

2018 ◽  
Vol 28 (12) ◽  
pp. 1850155 ◽  
Author(s):  
Seyyed Alireza Ahmadi ◽  
Xinxing Wu ◽  
Zonghong Feng ◽  
Xin Ma ◽  
Tianxiu Lu
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Uniform Spaces ◽  
Shadowing Property

We introduce and study the topological concepts of entropy points, expansivity and shadowing property for dynamical systems on noncompact nonmetrizable spaces, which generalize the relevant concepts for metric spaces. We also obtain various properties on uniform entropy on noncompact nonmetrizable spaces. The main result is a theorem which yields a relation between topological shadowing property and positive uniform entropy.

Shadowing, ergodic shadowing and uniform spaces

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5117-5124 ◽  
Author(s):  
Seyyed Ahmadi
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Uniform Spaces ◽  
Shadowing Property ◽  
Chain Transitivity ◽  
Metrizable Spaces

We introduce and study the topological concepts of ergodic shadowing, chain transitivity and topological ergodicity for dynamical systems on non-compact non-metrizable spaces. These notions generalize the relevant concepts for metric spaces. We prove that a dynamical system with topological ergodic shadowing property is topologically chain transitive, and that topological chain transitivity together with topological shadowing property implies topological ergodicity.

scholarly journals Shadowing as a structural property of the space of dynamical systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jonathan Meddaugh
Keyword(s):  
Dynamical Systems ◽  
Large Class ◽  
Structural Property ◽  
Metric Spaces ◽  
Shadowing Property ◽  
Compact Metric Spaces

<p style='text-indent:20px;'>We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that <i>continuously generated</i> pseudo-orbits can be shadowed.</p>

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

scholarly journals Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 158
Author(s):  
Liliana Guran ◽  
Monica-Felicia Bota
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Boundary Value ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Shadowing Property ◽  
Limit Shadowing ◽  
Type Boundary ◽  
Cyclic Type

The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, and the last part of the paper considers some applications of the main results. The first part of this section is devoted to the study of the existence of a solution to the boundary value problem. In the second part of this section, we study the existence of solutions to fractional boundary value problems with integral-type boundary conditions in the frame of some Caputo-type fractional operators.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

scholarly journals Strong Li–Yorke Chaos for Time-Varying Discrete Dynamical Systems with A-Coupled-Expansion

2015 ◽  
Vol 25 (13) ◽  
pp. 1550186 ◽  
Author(s):  
Hua Shao ◽  
Yuming Shi ◽  
Hao Zhu
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Strong Sense ◽  
Time Varying ◽  
Transition Matrices ◽  
Complete Metric Spaces ◽  
Compact Subsets

This paper is concerned with strong Li–Yorke chaos induced by [Formula: see text]-coupled-expansion for time-varying (i.e. nonautonomous) discrete systems in metric spaces. Some criteria of chaos in the strong sense of Li–Yorke are established via strict coupled-expansions for irreducible transition matrices in bounded and closed subsets of complete metric spaces and in compact subsets of metric spaces, respectively, where their conditions are weaker than those in the existing literature. One example is provided for illustration.

On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

2020 ◽  
Vol 36 (12) ◽  
pp. 1384-1394
Author(s):  
Xiao Fang Luo ◽  
Xiao Xiao Nie ◽  
Jian Dong Yin
Keyword(s):  
Dynamical Systems ◽  
Shadowing Property

Dynamics of G-processes

2020 ◽  
Vol 20 (01) ◽  
pp. 2050037
Author(s):  
W. Jung ◽  
K. Lee ◽  
C.A. Morales
Keyword(s):  
Dynamical Systems ◽  
Group Structure ◽  
Global Solutions ◽  
Topological Stability ◽  
Natural Group ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Mathematical Sciences ◽  
The Relationship

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group [Formula: see text]. These processes are broad enough to include the [Formula: see text]-actions (characterized as autonomous [Formula: see text]-processes) and the two-parameter flows (where [Formula: see text]). We endow the space of [Formula: see text]-processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for [Formula: see text]-processes and analyze the relationship with the [Formula: see text]-processes group structure. We study the equicontinuous [Formula: see text]-processes and use it to construct nonautonomous [Formula: see text]-processes with the shadowing property. We study the global solutions of the [Formula: see text]-processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the [Formula: see text]-processes. We prove that there are nonautonomous expansive [Formula: see text]-processes and characterize the existence of expansive equicontinuous [Formula: see text]-processes. We define the topological stability for [Formula: see text]-processes and prove that every expansive [Formula: see text]-process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable [Formula: see text]-processes are given.

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