Eigenfunctions of the Laplacian and associated Ruelle operator

TL;DR

This paper establishes a connection between eigenfunctions of the hyperbolic Laplacian on a co-compact Fuchsian surface and eigenfunctions of the associated Ruelle transfer operator, providing an explicit integral representation.

Contribution

It demonstrates the existence of a piecewise real analytic eigenfunction of the Ruelle operator linked to Laplacian eigenfunctions, extending previous work by Pollicott with an explicit integral formula.

Findings

Existence of a real analytic eigenfunction of the Ruelle operator for eigenvalue 1.

Explicit integral formula relating eigenfunctions of the Laplacian and Ruelle operator.

Connection between geometric structures and transfer operator eigenfunctions.

Abstract

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk $\DD$ and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $\dd_{f,s}$ (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity $\partial\DD=\SS^1$. The geodesic flow on the compact surface $\DD/\Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:\SS^1\to\SS^1$, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that $\dd_{f,s}$ is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \[ \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \] \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $\DD/\Gamma$.