Ruelle Operator Duality for Coupled Smooth Markov Maps of the Circle

Let TL and TR be two smooth surjective Markov maps of the circle, with TR expansive, coupled in such a way that there exists an extension (C, TC ) whose first factor is TL−1 and the second factor of TC is TR . Let AL piecewise continuous and AR piecewise absolutelycontinuous be two respective potentials. We show that, when those potentials are in involutionby a smooth kernel W on C, there is an explicit isomorphism between eigenfunctions of the Ruelle operator of (TL , AL ) and eigendistributions of the Ruelle operator of (TR , AR ) for the same eigenvalue. This gives a regularity result for eigendistributions of transfer operators associated with non-maximal eigenvalues.