Gibbs Measures for Fibred Systems

AbstractWe consider a topological dynamical system T:Y→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 ϕ 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 ϕ. 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