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arXiv · 2407.17304

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

Vesselin Petkov
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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.

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Vesselin Petkov. 2024-07-24. On the number of poles of the dynamical zeta functions for billiard flow. https://doi.org/10.3934/dcds.2025015

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