Dirichlet dynamical zeta function for billiard flow

Abstract

We study the Dirichlet dynamical zeta function $\hat{I}{\cdot }_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large $\mathrm{Re}s$ we have $\hat{I}{\cdot }_D(s)={\sum }_{n=1}^{\infty }{a}_n{e}^{-\hat{I}{»}_{n}s},a_n\in \mathbb{R}$ and $\hat{I}{\cdot }_D$ admits a meromorphic continuation to $\mathbb{C}$. We obtain some conditions of the frequencies $\hat{I}{»}_n$ and some sums of coefficients $a_n$ which imply that $\hat{I}{\cdot }_D$ cannot be prolonged as entire function.

In the new version there are some minor changes and a new Corollary 4.2. The paper is accepted for publication in Archiv der Mathematik