A fixed point theorem for branched covering maps of the plane

Details A fixed point theorem for branched covering maps of the plane A fixed point theorem for branched covering maps of the plane

2009-04-20

arxiv.org/abs/0904.2944v1

Fundamenta Mathematicae , 206 (2009), 77-111

Mathematics

General Topology

37 pages; one figure

Scientific paper

It is known that every homeomorphism of the plane has a fixed point in a non-separating, invariant subcontinuum. Easy examples show that a branched covering map of the plane can be periodic point free. In this paper we show that any branched covering map of the plane of degree with absolute value at most two, which has an invariant, non-separating continuum $Y$, either has a fixed point in $Y$, or $Y$ contains a \emph{minimal (by 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 \emph{outchannel} and the other two to \emph{inchannels}.

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