The Thurston Algorithm for quadratic matings

Abstract

Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichm\"uller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless the orbifold of f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of postcritical ray-equivalence classes for the modified mating, but it is enough to implement the pullback map for the formal mating. The proof combines the Selinger extension to augmented Teichm\"uller space with local estimates. Moreover, the Thurston Algorithm is implemented by pulling back a path in moduli space. This approach is due to Bartholdi--Nekrashevych in relation to one-dimensional moduli space maps, and to Buff--Ch\'eritat for slow mating. Here it is shown that slow mating is equivalent to the Thurston Algorithm for the formal mating. An initialization of the path is obtained for repelling-preperiodic captures as well, which provide an alternative construction of matings.