Dynamical systems arising from random substitutions

TL;DR

This paper investigates the dynamical and ergodic properties of systems generated by random substitutions, revealing their complex behaviors such as topological transitivity, entropy, and periodic point structure.

Contribution

It provides a foundational analysis of the dynamical systems arising from random substitutions, including conditions for transitivity, ergodicity, and entropy, expanding understanding beyond deterministic cases.

Findings

Systems are topologically transitive under certain conditions.

They have either no or dense periodic points.

Systems exhibit positive topological entropy.

Abstract

Random substitutions are a natural generalisation of their classical `deterministic' counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently replaced by a word from a finite set of possible words according to a probability distribution. We discuss the subshifts associated with such substitutions and explore the dynamical and ergodic properties of these systems in order to establish the groundwork for their systematic study. Among other results, we show under reasonable conditions that such systems are topologically transitive, have either empty or dense sets of periodic points, have dense sets of linearly repetitive elements, are rarely strictly ergodic, and have positive topological entropy.