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In this paper we present a combinatorial machinery, consisting of a  graph towerΓ  ←    and  vector towersv  ←    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  Nis determined, when   φ   is represented by a train track map.

Graph towers, laminations and their invariant measures