Configuration of the crucial set for a quadratic rational map

John R. Doyle, Kenneth Scott Jacobs*, Robert Rumely

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Let K be a complete, algebraically closed non-Archimedean valued field, and let φ(z) ∈ K(z) have degree two. We describe the crucial set of φ in terms of the multipliers of φ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space M2(K) related to the reduction type of φ. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the classical non-Archimedean Julia set.

Original languageEnglish (US)
Article number11
JournalResearch in Number Theory
Issue number1
StatePublished - Dec 1 2016


  • Crucial set
  • Moduli space
  • Potential good reduction
  • Quadratic map
  • Stratification

ASJC Scopus subject areas

  • Algebra and Number Theory


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