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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Invariant relations for the Bowen-Series transform
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by Vincent Pit PDF
Conform. Geom. Dyn. 16 (2012), 103-123 Request permission

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 $\mathbb {D} / \Gamma$.
References
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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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2910743