In any degenerating one-parameter family of rational maps, either all periodic multipliers stay uniformly bounded or almost all of them blow up at a power rate, and degenerating cubic families always contain a short repelling cycle with blowing-up multiplier.
Ramification loci of non-archimedean cubic rational functions
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abstract
For a cubic rational function with coefficients in a non-archimedean field $K$ whose residue characteristic is $0$ or greater than $3$, there are $2$ possibilities for the shape of its Berkovich ramification locus, considered as an endomorphism of the Berkovich projective line: one is the connected hull of all the critical points, and the other is consisting of $2$ disjoint segments. In this paper, we list up all the possible forms of cubic rational functions and calculate their ramification loci.
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Blow-up of multipliers in meromorphic families of rational maps
In any degenerating one-parameter family of rational maps, either all periodic multipliers stay uniformly bounded or almost all of them blow up at a power rate, and degenerating cubic families always contain a short repelling cycle with blowing-up multiplier.