6533b7d4fe1ef96bd1261b9b

RESEARCH PRODUCT

Laminations and tilings of the Hyperbolic upper half plane

subject

description

This thesis is devoted to the study of dynamical systems associated with tilings of theEuclidean plane or of the Hyperbolic half-plane. A such tiling codes an action of a group ofisometries (namely the group of translations of the plane or the group of affine maps) on a compactmetric space $\Omega$ such that the properties of this action are related with the combinatoricproperties of the tiling. The behaviors of the actions obtained by this way are really various. Insome cases, like for example for the Penrose's tiling, this action is free and minimal. This givesto the set $\Omega$ a structure of a specific lamination called {\it solenoid}. This space islocally the product of a Cantor set with an open subset of the Euclidean (resp. Hyperbolic) plane.In this thesis, we study the statistical behaviors of the orbits for this action. We give acombinatoric characterization of the invariant probability measures and of the harmonic measures ofthe associated solenoid. We observe here a fundamental difference between the Euclidean and theHyperbolic case. Finally, for every integer $r \geq 1$, we give explicit examples of tilings of theHyperbolic half-plane whose the associated dynamical system is a minimal and free action withexactly $r$ ergodic invariant finite measures.