Perron-Frobenius theory and frequency convergence for reducible substitutions

TL;DR

This paper extends Perron-Frobenius theory to reducible matrices and applies it to reducible substitutions, enabling the determination of limit frequencies and invariant measures in subshifts, filling a gap in the existing theory.

Contribution

It generalizes Perron-Frobenius convergence to reducible matrices and develops tools for analyzing reducible substitutions, which were previously lacking.

Findings

Established a general Perron-Frobenius convergence result for reducible matrices.

Applied the theory to produce limit frequencies for factors in reducible substitutions.

Provided a framework for invariant measures on associated subshifts for reducible cases.

Abstract

We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.