Abstract
In this paper, we consider tilings of the hyperbolic 2-space inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that 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.
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