Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems

TL;DR

This paper proves the meromorphic continuation of weighted zeta functions for open hyperbolic systems, linking residues to invariant distributions, and demonstrates numerical computation methods for specific scattering systems.

Contribution

It establishes the meromorphic continuation of weighted zeta functions using advanced resolvent techniques and connects residues to invariant Ruelle and Patterson-Sullivan distributions.

Findings

Residue formula linking zeta residues to Ruelle distributions

Meromorphic continuation of weighted zeta functions in hyperbolic systems

Numerical methods for invariant Ruelle distributions in 3-disc systems

Abstract

In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.