Surgery on postcritically finite rational maps by blowing up an arc

Abstract

Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a rational map and suppose that the postcritical set $P(f)$ is finite. Let $α$ be an embedded closed arc in the sphere and suppose that $f|α$ is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere open along $α$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$ and $\Int (D)$ by a homeomorphism to the complement of $f(α)$. We prove theorems which give combinatorial conditions on $f$ and $α$ for $g$ to be equivalent in the sense of Thurston to a rational map. The main idea in our proofs is a general theorem which forces a possible obstruction for $g$ away from the disc $D$ on which the new dynamics is defined.