Length spectrum of periodic rays for billard flow

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 $η_D(s)$ 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.