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We consider the classical dynamics given by a one sided shift on the Bernoulli space of d symbols. We study, on the space of Hölder functions, the eigendistributions of the Ruelle operator with a given potential. Our main theorem shows that for any isolated eigenvalue, the eigendistributions of such Ruelle operator are dual to eigenvectors of a Ruelle operator with a conjugate potential. We also show that the eigenfunctions and eigendistributions of the Koopman operator satisfy a similar relationship. To show such results we employ an integral kernel technique, where the kernel used is the involution kernel.

Abstract. Consider the Bowen-Series transform T associated with an even corners fundamental domain of finite volume for some Fuchsian group Γ. We prove a generic invariance result that abstracts Series' orbit-equivalence theorem to families of relations on the unit circle. Two applications of this result are developed. We first prove that T satisfies a strong-orbit equivalence property, which allows to identify its hyperbolic periodic orbits with primitive hyperbolic conjugacy classes of Γ. Then, we show thanks to the invariance theorem that the eigendistributions for the eigenvalue 1 of the transfer operator of T with spectral parameter s ∈ C are in bijection with smooth bounded eigenfunctions for the eigenvalue s(1 − s) of the hyperbolic Laplacian on the quotient D/Γ.