## ICMS Colloquium

### 19 March @ 16:00 - 17:30

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.