A fixed point theorem for branched covering maps of the plane
Tom 206 / 2009
Fundamenta Mathematicae 206 (2009), 77-111
MSC: Primary 54H25; Secondary 37C25, 37B45.
DOI: 10.4064/fm206-0-6
Streszczenie
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 $d, |d|\le 2$, 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 .
