Invariant measures for subshifts arising from substitutions of some primitive components

TL;DR

This paper introduces a new class of substitutions called primitive components, decomposes their associated subshifts into invariant sets, and characterizes invariant measures using eigenvalues of an incidence matrix.

Contribution

It defines substitutions of primitive components, analyzes their subshifts, and characterizes invariant measures through eigenvalues, advancing understanding of their measure-theoretic properties.

Findings

Decomposition of subshifts into disjoint invariant sets

Uniqueness of invariant Radon measures up to scaling

Characterization of finite invariant measures via eigenvalues

Abstract

The class of substitutions of some primitive components is introduced. A bilateral subshift arising from a substitution of some primitive components is decomposed into pairwise disjoint, locally compact, shift-invariant sets, on each of which an invariant Radon measure is unique up to scaling. It is completely characterized in terms of eigenvalues of an incidence matrix when the unique invariant measure is finite.