arXiv:1710.03841 [math.DS]AbstractReferencesReviewsResources
The Double Transpose of the Ruelle Operator
L. Cioletti, A. C. D. van Enter, R. Ruviaro
Published 2017-10-10Version 1
In this paper we study the double transpose extension of the Ruelle transfer operator $\mathscr{L}_{f}$ associated to a general real continuous potential $f\in C(\Omega)$, where $\Omega=E^{\mathbb{N}}$ and $E$ is any compact metric space. For this extension, we prove the existence of non-negative eigenfunctions, in the Banach lattice sense, associated to $\lambda_{f}$, the spectral radius of the Ruelle operator acting on $C(\Omega)$. As an application, we show that the natural extension of the Ruelle operator to $L^1(\Omega,\mathscr{B}(\Omega),\nu)$ (for a suitable Borel probability measure $\nu$) always has an eigenfunction associated to $\lambda_{f}$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ is either in H\"older or Walters spaces. We also constructed solutions to the classical (finite-state spaces) and generalized (general compact metric spaces) variational problem avoiding the standard normalization technique.