The Double Transpose of the Ruelle Operator

Abstract

In this paper we study the double transpose of the $L^1(X,\mathcal{B}(X),\nu )$-extensions of the Ruelle transfer operator ${\mathcal{L}}_f$ associated to a general real continuous potential $f\in C(X)$, where $X=E^{\mathbb{N}}$, the alphabet $E$ is any compact metric space and $\nu$ is a maximal eigenmeasure. For this operator, denoted by ${\mathbb{L}}_f^{\ast \ast }$, we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $\rho ({\mathcal{L}}_f)$, the spectral radius of the Ruelle operator acting on $C(X)$. As an application, we obtain a sufficient condition ensuring that the natural extension of the Ruelle operator to $L^1(X,\mathcal{B}(X),\nu )$ has an eigenfunction associated to $\rho ({\mathcal{L}}_f)$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

arXiv commentWe corrected a mistake; added some clarifications regarding the action of the Double transpose operator, and directions on future research. 19 pages