  We consider the classical dynamics given by a one sided shift on the
Bernoulli space of $d$ symbols. We study, on the space of H\"older 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.
