Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow{\mathrm{\Gamma }}$ and vector towers $\overleftarrow{v}$ on $\overleftarrow{\mathrm{\Gamma }}$, 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 $\phi :F_N\to F_N$ is determined, when $\phi$ is represented by a train track map.