Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$k$-type chaos of $\mathbb{Z}^d$ actions

Anshid Aboobacker, Sharan Gopal

TL;DR

This work addresses how to define and analyze chaos for dynamical systems with $\mathbb{Z}^d$ actions by introducing $k$-type proximal pairs, $k$-type asymptotic pairs, and $k$-type Li–Yorke sensitivity. It then proves an Auslander–Yorke dichotomy for these $k$-type notions, studies their preservation under uniform conjugacy, and relates them to the classical $\mathbb{Z}$-action notions. Additionally, it analyzes induced $\mathbb{Z}^d$ actions, showing that sensitivity and periodicity properties transfer from base systems to the induced actions and that chaos notions lift accordingly. Overall, the work provides a unified framework for multidimensional discrete-time chaos with practical implications for analysis of $\mathbb{Z}^d$-actions and their conjugacies.

Abstract

In this paper, we define and study the notions of $k$-type proximal pairs, $k$-type asymptotic pairs and $k$-type Li Yorke sensitivity for dynamical systems given by $\mathbb{Z}^d$ actions on compact metric spaces. We prove the Auslander-Yorke dichotomy theorem for $k$-type notions. The preservation of some of these notions under uniform conjugacy is also studied. We also study relations between these notions and their analogous notions in the usual dynamical systems.

$k$-type chaos of $\mathbb{Z}^d$ actions

TL;DR

This work addresses how to define and analyze chaos for dynamical systems with actions by introducing -type proximal pairs, -type asymptotic pairs, and -type Li–Yorke sensitivity. It then proves an Auslander–Yorke dichotomy for these -type notions, studies their preservation under uniform conjugacy, and relates them to the classical -action notions. Additionally, it analyzes induced actions, showing that sensitivity and periodicity properties transfer from base systems to the induced actions and that chaos notions lift accordingly. Overall, the work provides a unified framework for multidimensional discrete-time chaos with practical implications for analysis of -actions and their conjugacies.

Abstract

In this paper, we define and study the notions of -type proximal pairs, -type asymptotic pairs and -type Li Yorke sensitivity for dynamical systems given by actions on compact metric spaces. We prove the Auslander-Yorke dichotomy theorem for -type notions. The preservation of some of these notions under uniform conjugacy is also studied. We also study relations between these notions and their analogous notions in the usual dynamical systems.
Paper Structure (7 sections, 20 theorems, 8 equations)

This paper contains 7 sections, 20 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Auslander-Yorke Dichotomy theorem
  5. $k-$type Li Yorke Sensitivity
  6. Preservation under conjugacy
  7. Induced $\mathbb{Z}^d$ actions

Key Result

Theorem 1

Let $(X, f)$ be topologically transitive. If $(X, f)$ is almost equicontinuous, then the set of equicontinuous points coincides with the set of transitive points (and so the set of equicontinuous points is a dense $G_{\delta}$). In particular, a minimal almost equicontinuous dynamical system is equi

Theorems & Definitions (43)

  • Theorem
  • Theorem
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 33 more