Dynamics of certain nonconformal degree-two maps of the plane

Abstract

We consider the rational maps given by $z \mapsto |z|^{2\alpha-2}z^2+c$, for $z$ and $c$ complex and $\alpha > {1\over 2}$ fixed and real. The case $\alpha=1$ corresponds to quadratic polynomials: some of the well-known results for this conformal case still hold for $\alpha$ near $1$, while others break down. Among the differences between the two cases are the possibility, for $\alpha\ne1$, of periodic attractors that do not attract the critical point, and the fact that for $\alpha >1$ the Julia set is smooth for an open set of values of $c$. Numerical evidence suggests that the analogue of the Mandelbrot set for this family is connected, but not locally connected if $\alpha \ne 1$.