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