Polynomial basins of infinity

Details Polynomial basins of infinity Polynomial basins of infinity

2009-08-04

arxiv.org/abs/0908.0380v3

Mathematics

Dynamical Systems

v3: Reorganized, with more detailed proofs. To appear, Geom. Funct. Analysis

Scientific paper

We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with a dynamically natural Gromov-Hausdorff topology, the map $\pi$ becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of $\pi$ are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which model the local behavior of a polynomial.

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