Surgery on postcritically finite rational maps by blowing up an arc

Details Surgery on postcritically finite rational maps by blowing up an arc Surgery on postcritically finite rational maps by blowing up an arc

1995-12-18

arxiv.org/abs/math/9512221v1

Mathematics

Dynamical Systems

Scientific paper

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 $\alpha$ be an embedded closed arc in the sphere and suppose that $f|{\alpha}$ is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere open along $\alpha$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$ and $\Int (D)$ by a homeomorphism to the complement of $f(\alpha)$. We prove theorems which give combinatorial conditions on $f$ and $\alpha$ 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.

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