Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces

Abstract

Recently the Ruelle-Perron-Fröbenius theorem was proved for Hölder potentials defined on the symbolic space $Ω=M^{\mathbb{N}}$, where (the alphabet) $M$ is any compact metric space. In this paper, we extend this theorem to the Walters space $W(Ω)$, in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet-analyticity of the Ruelle operator under quite general conditions and we apply this result to prove the analytic dependence of this operator on both Walters and Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established. An exponential decay of the correlations is shown when the Ruelle operator has the spectral gap property. A new (and natural) family of Walters potentials (on a finite alphabet derived from the Ising model) not having an exponential decay of the correlations is presented. Because of the lack of exponential decay, for such potentials we have the absence of the spectral gap for the Ruelle operator. The key idea to prove the lack of exponential decay of the correlations are the Griffiths-Kelly-Sherman inequalities.