Dirichlet dynamical zeta function for billiard flow

TL;DR

This paper investigates the Dirichlet dynamical zeta function for billiard flows around convex obstacles, establishing conditions under which it cannot be extended as an entire function, thus revealing complex analytic properties of these dynamical systems.

Contribution

It provides new insights into the meromorphic continuation and analytic obstructions of the Dirichlet dynamical zeta function for billiard flows with convex obstacles.

Findings

ta_D(s) has a meromorphic continuation to .

Certain conditions on frequencies n and coefficients a_n prevent ta_D(s) from being entire.

The study links geometric billiard properties with complex analysis of zeta functions.

Abstract

We study the Dirichlet dynamical zeta function $\eta_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large ${\rm Re}\: s$ 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 entire function.