Description
The Ruelle zeta function is a way of organizing the closed geodesics of a compact locally symmetric space into a single dynamical object. In this talk, I will explain a version of this construction incorporating Hecke operators. Hecke operators are natural notions of symmetries of locally symmetric spaces, and define correspondences between every point and finitely many other points. In the classical setting, a closed geodesic returns to itself because of a deck transformation, or equivalently an element of the fundamental group. In the Hecke setting, we allow the return to happen through a Hecke correspondence. This leads to a new length spectrum: instead of only seeing whole multiples of the length of a primitive closed geodesic, we can see new return lengths, sometimes even fractional ones.
The main part of the talk will focus on how this Hecke Ruelle zeta function is constructed and why its definition naturally involves certain multiplicities. These multiplicities record how many Hecke returns give the same geometric data, and how the corresponding periodic geodesic loci cover each other. I will then discuss two applications. The first is a Hecke prime geodesic theorem, which gives an asymptotic count of closed geodesics returning under a Hecke correspondence. The second is a Hecke version of Fried's conjecture, relating the special value of the Hecke Ruelle zeta function to a Hecke analogue of analytic torsion in the acyclic rank one case. Part of the talk is based on an ongoing joint work with Peter Hochs and Polyxeni Spilioti.