## Fixed points for branched covering maps of the plane

### Volume 254 / 2021

#### Abstract

A well-known result of Brouwer states that any orientation preserving homeomorphism of the plane with no fixed points has an empty non-wandering set. In particular, the existence of 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:

$\bullet$ A totally invariant compact subset that does not separate the critical point from its image.

$\bullet$ An invariant compact subset with a connected neighbourhood $B$ such that $\mathrm {Fill}(B \cup f(B))$ does not contain the critical point nor its image.

$\bullet$ An invariant continuum such that the critical point and its image belong to the same connected component of its complement.