Gibbs Measures for Fibred Systems

✍ Scribed by Manfred Denker; Mikhail Gordin

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 319 KB
• Volume: 148
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1843

✦ Synopsis

We consider a topological dynamical system T : Y Ä Y 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 Ho lder 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. 1999 Academic Press y: f ( y)=x g( y) exp[.( y)],