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

Recently the Ruelle–Perron–Frobenius theorem was proved for Holder potentials defined on the symbolic space , where (the alphabet) M is any compact metric space. In this paper, we extend this theorem to the Walters space , in similar general alphabets. We also describe in detail an abstract procedure to obtain the Frechet 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 Holder spaces. The analyticity of the pressure functional on Holder 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) which do not have an exponential decay of the correlations is presented. Because of the lack of exponential decay, for such potentials there is an absence of the spectral gap for the Ruelle operator. The key idea in proving the lack of exponential decay of the correlations is the Griffiths–Kelly–Sherman inequalities.