Dynamics of interval maps generated by erasing substitutions

TL;DR

This paper explores how erasing block substitutions influence the dynamics of generated interval maps, revealing that completely erasing substitutions produce the most complex behaviors like chaos and high entropy.

Contribution

It introduces a hierarchy of erasing substitutions and identifies the class that yields the richest dynamical phenomena in interval maps.

Findings

Completely erasing substitutions lead to Devaney and Li-Yorke chaos.

These maps exhibit infinite topological entropy.

Hierarchy classifies erasing substitutions by their dynamical complexity.

Abstract

We study discontinuous interval maps generated by the action of erasing block substitutions on the binary expansion. After establishing some general properties of these maps, we categorize erasing block substitutions in a hierarchy of classes displaying progressively stronger erasing character. We investigate how this affects the dynamics of the corresponding interval maps, showing that the richest dynamical behavior (Devaney and Li-Yorke chaos, infinite topological entropy) is achieved at a precise step in this hierarchy, which we name completely erasing substitutions. KEYWORDS: Topological dynamics, Erasing substitutions; Devaney chaos; Li-Yorke chaos; Topological entropy.