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Dynamical Zeta Function for Several Strictly Convex Obstacles

Published online by Cambridge University Press:  20 November 2018

Vesselin Petkov
Vesselin Petkov*
Affiliation:
Département de Mathématiques Appliquées, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence, France e-mail: petkov@math.u-bordeaux1.fr
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Abstract

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The behavior of the dynamical zeta function ${{Z}_{D}}(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s)\,=\,\frac{1}{\zeta (s)}$ of the Riemann zeta function $\zeta \left( s \right)$. Let $\prod \left( s \right)$ be the series obtained from ${{Z}_{D}}(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of ${{Z}_{D}}(s)$ and $\prod \left( s \right)$ close to the line $\Re s={{s}_{2}},$ where ${{s}_{2}}$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ${{Z}_{D}}(s),$$\prod \left( s \right)$ has a singularity at $s\,=\,{{s}_{2}}$.

Keywords

11M3658J50dynamical zeta functionperiodic rays
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 100 - 113
Copyright
Copyright © Canadian Mathematical Society 2008

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