6533b837fe1ef96bd12a3172

RESEARCH PRODUCT

On invariant measures of finite affine type tilings

Samuel Petite

subject

General MathematicsSubstitution tiling[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]30C85Dynamical Systems (math.DS)01 natural sciences37D40; 52C20; 30C85CombinatoricsAffine geometryAffine representationAffine hull0103 physical sciencesAffine groupFOS: MathematicsMathematics - Dynamical Systems0101 mathematicsMathematics37D40Applied Mathematics010102 general mathematics52C20Affine coordinate systemAffine shape adaptationAffine geometry of curves010307 mathematical physics

description

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.

yearjournalcountryeditionlanguage
2006-01-01
https://hal.archives-ouvertes.fr/hal-00131587