Eigenfunctions of the Laplacian and associated Ruelle operator

Abstract

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 formulawhere 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. * Instituto de Matemática, UFRGS, Porto Alegre 91501-970, Brasil. Partially supported by CNPq, Fapergs, Instituto do Milenio, beneficiário de auxílio financeiro CAPES -Brasil.

Duality between eigenfunctions and eigendistributions of Ruelle and Koopman operators via an integral kernel

“…Note that in [18], [20] and [14], where similar results are obtained for a specific family of Markov maps of the circle, it was possible to prove directly that the analogue of the Φ W map is injective thanks to the smooth structure on the circle which gives additional regularity to the eigendistributions. This is not the case in our symbolic setting, so we must resort to a dimension argument.…”

“…Distributions related to eigenfunctions and the involution kernel appeared in [18], [19], [20] and [14] (plus, in a non rigorous form, in [8] and [9]). Moreover, distributions have been intensively studied, starting with [3], through anisotropic 1 We will use the two notations y → M e A ⋆ (ya) ϕ(ya)da and M e A ⋆ (·a) ϕ(·a)da indistinctly to indicate the needed function.…”

Invariant relations for the Bowen-Series transform

“…The reverse implication is the easiest, and is actually the generalization to the finite covolume case of proposition 7 from [LT08]. Let f ∈ E b s(1−s) (Γ) and ν = D f,s .…”

“…Proposition 4 from [LT08] tells us how we can express the γ-invariance of f in terms of its Helgason boundary value.…”

Ergodic optimization, zero temperature limits and the max-plus algebra

“…In this section we will present the involution kernel which is a concept that is sometimes useful for understanding problems of different areas: large deviations ( [12]), issues of differentiability of the main eigenfunction and piecewise differentiability of the subaction [111] [110], optimal transport (see [49] [106], [99], [100]), etc... It is also related to the Gromov distance on hyperbolic spaces (see [101]).…”