Duality between Eigenfunctions and Eigendistributions of Ruelle and   Koopman operators via an integral kernel
  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. 
Paolo Giulietti, Artur O. Lopes and Vincent Pit
math.DS cond-mat.stat-mech math.PR