Abstract
We prove that ifFFis a degree33Thurston map with two fixed critical points, then any irreducible obstruction forFFcontains a Levy cycle. As a corollary, it will be shown that ifffandggare two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the matingf⊥⊥gf \perp \! \! \! \perp gcontains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.
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