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

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 $\{z\in \mathbb{C}:\mathrm{R}\mathrm{e}s>\beta \}$ 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 ${\mathrm{\Sigma }}_A^f$ related to the dynamical characteristics of the obstacle.