Self-Induced Systems

TL;DR

This paper explores self-induced minimal Cantor systems, characterizing their types, providing examples beyond classical cases, and establishing a connection with substitutions on finite or infinite alphabets.

Contribution

It classifies self-induced systems in equicontinuous and expansive cases and introduces new examples with zero or infinite entropy, expanding understanding of these systems.

Findings

Classifies self-induced systems as classical examples in specific cases

Provides examples of zero entropy non-expansive, non-equicontinuous systems

Characterizes self-induced systems via substitutions on various alphabets

Abstract

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous or expansive case. Nevertheless, we exhibit a zero entropy self-induced system that is neither equicontinuous nor expansive. We also provide non-uniquely ergodic self-induced systems with infinite entropy.Moreover, we give a characterization of self-induced minimal Cantor systems in terms of substitutions on finite or infinite alphabets.