Jordan tori and polynomial endomorphisms in ℂ 2

Manfred Denker; Stefan Heinemann

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 139-159
  • ISSN: 0016-2736

Abstract

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For a class of quadratic polynomial endomorphisms f : ℂ 2 → ℂ 2 close to the standard torus map ( x , y ) → ( 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).

How to cite

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Denker, Manfred, and Heinemann, Stefan. "Jordan tori and polynomial endomorphisms in $ℂ^2$." Fundamenta Mathematicae 157.2-3 (1998): 139-159. <http://eudml.org/doc/212282>.

@article{Denker1998,
abstract = {For a class of quadratic polynomial endomorphisms $f: ℂ^2 → ℂ^2$ close to the standard torus map $(x,y) → (x^2,y^2)$, 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).},
author = {Denker, Manfred, Heinemann, Stefan},
journal = {Fundamenta Mathematicae},
keywords = {quadratic polynomial mappings; expanding repeller; standard map; periodic points; Julia sets; harmonic measures},
language = {eng},
number = {2-3},
pages = {139-159},
title = {Jordan tori and polynomial endomorphisms in $ℂ^2$},
url = {http://eudml.org/doc/212282},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Denker, Manfred
AU - Heinemann, Stefan
TI - Jordan tori and polynomial endomorphisms in $ℂ^2$
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 139
EP - 159
AB - For a class of quadratic polynomial endomorphisms $f: ℂ^2 → ℂ^2$ close to the standard torus map $(x,y) → (x^2,y^2)$, 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).
LA - eng
KW - quadratic polynomial mappings; expanding repeller; standard map; periodic points; Julia sets; harmonic measures
UR - http://eudml.org/doc/212282
ER -

References

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