
TL;DR
This paper provides an explicit description of the shift locus of degree d polynomials, revealing its complex structure, homotopy type, and new classes of complex surfaces with interesting topological properties.
Contribution
It introduces a novel combinatorial and algebraic framework for understanding the shift locus, including explicit descriptions and topological insights.
Findings
${ mf S}_d$ has the homotopy type of a CW complex of real dimension $d-1$
${ mf S}_3$ and ${ mf S}_4$ are $K(\pi,1)$s
Discovery of new complex surfaces homotopic to locally CAT(0) complexes
Abstract
For each the shift locus of degree , denoted , is the space of normalized degree polynomials in one complex variable for which every critical point is in the attracting basin of infinity under iteration. It is a complex analytic manifold of complex dimension . We are able to give an explicit description of as a complex of spaces over a contractible building, and to describe the pieces in two quite different ways: 1. (combinatorial): in terms of dynamical extended laminations; or 2. (algebraic): in terms of certain explicit `discriminant-like' affine algebraic varieties. From this structure one may deduce numerous facts, including that has the homotopy type of a CW complex of real dimension ; and that and are s. The method of proof is rather…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Mathematical Dynamics and Fractals · Mathematics and Applications
