On the Ruelle-Mayer Transfer Operators for H\"older Continuous Functions

TL;DR

This paper investigates Ruelle-Mayer transfer operators related to geodesic flow, showing that key spectral properties are preserved when extending their domain to H"older continuous functions, with implications for Maass forms and the Riemann zeta function.

Contribution

It demonstrates that spectral information of transfer operators remains intact in the H"older continuous function space, extending previous spectral results to a broader functional setting.

Findings

Spectral properties are preserved for H"older continuous functions with specific exponents.

Point spectra related to Maass cusp forms remain unchanged in the H"older setting.

Partial classification of solutions to a H"older-based functional equation is provided.

Abstract

We consider a family of operators connected with the geodesic flow on the modular surface. We show certain spectral information is retained after expanding their domain to the space of $\alpha$-H\"older continuous functions on the unit interval. For example, the point spectra associated with the Maass cusp forms and non-trivial zeroes of the Riemann zeta function to the right of the critical line remain unchanged when the H\"older constant is $(1/2+\varepsilon)$ and $3/4$ respectively. We briefly consider a three-term functional equation introduced by Lewis in the H\"older setting and provide a partial classification of solutions in this setting.