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Codage du flot géodésique sur les surfaces hyperboliques de volume fini

Abstract : This thesis focuses on the study of the objects linked to the Bowen-Series coding of the geodesic flow for hyperbolic surfaces of finite volume. It is first proved that the geodesic billiard associated with an "even corners" fundamental domain for a cofinite fuchsian group is conjugated with a bijection of the torus, called extended coding, one factor of which is the Bowen-Series transform. The sharpest property of that conjugacy is that it always only involves a finite number of objects. Some classical results about the Bowen-Series coding are then rediscovered : it is orbit-equivalent with the group, its periodic points are dense, and its periodic orbits are in bijection with conjugacy classes of primitive hyperbolic isometries ; which eventually links its Ruelle zeta function to the Selberg zeta function. The proofs of those results use a combinatorial lemma that abstracts the orbit-equivalence property to families of relations that can be defined on every set on which the group acts. The extended coding is also proved to be conjugated with a subshift of finite type, except for a countable set of points. Finally, it is shown that eigendistributions of the transfer operator for the eigenvalue 1 are the Helgason boundary values of eigenfunction of laplacian on the surface, plus that one can associate to each such eigendistribution a non-trivial eigenfunction of the transfer operator and that this process has a reciprocal in some cases.
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Contributor : Vincent Pit <>
Submitted on : Thursday, January 6, 2011 - 3:47:37 PM
Last modification on : Thursday, January 11, 2018 - 6:21:22 AM
Long-term archiving on: : Friday, December 2, 2016 - 7:19:28 PM


  • HAL Id : tel-00553138, version 1




Vincent Pit. Codage du flot géodésique sur les surfaces hyperboliques de volume fini. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2010. Français. ⟨tel-00553138⟩



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