Polynomial Basins of Infinity |
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Authors: | Laura DeMarco Kevin M Pilgrim |
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Institution: | 1. Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St. (MC 249), Chicago, IL, 60607-7045, USA 2. Department of Mathematics, Rawles Hall, Indiana University, Bloomington, IN, 47405, USA
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Abstract: | We study the projection p: Md ? Bd{\pi : \mathcal{M}_d \rightarrow \mathcal{B}_d} which sends an affine conjugacy class of polynomial
f : \mathbbC ? \mathbbC{f : \mathbb{C} \rightarrow \mathbb{C}} to the holomorphic conjugacy class of the restriction of f to its basin of infinity. When Bd{\mathcal{B}_d} is equipped with a dynamically natural Gromov–Hausdorff topology, the map π becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of π 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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