Gibbs Measures for Fibred Systems

We consider a topological dynamical system T : Y ! Y on a metric space Y which forms a bre bundle over another dynamical system. If T is brewise expanding and exact along bres and if ' is a HH older 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 ((5]) when the base of the bred 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.