On Ruelle-Perron-Frobenius operators. I. Ruelle theorem

We study Ruelle-Perron-Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potenti als satisfying the Dini condition. In this paper, we give a proof of the Ru elle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be present ed in a forthcoming paper.