Configuration of the Crucial Set for a Quadratic Rational Map

John R. Doyle; Kenneth Jacobs; Robert Rumely

Configuration of the Crucial Set for a Quadratic Rational Map

Number Theory 2021-08-12 v1 Dynamical Systems

Authors: John R. Doyle , Kenneth Jacobs , Robert Rumely

Abstract

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of $\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $\varphi$ . We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.

Keywords

quadratic rational map crucial set non-archimedean field Julia set moduli space repelling periodic points

Cite

@article{arxiv.1507.03535,
  title  = {Configuration of the Crucial Set for a Quadratic Rational Map},
  author = {John R. Doyle and Kenneth Jacobs and Robert Rumely},
  journal= {arXiv preprint arXiv:1507.03535},
  year   = {2021}
}

Configuration of the Crucial Set for a Quadratic Rational Map

Abstract

Keywords

Cite

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