Topology of Fatou components for endomorphisms of $ℂ{\mathbb{P}}^k$: linking with the Green’s current

Suzanne Lynch Hruska, and Roland K. W. Roeder. "Topology of Fatou components for endomorphisms of $ℂℙ^k$: linking with the Green’s current." Fundamenta Mathematicae 210.1 (2010): 73-98. <http://eudml.org/doc/282886>.

@article{SuzanneLynchHruska2010,
abstract = {Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f: ℂℙ^k → ℂℙ^k$, when k >1. Classical theory describes U(f) as the complement in $ℂℙ^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂℙ^k$, we give a definition of linking number between closed loops in $ℂℙ^k ∖ supp S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁(ℂℙ^k ∖ supp S)$. As an application, we use these linking numbers to establish that many classes of endomorphisms of ℂℙ² have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of ℂℙ² for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of ℂℙ² has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.},
author = {Suzanne Lynch Hruska, Roland K. W. Roeder},
journal = {Fundamenta Mathematicae},
keywords = {Fatou components; linking numbers; closed currents},
language = {eng},
number = {1},
pages = {73-98},
title = {Topology of Fatou components for endomorphisms of $ℂℙ^k$: linking with the Green’s current},
url = {http://eudml.org/doc/282886},
volume = {210},
year = {2010},
}

TY - JOUR
AU - Suzanne Lynch Hruska
AU - Roland K. W. Roeder
TI - Topology of Fatou components for endomorphisms of $ℂℙ^k$: linking with the Green’s current
JO - Fundamenta Mathematicae
PY - 2010
VL - 210
IS - 1
SP - 73
EP - 98
AB - Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f: ℂℙ^k → ℂℙ^k$, when k >1. Classical theory describes U(f) as the complement in $ℂℙ^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂℙ^k$, we give a definition of linking number between closed loops in $ℂℙ^k ∖ supp S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁(ℂℙ^k ∖ supp S)$. As an application, we use these linking numbers to establish that many classes of endomorphisms of ℂℙ² have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of ℂℙ² for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of ℂℙ² has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.
LA - eng
KW - Fatou components; linking numbers; closed currents
UR - http://eudml.org/doc/282886
ER -