Discontinuity of a degenerating escape rate

TL;DR

This paper constructs examples of degenerating rational map families where the bifurcation current lacks bounded potential near a puncture, contrasting previous results for polynomial families and challenging existing conjectures.

Contribution

It provides the first known counterexamples to the bounded potential property of bifurcation currents in degenerating rational map families, highlighting differences from polynomial cases.

Findings

Counterexamples to bounded potential of bifurcation currents near punctures.

Demonstrates failure of continuity in certain rational families.

Explains why polynomial and rational families over number fields differ.

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 $\mathbb{Q}$.