Length spectrum of periodic rays for billard flow
Vesselin Petkov
TL;DR
The paper analyzes the length spectrum of primitive periodic billiard rays outside several strictly convex obstacles under a non-eclipse condition, linking this spectrum to lower bounds on Dirichlet-Laplacian resonances through the Dirichlet dynamical zeta function $η_D(s)$ and the LB condition. It derives sharp counts for iterated rays, analyzes the distribution of lengths in small intervals, and proves that a very small separation of even-reflection rays suffices to guarantee $(LB)$, yielding MLPC-type resonance lower bounds. A central technical contribution is the exponential separation of phase-space periodic orbits, established via detailed analysis of configurations and the billiard ball map, enabling control of contributions to $η_D(s)$. The work also outlines open questions for generic obstacles and boundary perturbations, aiming to extend the separation framework beyond current smoothness assumptions.
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.