The Bowen$\unicode{x2013}$Series coding and zeros of zeta functions

TL;DR

This paper explores the use of Bowen–Series coding to analyze zeta functions related to geodesic flows on surfaces, providing new insights into the zeros of these functions, especially for infinite area surfaces without cusps.

Contribution

It introduces a dynamical approach using Bowen–Series coding for studying zeta functions, extending analysis to infinite area surfaces without cusps, and offers numerical insights into zeros of these functions.

Findings

Zeros exhibit striking structures in certain limits

The Bowen–Series approach extends to infinite area surfaces without cusps

Numerical results on zeros of zeta functions for specific geometries

Abstract

We give a discussion of the classical Bowen$\unicode{x2013}$Series coding and, in particular, its application to the study of zeta functions associated to geodesic flows and their zeros. In the case of compact surfaces of constant negative curvature $-1$ the analytic extension of the Selberg zeta function to the entire complex plane is classical, and can be achieved using the Selberg trace formula. However, an alternative dynamical approach is to use the Bowen$\unicode{x2013}$Series coding on the boundary at infinity to obtain a piecewise analytic expanding map from which the extension of the zeta function can be obtained using properties of the associated transfer operator. This latter method has the advantage that it also applies in the case of infinite area surfaces provided they do not have cusps. For such examples the location of the zeros is somewhat more mysterious. However, in particularly simple examples there is a striking structure to the zeros when we take appropriate limits. We will try to give some insight into this phenomenon. The newer version, in addition to the study of pair of pants, also includes heuristic analysis of the zeta function associated to the geodesic flow on symmetric one-funneled tori; in particular, there is a number of pure numerical results on location of its small zeros.