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We consider the classical dynamics given by a one sided shift on theBernoulli space of $d$ symbols. We study, on the space of H\"older functions,the eigendistributions of the Ruelle operator with a given potential. Our maintheorem shows that for any isolated eigenvalue, the eigendistributions of suchRuelle operator are dual to eigenvectors of a Ruelle operator with a conjugatepotential. We also show that the eigenfunctions and eigendistributions of theKoopman operator satisfy a similar relationship. To show such results we employan integral kernel technique, where the kernel used is the involution kernel.

Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel