Constant-length substitutions and countable scrambled sets

TL;DR

This paper explores the existence of various types of scrambled sets in constant-length substitution dynamical systems, providing examples and conditions for Li-Yorke, asymptotic, and distal pairs, including systems with non-recurrent Li-Yorke pairs.

Contribution

It introduces new examples of topological dynamical systems with finite or countable scrambled sets and characterizes conditions for different pairs in substitution systems.

Findings

Existence of systems with finite or countable scrambled sets.

Conditions for Li-Yorke, asymptotic, and distal pairs in substitution systems.

Construction of a system with Li-Yorke pairs that are not recurrent.

Abstract

In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of Li-Yorke, asymptotic and distal pairs in constant--length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li--Yorke pairs, none of which is recurrent.