  In this paper, we consider tilings of the hyperbolic 2-space, built with a
finite number of polygonal tiles, up to affine transformation. To such a tiling
T, we associate a space of tilings: the continuous hull Omega(T) on which the
affine group acts. This space Omega(T) inherits a solenoid structure whose
leaves correspond to the orbits of the affine group. First we prove the finite
harmonic measures of this laminated space correspond to finite invariant
measures for the affine group action. Then we give a complete combinatorial
description of these finite invariant measures. Finally we give examples with
an arbitrary number of ergodic invariant probability measures.
