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

Recently the Ruelle-Perron-Fr\"obenius theorem was proved for H\"older potentials defined on the symbolic space $\Omega=M^{\mathbb{N}}$, where (the alphabet) $M$ is any compact metric space. In this paper, we extend this theorem to the Walters space $W(\Omega)$, in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fr\'echet-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 H\"older spaces. The analyticity of the pressure functional on H\"older 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.