Length spectrum of periodic rays for billard flow

TL;DR

This paper investigates the length spectrum of periodic rays in billiard flows with convex obstacles, establishing sequences that satisfy conditions linked to scattering resonances and demonstrating exponential separation of periodic orbits.

Contribution

It introduces new sequences satisfying the (LB) condition under weaker separation assumptions and proves exponential separation of periodic orbits in phase space.

Findings

Existence of sequences satisfying the (LB) condition for the length spectrum.

Lower bounds for scattering resonances of the Dirichlet Laplacian.

Exponential separation of periodic orbits in phase space.

Abstract

We study for several compact strictly convex disjoint obstacles the length spectrum $\mathcal L$ formed by the lengths of all primitive periodic reflecting rays. We prove the existence of sequences $\{\ell_j\},\: \{m_j\}$ with $\ell_j \in \mathcal L,\: m_j \in \mathbb N$ such that the condition (LB) related to the dynamical zeta function $\eta_D(s)$ is satisfied. This condition implies the existence of lower bounds for the number of the scattering resonances for Dirichlet Laplacian. We construct such sequences under some separation condition for a small subset of $\mathcal L$ corresponding to lengths of the periodic rays with even reflexions. Our separation condition is weaker than the assumption of exponentially separated length spectrum $\mathcal L.$ Moreover, we show that the periodic orbits in the phase space are exponentially separated.