Mathematics – Dynamical Systems
Scientific paper
1991-09-26
Experimental Math, vol 2 issue 4 (1993), 281-300
Mathematics
Dynamical Systems
Scientific paper
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 $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$ When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $\alpha$ 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 $\alpha$ is close to one, close to being quadratic).
Bielefeld Ben
Sutherland Scott
Tangerman Folkert
Veerman J. J. P.
No associations
LandOfFree
Dynamics of certain non-conformal degree two maps on the plane does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Dynamics of certain non-conformal degree two maps on the plane, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dynamics of certain non-conformal degree two maps on the plane will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-489052