# Etude du billard polyédral

Abstract : In this work, we study polyhedral billiard. We code the orbit of a point, with a letter by face. Then we study two sorts of problems: The complexity function of the billiard words, and the existence of periodic orbits. We show that the complexity function is related to the notion of generalized diagonal: A generalized diagonal is a billiard trajectory between two edges of the polyhedron. We show, first, a new proof of the computation of the complexity of a rotation of the torus $\mathbb(T)^2$. This proof allows us, to obtain estimates for the complexity in some right prisms. Then we search for bounds of the global complexity in the case of the cube. Then we prove that the billiard in any convex polyhedron is of zero topological entropy. Then we study the periodic orbits. We find a sufficiant condition for the existence of stable periodic words, and we obtain periodic trajectories of length four in a tetrahedron. Then we obtain ergodicity and estimate for the complexity of a subclass of rectangle exchanges. This is related to the billiard map, since they represent the first return map to a transverse set.
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https://tel.archives-ouvertes.fr/tel-00009363
Contributor : Nicolas Bedaride <>
Submitted on : Wednesday, June 1, 2005 - 3:13:27 PM
Last modification on : Thursday, January 18, 2018 - 2:09:52 AM
Long-term archiving on: : Friday, April 2, 2010 - 9:57:17 PM

### Identifiers

• HAL Id : tel-00009363, version 1

### Citation

Nicolas Bedaride. Etude du billard polyédral. Mathématiques [math]. Université de la Méditerranée - Aix-Marseille II, 2005. Français. ⟨tel-00009363⟩

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