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Dynamics and zeta functions on conformally compact manifolds

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 Added by Julie Rowlett
 Publication date 2011
  fields
and research's language is English




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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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In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $dge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $dge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.
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