# Fixed points for branched covering maps of the plane

## TL;DR

This paper establishes new sufficient conditions for degree 2 branched covering maps of the plane to have fixed points, extending classical fixed point results for plane homeomorphisms.

## Contribution

It introduces specific criteria involving invariant sets and their topological properties that guarantee the existence of fixed points in branched covering maps.

## Key findings

- Provides conditions involving totally invariant sets and their separation properties.
- Shows that certain invariant continua containing critical points ensure fixed points.
- Extends Brouwer's fixed point theory to branched covering maps with critical points.

## Abstract

A well-known result from Brouwer states that any orientation preserving homeomorphism of the plane with no fixed points has an empty non-wandering set. In particular, an invariant compact set implies the existence of a fixed point. In this paper we give sufficient conditions for degree 2 branched covering maps of the plane to have a fixed point, namely: A totally invariant compact subset such that it does not separate the critical point from its image An invariant compact subset with a connected neighbourhood $U$, such that $\mathrm{Fill}(U \cup f(U))$ does not contain the critical point nor its image. An invariant continuum such that the critical point and its image belong to the same connected component of its complement.