Time: 19.03.2024, Wednesday, 4:00 pm

Place: Zoom only

Lecturer: Vesselin Petkov (Université Bordeaux)

Title: Dynamical zeta function for billiard flow

Abstract: We will present briefly the connection between Riemann zeta function, Ruellle zeta function and dynamical zeta function. The last one is related to the billiard flow for the union $D \subset R^d$ of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in C$, $\Im \mu_j > 0$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$. For $d$ odd, $u(t) = \sum_j e^{i |t| \mu_j}$ is a distribution in $\mathcal{D}'(R \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $\eta_{\mathrm N}(s),\: \eta_{\mathrm D}(s)$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. In this talk we discuss two results. (1) Under a non-eclipse condition, we show that $\eta_{\mathrm N}$ and $\eta_\mathrm D$ admit a <em>meromorphic continuation</em> in the whole complex plane with simple poles and integer residues. (2) In the case when the boundary $\partial D$ is real analytic, we prove that the function $\eta_\mathrm{D}$ cannot be entire and the modified Lax Phillips conjecture (MLPC) holds. This conjecture introduced in 1990 says that there exists of a strip $\{z \in C: \: 0 < \Im z \leqslant\alpha\}$ containing an infinite number of resonances $\mu_j$ for the Dirichlet problem. The above results are obtained in a joint work with Yann Chaubet.