Dirichlet dynamical zeta function for billiard flow

Abstract

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