Graph towers, laminations and their invariant measures
Nicolas Bédaride, Arnaud Hilion, Martin Lustig · Journal of the London Mathematical Society · 2020
In this paper we present a combinatorial machinery, consisting of a graph tower Γ ← and vector towers v ← on Γ ← , which allows us to efficiently describe all invariant measures μ = μ 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 φ : F N → F N is determined, when φ is represented by a train track map.