2008/07/31 by Artur O. Lopes, Philippe Thieullen, Ph. Thieullen
Mathematics · #Advanced Differential Equations and Dynamical Systems #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.AP #math.DS #msc:11F12 #msc:11F72 #msc:37C30 #msc:46F12
paper · pdf · doi:10.1088/0951-7715/21/10/003
openalex publication_date 2008/09/12 · arxiv created 2008/09/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk and Δ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of Δ, equivariant by Γ with real eigenvalue λ = − s (1 − s ), where , admits an integral representation by a distribution (the Helgason distribution) which is equivariant by Γ and supported at infinity . The geodesic flow on the compact surface is conjugate to a suspension over a natural extension of a piecewise analytic map , the so-called Bowen–Series transformation. Let be the complex Ruelle transfer operator associated with the Jacobian − s ln | T '|. Pollicott showed that is an eigenfunction of the dual operator for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction ψ f , s of for the eigenvalue 1, given by an integral formula 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 .