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Dynamics and Zeta functions on conformally compact manifolds


Authors: Julie Rowlett, Pablo Suárez-Serrato and Samuel Tapie
Journal: Trans. Amer. Math. Soc. 367 (2015), 2459-2486
MSC (2010): Primary 37D40, 58J50, 53C22, 53D25, 35P15
DOI: https://doi.org/10.1090/S0002-9947-2014-05999-0
Published electronically: November 24, 2014
MathSciNet review: 3301870
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Abstract: In this note, we study the dynamics and associated Zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical Zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical Zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.


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Additional Information

Julie Rowlett
Affiliation: Max Planck Institut für Mathematik, Vivatgasse 7, D-53111 Bonn, Germany
Email: rowlett@mpim-bonn.mpg.de

Pablo Suárez-Serrato
Affiliation: Instituto de Matemáticas, Universidad Nacional Autonóma de México, Ciudad Universitaria, Coyoacán, 04510, México, D. F.
Email: p.suarez-serrato@matem.unam.mx

Samuel Tapie
Affiliation: Laboratoire Jean Leray, Université de Nantes, 2, rue de la Houssinière - BP 92208, F-44322 Nantes Cedex 3, France
Email: samuel.tapie@univ-nantes.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-05999-0
Keywords: Convex co-compact, conformally compact, negative curvature, geodesic length spectrum, topological entropy, dynamics, geodesic flow, prime orbit theorem, Laplacian, pure point spectrum
Received by editor(s): June 27, 2011
Received by editor(s) in revised form: August 23, 2012
Published electronically: November 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society