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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-6,
author = {Alexander Blokh and Lex Oversteegen},
title = {A fixed point theorem for branched covering maps of the plane},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {77-111},
zbl = {1197.54058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-6}
}
Alexander Blokh; Lex Oversteegen. A fixed point theorem for branched covering maps of the plane. Fundamenta Mathematicae, Tome 205 (2009) pp. 77-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-6/