A family of cubic rational maps and matings of cubic polynomials

Abstract

We study a family of cubic branched coverings and matings of cubic polynomials of the form $g\mate f$, with $g=g_a:z\mapsto z^3+a$ and $f=P_i$ for $i=1$, 2, 3 or $4$. We give criteria for matability or not of critically finite $g_a$ with each $P_i$. The maps $g_a\mate P_1$ illustrate features that do not occur for matings of quadratic polynomials: they never have Levy cycles but do sometimes have Thurston obstructions.