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Distributional chaotic generalized shifts

Zahra Nili Ahmadabadi, Fatemah Ayatollah Zadeh Shirazi

Abstract

Suppose $X$ is a finite discrete space with at least two elements, $Γ$ is a nonempty countable set, and consider self--map $\varphi:Γ\toΓ$. We prove that the generalized shift $σ_\varphi:X^Γ\to X^Γ$ with $σ_\varphi((x_α)_{α\inΓ})=(x_{\varphi(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is: $\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $\varphi:Γ\toΓ$ has at least a non-quasi-periodic point, $\bullet$ dense distributional chaotic if and only if $\varphi:Γ\toΓ$ does not have any periodic point, $\bullet$ transitive distributional chaotic if and only if $\varphi:Γ\toΓ$ is one--to--one without any periodic point. We complete the text by counterexamples.

Distributional chaotic generalized shifts

Abstract

Suppose is a finite discrete space with at least two elements, is a nonempty countable set, and consider self--map . We prove that the generalized shift with (for ) is: distributional chaotic (uniform, type 1, type 2) if and only if has at least a non-quasi-periodic point, dense distributional chaotic if and only if does not have any periodic point, transitive distributional chaotic if and only if is one--to--one without any periodic point. We complete the text by counterexamples.

Paper Structure

This paper contains 6 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. DC1 and DC2 generalized shifts
  4. Dense distributional chaotic generalized shifts
  5. Transitive distributional chaotic generalized shifts
  6. Counterexamples

Key Result

Lemma 3.1

For $x,y\in X^\Gamma$, $n\geq1$, $\alpha\in{\mathcal{F}}$ and $f:X^\Gamma\to X^\Gamma$ let: and Now we have:

Theorems & Definitions (22)

  • Remark 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • ...and 12 more