Ergodic Theory and Dynamical Systems

Research Article

Julia sets for holomorphic endomorphisms of n

Stefan-M. Heinemanna1

a1 Institut für Mathematische Stochastik, Lotzestraβe 13, D-37083 Göttingen, Germany

Abstract

We give a definition for a Julia set J(f) for generic classes of polynomial endomorphisms f: nn. For n = 1, our definition is equivalent to the usual one, which gives the points where the iterates of f do not form a normal family. Moreover, the Julia set J(f1 × … × fn) n for a product of one-dimensional polynomials fi: turns out to be the product J(f1) × … × J(fn) of the associated Julia sets J(fi) . For a special class of mappings f: 22 which is not of this simple type, the so-called Cantor skews, we investigate topological structure as well as measure theoretic aspects of the Julia sets obtained using our definition.

(Received October 11 1994)

(Revised January 30 1995)

Footnotes

† Research supported by SFB 170 ‘Geometrie und Analysis’ and SPP ‘Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme’ Georg-August-Universität Göttingen.