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

TL;DR

This paper introduces and analyzes new $k$-type chaos concepts in $bZ^d$ actions, establishing foundational theorems and exploring their invariance and relationships with classical dynamical notions.

Contribution

It defines $k$-type proximal, asymptotic, and Li Yorke sensitivity notions for $bZ^d$ actions and proves a dichotomy theorem, extending chaos theory in higher-dimensional group actions.

Findings

Proved the Auslander-Yorke dichotomy for $k$-type notions.

Studied invariance of these notions under uniform conjugacy.

Explored relations between $k$-type notions and classical dynamical concepts.

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.