Dynamics and Zeta functions on conformally compact manifolds

Rowlett, Julie; Suárez-Serrato, Pablo; Tapie, Samuel

doi:10.1090/S0002-9947-2014-05999-0

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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