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Conformal Geometry and Dynamics

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Discontinuity of a degenerating escape rate


Authors: Laura DeMarco and Yûsuke Okuyama
Journal: Conform. Geom. Dyn. 22 (2018), 33-44
MSC (2010): Primary 37F45; Secondary 37P30
DOI: https://doi.org/10.1090/ecgd/318
Published electronically: May 8, 2018
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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}$.


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Additional Information

Laura DeMarco
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
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

DOI: https://doi.org/10.1090/ecgd/318
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.
Article copyright: © Copyright 2018 American Mathematical Society

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