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Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts

Fatemah Ayatollah Zadeh Shirazi, Elaheh Hakimi, Arezoo Hosseini, Reza Rezavand

TL;DR

The paper classifies Li--Yorke and Devaney chaos for weighted generalized shifts $\sigma_{\varphi,\mathfrak{w}}$ on $R^\Gamma$ with $R$ a finite field. It derives necessary and sufficient conditions for sensitivity, scrambled pairs, and Li--Yorke chaos in terms of a non--quasi--periodic orbit $\theta$ with nonvanishing weights along $\varphi$-orbits, and it connects dense periodic points to onto-ness and invertibility of weights. The Devaney chaotic regime is shown to occur exactly when $\varphi$ is injective without periodic points and all weights are invertible, linking topological transitivity, sensitivity, and dense periodic points to precise algebraic-dynamical criteria. Overall, the results establish tight equivalences between chaos notions in the uniform setting of weighted shifts and the combinatorial/dynamical structure of the base map $\varphi$ and weight vector $\mathfrak{w}$. These findings provide a clear, testable framework for classifying chaotic behavior in uniform dynamical systems built from weighted shifts on finite-field product spaces.

Abstract

In this paper, for finite discrete field $F$, nonempty set $Γ$, weight vector $\mathfrak{w}=({\mathfrak w}_α)_{α\inΓ}\in F^Γ$ and weighted generalized shift $σ_{\varphi,{\mathfrak w}}:F^Γ\to F^Γ$, we find necessary and sufficient conditions for uniform dynamical system $(F^Γ,σ_{\varphi,{\mathfrak w}})$ to be Li--Yorke chaotic. Next we find necessary and sufficient conditions for $(F^Γ,σ_{\varphi,{\mathfrak w}})$ to be Devaney chaotic.

Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts

TL;DR

The paper classifies Li--Yorke and Devaney chaos for weighted generalized shifts on with a finite field. It derives necessary and sufficient conditions for sensitivity, scrambled pairs, and Li--Yorke chaos in terms of a non--quasi--periodic orbit with nonvanishing weights along -orbits, and it connects dense periodic points to onto-ness and invertibility of weights. The Devaney chaotic regime is shown to occur exactly when is injective without periodic points and all weights are invertible, linking topological transitivity, sensitivity, and dense periodic points to precise algebraic-dynamical criteria. Overall, the results establish tight equivalences between chaos notions in the uniform setting of weighted shifts and the combinatorial/dynamical structure of the base map and weight vector . These findings provide a clear, testable framework for classifying chaotic behavior in uniform dynamical systems built from weighted shifts on finite-field product spaces.

Abstract

In this paper, for finite discrete field , nonempty set , weight vector and weighted generalized shift , we find necessary and sufficient conditions for uniform dynamical system to be Li--Yorke chaotic. Next we find necessary and sufficient conditions for to be Devaney chaotic.
Paper Structure (7 sections, 11 theorems, 37 equations)

This paper contains 7 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Sensitivity in $(R^\Gamma,\sigma_{\varphi,{\mathfrak w}})$
  3. Li--Yorke chaotic $(R^\Gamma,\sigma_{\varphi,{\mathfrak w}})$, for finite field $R$
  4. Lemmas on a special case
  5. Main theorem on Li--Yorke chaoticity of $(R^\Gamma,\sigma_{\varphi,\mathfrak{w}})$
  6. When does $(R^\Gamma,\sigma_{\varphi,{\mathfrak w}})$ have dense periodic points?
  7. Devaney chaotic $(R^\Gamma,\sigma_{\varphi,{\mathfrak w}})$, for finite field $R$

Key Result

Lemma 2.3

If for all $\alpha\in\Gamma$: $\bullet$ either $\alpha$ is a quasi--periodic point of $\varphi$, $\bullet$ or $\alpha$ is a non--quasi--periodic point of $\varphi$ and there exists $n\geq0$ with ${\mathfrak w}_{\varphi^n(\alpha)}=0$, then $(R^\Gamma,\sigma_{\varphi,\mathfrak{w}})$ is not sensitive.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 3.2
  • ...and 17 more