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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Discontinuity of a degenerating escape rate
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by Laura DeMarco and Yûsuke Okuyama PDF
Conform. Geom. Dyn. 22 (2018), 33-44 Request permission

Abstract:

We look at degenerating meromorphic families of rational maps on $\mathbb {P}^1$—holomorphically parameterized by a punctured disk—and we provide examples where the bifurcation current fails to have a bounded potential in a neighborhood of the puncture. This is in contrast to the recent result of Favre-Gauthier that we always have continuity across the puncture for families of polynomials; and it provides a counterexample to a conjecture posed by Favre in 2016. We explain why our construction fails for polynomial families and for families of rational maps defined over finite extensions of the rationals $\mathbb {Q}$.
References
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Additional Information
  • Laura DeMarco
  • Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
  • MR Author ID: 677013
  • Email: demarco@math.northwestern.edu
  • Yûsuke Okuyama
  • Affiliation: Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan
  • Email: okuyama@kit.ac.jp
  • Received by editor(s): October 4, 2017
  • Received by editor(s) in revised form: January 31, 2018
  • Published electronically: May 8, 2018
  • Additional Notes: This research was partially supported by JSPS Grant-in-Aid for Scientific Research (C), 15K04924, and the National Science Foundation DMS-1600718.
  • © Copyright 2018 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 22 (2018), 33-44
  • MSC (2010): Primary 37F45; Secondary 37P30
  • DOI: https://doi.org/10.1090/ecgd/318
  • MathSciNet review: 3798915