Dynamical Zeta Function for Several Strictly Convex Obstacles

Abstract

Abstract. The behavior of the dynamical zeta function Z D (s) related to several strictly convex disjoint obstacles is similar to that of the inverseof the Riemann zeta function ζ(s). Let Π(s) be the series obtained from Z D (s) summing only over primitive periodic rays. In this paper we examine the analytic singularities of Z D (s) and Π(s) close to the line ℜs = s 2 , where s 2 is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions Z D (s), Π(s) has a singularity at s = s 2 .

Dynamical zeta functions for billiards

“…The analytic singularities of η N (s) and η D (s) are important for the analysis of the distribution of the resonances (see [Ika88,Ika90a,Ika90b,Ika92,Sto09,Pet08] and the papers cited there). By using the Ruelle transfer operator and symbolic dynamics (see [Ika90a,Pet99,Sto09,Mor91]), a meromorphic continuation of η N (s), η D (s) has been proved in a domain s 0 − α Re s with a suitable α > 0, where s 0 is the abscissa of absolute convergence of the Dirichlet series η N (s), η D (s).…”

“…The meromorphic continuation of the Ruelle zeta function for general Anosov flows was established by Giulietti-Liverani-Pollicott [GLP13] (see also the work of Dyatlov-Zworski [DZ16] for another microlocal proof). In this paper the series η N (s), η D (s) are simply called dynamical zeta functions following previous works [Pet99,Pet08] and we refer to the book of Baladi [Bal18] for more references concerning zeta functions for hyperbolic dynamical systems.…”