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Conformal Geometry and Dynamics

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Invariant relations for the Bowen-Series transform


Author: Vincent Pit
Journal: Conform. Geom. Dyn. 16 (2012), 103-123
MSC (2010): Primary 37D40; Secondary 37C30, 58C40
DOI: https://doi.org/10.1090/S1088-4173-2012-00238-X
Published electronically: April 16, 2012
MathSciNet review: 2910743
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Abstract: Consider the Bowen-Series transform $T$ associated with an even corners fundamental domain of finite volume for some Fuchsian group $\Gamma$. We prove a generic invariance result that abstracts Series’ orbit-equivalence theorem to families of relations on the unit circle. Two applications of this result are developed. We first prove that $T$ satisfies a strong-orbit equivalence property, which allows to identify its hyperbolic periodic orbits with primitive hyperbolic conjugacy classes of $\Gamma$. Then, we show thanks to the invariance theorem that the eigendistributions for the eigenvalue $1$ of the transfer operator of $T$ with spectral parameter $s \in \mathbb {C}$ are in bijection with smooth bounded eigenfunctions for the eigenvalue $s(1-s)$ of the hyperbolic Laplacian on the quotient $\mathbb {D} / \Gamma$.


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Additional Information

Vincent Pit
Affiliation: Département de Mathématiques d’Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France
Email: vincent.pit@math.u-psud.fr

Received by editor(s): December 7, 2011
Published electronically: April 16, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.