Invariant measures for train track towers

Nicolas Bedaride (1) , Arnaud Hilion (1) , Martin Lustig (1)

1 I2M - Institut de Mathématiques de Marseille

In this paper we present a combinatorial machinery, consisting of a graph tower Γ← and a weight towers ω← on Γ←, which allow us to efficiently describe invariant measures μ=μω← on rather general discrete dynamicals system over a finite alphabet. A train track map f:Γ→Γ defines canonically a stationary such graph tower Γf←. In the most important two special cases the measure μ specializes to a (typically ergodic) invariant measure on a substitution subshift, or to a projectively f∗-invariant current on the free group π1Γ. Our main result establishes a 1-1 correspondence between such measures μ and the non-negative eigenvectors of the incidence ("transition") matrix of f.

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

Nicolas Bedaride, Arnaud Hilion, Martin Lustig. Invariant measures for train track towers. 2015. ⟨hal-01218333⟩