Matings of cubic polynomials with a fixed critical point, Part I: Thurston obstructions

Abstract

We prove that if F F is a degree 3 3 Thurston map with two fixed critical points, then any irreducible obstruction for F F contains a Levy cycle. As a corollary, it will be shown that if f f and g g are two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the mating f ⊥ ⊥ g f \perp \! \! \! \perp g contains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.