# Laminations et pavages du demi-plan hyperbolique

Abstract : This thesis is devoted to the study of dynamical systems associated with tilings of the
Euclidean plane or of the Hyperbolic half-plane. A such tiling codes an action of a group of
isometries (namely the group of translations of the plane or the group of affine maps) on a compact
metric space $\Omega$ such that the properties of this action are related with the combinatoric
properties of the tiling. The behaviors of the actions obtained by this way are really various. In
some cases, like for example for the Penrose's tiling, this action is free and minimal. This gives
to the set $\Omega$ a structure of a specific lamination called {\it solenoid}. This space is
locally 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 a
combinatoric characterization of the invariant probability measures and of the harmonic measures of
the associated solenoid. We observe here a fundamental difference between the Euclidean and the
Hyperbolic case. Finally, for every integer $r \geq 1$, we give explicit examples of tilings of the
Hyperbolic half-plane whose the associated dynamical system is a minimal and free action with
exactly $r$ ergodic invariant finite measures.
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00011423
Contributor : Samuel Petite <>
Submitted on : Thursday, January 19, 2006 - 2:29:04 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Monday, September 17, 2012 - 11:00:31 AM

### Identifiers

• HAL Id : tel-00011423, version 1

### Citation

Samuel Petite. Laminations et pavages du demi-plan hyperbolique. Mathématiques [math]. Université de Bourgogne, 2005. Français. ⟨tel-00011423⟩

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