Eigenfunctions of the Laplacian and Associated Ruelle Operator

Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk D and ∆ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of ∆, equivariant by Γ with real eigenvalue λ = −s(1 − s), where s = 1 2 + it, admits an integral representation by a distribution D f,s (the Helgason distribution) which is equivariant by Γ and supported at infinity ∂D = S 1. The geodesic flow on the compact surface D/Γ is conjugate to a suspension over a natural extension of a piecewise analytic map T : S 1 → S 1 , the so-called Bowen-Series transformation. Let L s be the complex Ruelle transfer operator associated to the jacobian −s ln |T ′ |. M. Pollicott showed that D f,s is an eigenfunction of the dual operator L * s for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction ψ f,s of L s for the eigenvalue 1, given by an integral formula ψ f,s (ξ) = J(ξ, η) |ξ − η| 2s D f,s (dη), where J(ξ, η) is a {0, 1}-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface D/Γ.