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Gibbs Measures for Fibred Systems

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We consider a topological dynamical system T: YY on a metric space Y which forms a fibre bundle over another dynamical system. If T is fibrewise expanding and exact along fibres and if is a Hölder continuous function we prove the existence of a system of conditional measures (called a family of Gibbs measures) where the Jacobian is determined by . This theorem reduces to Ruelle's Perron–Frobenius theorem when the base of the fibred system consists of a single point. The method of proof does not use any form of symbolic representation. We also study continuity properties of a family of Gibbs measures (over the base) and give applications to the equilibrium theory of higher dimensional complex dynamics. Copyright 1999 Academic Press.

Document Type: Research Article

Affiliations: 1: Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, Göttingen, 37083, Germany 2: St. Petersburg Division, V. A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 197011, Russia

Publication date: December 1, 1999

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