On the number of poles of the dynamical zeta functions for billiard flow

Abstract

We study the number of the poles of the meromorphic continuation of the dynamical zeta functions ${\eta }_N$ and ${\eta }_D$ for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip $\{z\in \mathbb{C}:\mathrm{Re}s>\beta \}$ with infinite number of poles. For ${\eta }_D$ we prove the same result assuming the boundary real analytic. Moreover, for ${\eta }_N$ we obtain a characterisation of $\beta$ by the pressure $P(2G)$ of some function $G$ on the space ${\Sigma }_A^f$ related to the dynamical characteristics of the obstacle.

This is revised version including shorter proofs of some results. The paper is accepted for publication in Discrete and Continuous Dynamical Systems

References in corpus (1)

• Dynamical zeta functions for billiards