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

TL;DR

This paper investigates the distribution of poles in the meromorphic continuation of dynamical zeta functions for convex billiard obstacles, revealing an infinite number of poles in a specific complex strip and linking their properties to dynamical pressures.

Contribution

It establishes the existence of infinitely many poles in a certain strip for these zeta functions and characterizes the boundary of this strip using dynamical pressure, extending understanding of billiard flow spectral properties.

Findings

Infinite poles in a specific complex strip for $eta$

Boundary of the pole strip characterized by pressure $P(2G)$

Results hold under non-eclipse and real analytic boundary conditions

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 $\{z \in \mathbb C:\: {\rm Re}\: s > \beta\}$ 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.