  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.
