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Conformal Geometry and Dynamics

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Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

Authors: Araceli Bonifant, Jan Kiwi and John Milnor
Journal: Conform. Geom. Dyn. 14 (2010), 68-112
MSC (2010): Primary 37F10, 30C10, 30D05
Published electronically: March 9, 2010
Erratum: Conform. Geom. Dyn. 14 (2010), 190-193.
MathSciNet review: 2600536
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Abstract: The parameter space $\mathcal {S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $p$ is a smooth affine algebraic curve whose genus increases rapidly with $p$. Each $\mathcal {S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $\mathcal {S}_p$, and of its smooth compactification, in terms of these escape regions. In particular, it computes the Euler characteristic. It concludes with a discussion of the real sub-locus of $\mathcal {S}_p$.

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Additional Information

Araceli Bonifant
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881
MR Author ID: 600241

Jan Kiwi
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica, Casilla 306, Correo 22, Santiago de Chile, Chile

John Milnor
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
MR Author ID: 125060

Received by editor(s): September 3, 2009
Published electronically: March 9, 2010
Additional Notes: The first author was partially supported by the Simons Foundation
The second author was supported by Research Network on Low Dimensional Dynamics PBCT/CONICYT, Chile
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.