Cluster points and asymptotic values of planar harmonic functions
  A sufficient condition for a cluster point of a planar harmonic function to be an asymptotic value is given, based on a partitioning into regions of constant valence. A sufficient condition for the cluster set of a planar harmonic function to have non-empty interior is given. An example is given of a planar harmonic function where the image of the critical set is not closed and such that the cluster set has non-empty interior and is a proper subset of the image. 
Genevra Neumann