Degenerations of complex dynamical systems II: analytic and algebraic stability: With an appendix by Jan Kiwi

Laura DeMarco, Xander Faber*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We study pairs (f, Γ) consisting of a non-Archimedean rational function f and a finite set of vertices Γ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set Γ. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we construct a convergent countable-state Markov chain that computes the limit measure. A classification of the periodic Fatou components for non-Archimedean rational functions, due to Rivera-Letelier, plays a key role in the proofs of our main theorems. The appendix contains a proof of this classification for all tame rational functions.

Original languageEnglish (US)
Pages (from-to)1669-1699
Number of pages31
JournalMathematische Annalen
Volume365
Issue number3-4
DOIs
StatePublished - Aug 1 2016

Funding

We would like to thank Rob Benedetto, Jeff Diller, Charles Favre, Liang-Chung Hsia, and Mattias Jonsson for valuable conversations. Finally, we are indebted to the anonymous referees for discovering small errors in earlier versions of our manuscript. This research was supported by the US National Science Foundation DMS-1302929 and DMS-1517080; Jan Kiwi was supported by the Chile Fondecyt 1110448.

ASJC Scopus subject areas

  • General Mathematics

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