Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions

TL;DR

This paper introduces a numerical method to compute invariant Ruelle distributions on convex co-compact hyperbolic surfaces by leveraging weighted zeta functions and Fredholm determinants, with improvements for symmetric cases.

Contribution

The authors develop a novel numerical algorithm that connects Ruelle distributions to weighted zeta functions using Fredholm determinants, including symmetry reduction techniques.

Findings

Effective computation of Ruelle distributions demonstrated

Enhanced convergence speed with symmetry reduction

Algorithm applicable to surfaces with additional symmetries

Abstract

We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of meromorphically continued weighted zeta functions established by the authors together with Barkhofen (2021). To make this applicable for numerics we express the weighted zeta as the logarithmic derivative of a suitable parameter dependent Fredholm determinant similar to Borthwick (2014). As an additional difficulty our transfer operator has to include a contracting direction which we account for with techniques developed by Rugh (1992). We achieve a further improvement in convergence speed for our algorithm in the case of surfaces with additional symmetries by proving and applying a symmetry reduction of weighted zeta functions.