Jordan tori and polynomial endomorphisms in $ℂ^2$

Manfred Denker; Stefan Heinemann

ArticleOriginal scientific text

Title

Jordan tori and polynomial endomorphisms in $ℂ^{2}$

Authors ¹, ¹

Affiliations

Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Lotzestraße 13, D-37083 Göttingen, Germany

Abstract

For a class of quadratic polynomial endomorphisms

f : ℂ^{2} \to ℂ^{2}

close to the standard torus map

(x, y) \to (x^{2}, y^{2})

, we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).

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Title

Jordan tori and polynomial endomorphisms in ℂ2

Affiliations

Abstract

Bibliography

Jordan tori and polynomial endomorphisms in $ℂ^{2}$