On the Ruelle-Mayer Transfer Operators for Hölder Continuous Functions
Alexander Baumgartner
TL;DR
The paper analyzes Mayer transfer operators for the Gauss map acting on Hölder spaces $C^{\alpha}([0,1])$, establishing a uniform bound on the essential spectrum via $\rho_e(L_\beta) \le \rho(\mathcal{L}_{\Re(\beta)+\alpha})$ and showing that eigenvalues above this threshold persist as generalized eigenfunctions, closely linked to $\mathcal{L}_{\Re(\beta)+\alpha}$. This yields preservation of spectral data associated with Maass cusp forms and nontrivial zeros of the Riemann zeta function to the right of the critical line for suitable $\alpha$ (namely $\alpha>1/2$ for cusp forms and $\alpha=3/4$ for zeta zeros). The work connects these spectral properties to Lewis' three-term functional equation, proving a Hölder-based bootstrapping result that extends real-analytic solutions to holomorphic ones on $\mathbb{C}\setminus(-\infty,-1]$. It also outlines potential generalizations to broader transfer-operator families and higher-dimensional analogues, highlighting the relevance to geodesic flows on modular surfaces and to resolvent/zeta-function analyses.
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 $α$-Hölder 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 Hölder constant is $(1/2+\varepsilon)$ and $3/4$ respectively. We briefly consider a three-term functional equation introduced by Lewis in the Hölder setting and provide a partial classification of solutions in this setting.