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Conformal Geometry and Dynamics

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ISSN 1088-4173

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Orbit portraits of unicritical antiholomorphic polynomials
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by Sabyasachi Mukherjee PDF
Conform. Geom. Dyn. 19 (2015), 35-50 Request permission

Abstract:

Orbit portraits were introduced by Goldberg and Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical antiholomorphic polynomials, and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial objects. The results obtained in this paper serve as a combinatorial foundation for a detailed understanding of the combinatorics and topology of the parameter spaces of unicritical antiholomorphic polynomials and their connectedness loci, known as the multicorns.
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Additional Information
  • Sabyasachi Mukherjee
  • Affiliation: Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany
  • MR Author ID: 1098266
  • ORCID: 0000-0002-6868-6761
  • Email: s.mukherjee@jacobs-university.de
  • Received by editor(s): June 15, 2014
  • Received by editor(s) in revised form: February 1, 2015, and February 2, 2015
  • Published electronically: March 3, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 19 (2015), 35-50
  • MSC (2010): Primary 37E15, 37E10, 37F10, 37F20
  • DOI: https://doi.org/10.1090/S1088-4173-2015-00276-3
  • MathSciNet review: 3317234