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On Li--Yorke chaotic transformation groups modulo an ideal

Mehrnaz Pourattar, Fatemah Ayatollah Zadeh Shirazi

Abstract

In the following text we introduce the notion of chaoticity modulo an ideal in the sense of Li-Yorke in topological transformation semigroups and bring some of its elementary properties. We continue our study by characterizing a class of abelian infinite Li-Yorke chaotic Fort transformation groups and show all of the elements of the above class is co-decomposable to non-Li-Yorke chaotic transformation groups.

On Li--Yorke chaotic transformation groups modulo an ideal

Abstract

In the following text we introduce the notion of chaoticity modulo an ideal in the sense of Li-Yorke in topological transformation semigroups and bring some of its elementary properties. We continue our study by characterizing a class of abelian infinite Li-Yorke chaotic Fort transformation groups and show all of the elements of the above class is co-decomposable to non-Li-Yorke chaotic transformation groups.

Paper Structure

This paper contains 8 sections, 14 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on uniform spaces
  4. Ideals and Fort spaces
  5. Background on Li--Yorke chaos in dynamical systems
  6. Background on transformation semigroup
  7. Asymptoticity and Li--Yorke chaoticity modulo an ideal
  8. Li--Yorke chaotic Fort transformation groups

Key Result

Theorem 3.1

In transformation semigroup $(X,S)$ suppose $\mathcal{I}$ and $\mathcal{J}$ are ideals on $S$ with $\mathcal{I}\subseteq\mathcal{J}$. We have: $\bullet$$Asym_{\mathcal{I}}(X,S)\subseteq Asym_{\mathcal{J}}(X,S)$, $\bullet$ if $D\subseteq X$ is an scrambled set modulo $\mathcal{J}$, then it is an scra

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 22 more