Dynamics of certain non-conformal degree two maps on the plane

Abstract

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $α$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{iθ})\;=\;r^{2α}\,e^{2iθ}\,+\,c$$ When $α=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $α$ is different from one, the dynamics is no longer conformal. In particular, the dynamics is not completely determined by the orbit of the critical point. Nevertheless, for many values of the parameter c, the dynamics has strong similarities to that of the quadratic family. For other parameter values the dynamics is dominated by 2 dimensional behavior: saddles and the like. The objects of study are Julia sets, filled-in Julia sets and the connectedness locus. These are defined in analogy to the conformal case. The main drive in this study is to see to what extent the results in the conformal case generalize to that of maps which are topologically like quadratic maps (and when $α$ is close to one, close to being quadratic).