Communications in Mathematical Physics

, Volume 223, Issue 1, pp 125-141

On Ruelle–Perron–Frobenius Operators.¶I. Ruelle Theorem

  • Aihua FanAffiliated withDépartement de Mathématiques et Informatique, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France. E-mail:
  • , Yunping JiangAffiliated withDepartment of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA and Department of Mathematics, CUNY Graduate School, 635 Fifth Avenue, New York, NY 10016, USA.¶E-mail:

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We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle 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 presented in a forthcoming paper.