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

Vesselin Petkov

TL;DR

This work analyzes the pole structure of meromorphically continued dynamical zeta functions $\\eta_N$, $\\eta_D$ for billiard flows around multiple strictly convex obstacles under a non-eclipse condition. It builds a vector-bundle open-hyperbolic framework and a local trace formula that links resonances of transfer operators to sums over periodic rays, leveraging symbolic dynamics on a suspended space to express dynamical thresholds as pressures $P(G)$ and $P(2G)$. The authors prove that each zeta function $\\eta_q$ has infinitely many poles in a rightward strip $\\Re s > \\beta$, with an analogous result for $\\eta_D$ when the boundary is real-analytic, and they identify $a_1 = P(G)$ and $b_1 = P(2G)$ as the critical abscissas governing convergence and resonance distribution. They further obtain density-type lower bounds for resonances akin to Naud's methods, revealing a deep connection between billiard dynamics, symbolic dynamics, and scattering resonances. These results provide insight into the spectral structure of multi-obstacle scattering and contribute to understanding potential essential spectral gaps in this geometric setting.

Abstract

We study the number of the poles of the meromorphic continuation of the dynamical zeta functions $η_N$ and $η_D$ for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip $\{z \in \mathbb C:\: {\rm Re}\: s > β\}$ with infinite number of poles. For $η_D$ we prove the same result assuming the boundary real analytic. Moreover, for $η_N$ we obtain a characterisation of $β$ by the pressure $P(2G)$ of some function $G$ on the space $Σ_A^f$ related to the dynamical characteristics of the obstacle.

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

TL;DR

This work analyzes the pole structure of meromorphically continued dynamical zeta functions , for billiard flows around multiple strictly convex obstacles under a non-eclipse condition. It builds a vector-bundle open-hyperbolic framework and a local trace formula that links resonances of transfer operators to sums over periodic rays, leveraging symbolic dynamics on a suspended space to express dynamical thresholds as pressures and . The authors prove that each zeta function has infinitely many poles in a rightward strip , with an analogous result for when the boundary is real-analytic, and they identify and as the critical abscissas governing convergence and resonance distribution. They further obtain density-type lower bounds for resonances akin to Naud's methods, revealing a deep connection between billiard dynamics, symbolic dynamics, and scattering resonances. These results provide insight into the spectral structure of multi-obstacle scattering and contribute to understanding potential essential spectral gaps in this geometric setting.

Abstract

We study the number of the poles of the meromorphic continuation of the dynamical zeta functions and for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip with infinite number of poles. For we prove the same result assuming the boundary real analytic. Moreover, for we obtain a characterisation of by the pressure of some function on the space related to the dynamical characteristics of the obstacle.
Paper Structure (4 sections, 8 theorems, 173 equations)

This paper contains 4 sections, 8 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local trace formula
  4. Strip with infinite number poles

Key Result

Theorem 1.1

For every $0 < \delta < 1$ there exists $\alpha_{\delta, q} < a_q$ such that for $\alpha < \alpha_{\delta, q}$ we have If $\eta_D$ cannot be prolonged as entire function, the same result holds for ${\rm Res}\: \eta_D.$

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Proposition 1.2
  • Remark 1.3
  • Lemma 2.1: Lemma 3.1, chaubet2022
  • Theorem 3.1: Theorem 1.5 and (6.5), jin2023resonances
  • Theorem 3.2
  • proof : Proof of Theorem 1.1
  • Proposition 4.1: Prop. 2.3, ikawa1990poles
  • ...and 4 more