The Double Transpose of the Ruelle Operator

TL;DR

This paper investigates the double transpose of the Ruelle transfer operator on a broad class of spaces, establishing the existence of eigenfunctions linked to the spectral radius and applying these results to variational problems.

Contribution

It introduces the analysis of the double transpose of the Ruelle operator on $L^1$ spaces, extending the understanding of eigenfunctions beyond classical classes like Hölder or Walters.

Findings

Existence of non-negative eigenfunctions for the double transpose operator.

Conditions under which the extended Ruelle operator has eigenfunctions matching classical maximal eigenfunctions.

Construction of solutions to classical and generalized variational problems using these eigenfunctions.

Abstract

In this paper we study the double transpose of the $L^1(X,\mathscr{B}(X),\nu)$-extensions of the Ruelle transfer operator $\mathscr{L}_{f}$ 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 $\mathbb{L}^{**}_{f}$, we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $\rho(\mathscr{L}_{f})$, 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 $L^1(X,\mathscr{B}(X),\nu)$ has an eigenfunction associated to $\rho(\mathscr{L}_{f})$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ belongs to the H\"older, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.