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Julia sets for holomorphic endomorphisms of ℂn

  • Stefan-M. Heinemann (a1)

Abstract

We give a definition for a Julia set J(f) for generic classes of polynomial endomorphisms f: ℂn→ ℂn. 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) ⊂ ℂn for a product of one-dimensional polynomials fi: ℂ → ℂ turns out to be the product J(f1) × … × J(fn) of the associated Julia sets J(fi) ⊂ ℂ. For a special class of mappings f: ℂ2 → ℂ2 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.

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