A fixed point theorem for branched covering maps of the plane

TL;DR

This paper establishes a fixed point theorem for certain branched covering maps of the plane, showing that such maps either have a fixed point in an invariant continuum or a specific minimal invariant subcontinuum with detailed boundary properties.

Contribution

It extends fixed point results to degree -2 branched coverings, identifying conditions under which invariant continua contain fixed points or specific minimal subcontinua.

Findings

Maps of degree ±1 or 2 with invariant continua have fixed points or minimal invariant subcontinua.

Degree -2 maps with minimal invariant continua have exactly three fixed prime ends.

Characterization of boundary structure in minimal invariant subcontinua.

Abstract

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}.