On invariant measures of finite affine type tilings

Abstract

In this paper, we consider tilings of the hyperbolic 2-space H 2 , 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 (T ) on which the affine group acts. This space (T ) 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.

Affability of Euclidean tilings

“…This theorem is not true for the hyperbolic tilings constructed in [12] because their continuous hulls do not have transverse invariant measures. In an analogous way, to construct a non-uniquely ergodic example arising from a planar tiling, it is enough to decorate the usual tessellation by unit squares with the product of two Oxtoby's sequences [11].…”

Geometric Realizations of Two-Dimensional Substitutive Tilings

“…The set Σ P can be equipped with a metric topology. The idea is that two tilings x, y in Σ P are close to each other if, up to moving y by an element g ∈ Γ which "does not move a lot", x and y agree on a large ball of M centered at the origin (see for instance [BBG06,Rob04,Sad08,Pet06].…”

C∗-algebras of Penrose hyperbolic tilings

“…Actually, it is shown in [14] , that on X G P(w) the notions of harmonic and invariant measures are the same and such measures can be described in terms of inverse limit of vectoriel cones.…”

“…Nevertheless, the affine group is amenable, so the hull admits at least one invariant probability measure. These measures are actually in one-to-one correspondence with harmonic currents [14], and they provide 3-cyclic cocycles on the smooth algebra of the tiling.…”