# Jordan tori and polynomial endomorphisms in ${\u2102}^{2}$

Manfred Denker; Stefan Heinemann

Fundamenta Mathematicae (1998)

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

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topDenker, 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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