Invariant measures for train track towers

Nicolas Bédaride, Arnaud Hilion, Martin Lustig

Published 2015-03-27Version 1

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow\Gamma$ and a weight towers $\overleftarrow\omega$ on $\overleftarrow\Gamma$, which allow us to efficiently describe invariant measures $\mu = \mu^{\overleftarrow\omega}$ on rather general discrete dynamicals system over a finite alphabet. A train track map $f: \Gamma \to \Gamma$ defines canonically a stationary such graph tower $\overleftarrow{\Gamma_f}$. In the most important two special cases the measure $\mu$ specializes to a (typically ergodic) invariant measure on a substitution subshift, or to a projectively $f_*$-invariant current on the free group $\pi_1 \Gamma$. Our main result establishes a 1-1 correspondence between such measures $\mu$ and the non-negative eigenvectors of the incidence ("transition") matrix of $f$.