Shifts of finite type and random substitutions

TL;DR

This paper demonstrates that all one-dimensional topologically transitive shifts of finite type are conjugate to subshifts from primitive random substitutions, and shows that their entropy values are dense in positive reals, including all Perron numbers.

Contribution

It establishes a conjugacy between shifts of finite type and random substitution subshifts, and proves the density of their entropy values in positive real numbers.

Findings

All topologically transitive shifts of finite type are conjugate to primitive random substitution subshifts.

The set of entropy values for these subshifts includes all Perron numbers and is dense in positive reals.

An elementary proof confirms the density of entropy values.

Abstract

We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.