A dynamical approach to Maass cusp forms

TL;DR

This paper establishes a constructive isomorphism between Maass cusp forms for certain Fuchsian groups and eigenfunctions of a transfer operator, providing a new dynamical perspective on these forms.

Contribution

It introduces a dynamical approach linking Maass cusp forms to transfer operators for specific Fuchsian groups, enhancing understanding of their structure.

Findings

Maass cusp forms are isomorphic to 1-eigenfunctions of a transfer operator

The isomorphism is explicitly constructed

Applicable to Fuchsian groups satisfying a geometric condition

Abstract

For nonuniform cofinite Fuchsian groups $\Gamma$ which satisfy a certain additional geometric condition, we show that the Maass cusp forms for $\Gamma$ are isomorphic to 1-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.