Discontinuity of a degenerating escape rate

DeMarco, Laura; Okuyama, Yûsuke

doi:10.1090/ecgd/318

Discontinuity of a degenerating escape rate
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by Laura DeMarco and Yûsuke Okuyama

Conform. Geom. Dyn. 22 (2018), 33-44

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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