Skew products in C²
by
Manfred Denker
University of Göttingen, Germany

A fibred system is a dynamical system (Y, T) together with a factor \pi:(Y, T) --> (X, S). We assume that all maps are continuous. We first discuss the existence of Gibbs measures in the form of conditional measures given \pi and their importance for equilibrium theory. A particular example is given for skew products of polynomial mappings in C², T(x, y)=(p(x), q(x, y)) (x, y in C). We describe a large class of such maps which have a completely invariant set \Cal H subset C² of maximal topological entropy, and which are a fibred system when restricted to this set. It follows that for certain Hölder continuous functions on \Cal H there exist unique equilibrium measures. We also characterize \Cal H in different ways: by normal convergence, fixed points and boundary behavior.

Date received: March 2, 1998