A fixed point theorem for branched covering maps of the plane

Alexander Blokh, and Lex Oversteegen. "A fixed point theorem for branched covering maps of the plane." Fundamenta Mathematicae 206.1 (2009): 77-111. <http://eudml.org/doc/283377>.

@article{AlexanderBlokh2009,
abstract = {It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has exactly three fixed prime ends, one corresponding to an outchannel and the other two to inchannels.},
author = {Alexander Blokh, Lex Oversteegen},
journal = {Fundamenta Mathematicae},
keywords = {fixed points; tree-like continuum; branched covering map},
language = {eng},
number = {1},
pages = {77-111},
title = {A fixed point theorem for branched covering maps of the plane},
url = {http://eudml.org/doc/283377},
volume = {206},
year = {2009},
}

TY - JOUR
AU - Alexander Blokh
AU - Lex Oversteegen
TI - A fixed point theorem for branched covering maps of the plane
JO - Fundamenta Mathematicae
PY - 2009
VL - 206
IS - 1
SP - 77
EP - 111
AB - It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has exactly three fixed prime ends, one corresponding to an outchannel and the other two to inchannels.
LA - eng
KW - fixed points; tree-like continuum; branched covering map
UR - http://eudml.org/doc/283377
ER -