A FAMILY OF CUBIC RATIONAL MAPS AND MATINGS OF CUBIC POLYNOMIALS

MITSUHIRO SHISHIKURA AND TAN LEI

Abstract. We study a family of cubic branched coverings and matings of cubic polynomials of the form g ⊥ ⊥ f, with g = g_{a} : z ↦ 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} ⊥ ⊥ P_{1} illustrate features that do not occur for matings of quadratic polynomials: they never have Levy cycles but do sometimes have Thurston obstructions.