Graph towers, laminations and their invariant measures

TL;DR

This paper introduces a combinatorial framework using graph towers and vector towers to efficiently describe invariant measures on shift spaces, with applications to substitution subshifts, minimal subshifts, and currents on free groups.

Contribution

It develops a new combinatorial machinery that simplifies the analysis of invariant measures and applies to various dynamical systems and free group currents.

Findings

Provides an efficient method to compute invariant measures on shift spaces.

Characterizes projectively fixed currents under free group endomorphisms.

Applies to non-primitive substitution subshifts and minimal subshifts with many ergodic measures.

Abstract

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.