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 classical non-Archimedean Julia set.
Highlights
Let K be an algebraically closed field, complete with respect to a non-Archimedean absolute value | · |v
The crucial set appears to classify the type of bad reduction that φ has; this paper provides quantitative support for that idea when φ is a quadratic rational map
The main result of this paper is the following theorem, which says that for quadratic functions, the crucial set determines a stratification of M2(K ) compatible with specialization of [φ] to M2 (k ): Theorem 1.1 Let K be a complete, algebraically closed non-Archimedean field, and let φ be a degree two rational map over K
Summary
Let K be an algebraically closed field, complete with respect to a non-Archimedean absolute value | · |v. The main result of this paper is the following theorem, which says that for quadratic functions, the crucial set determines a stratification of M2(K ) compatible with specialization of [φ] to M2 (k ): Theorem 1.1 Let K be a complete, algebraically closed non-Archimedean field, and let φ be a degree two rational map over K. If one is given an arbitrary quadratic rational map φ, it is possible to determine the unique point ξ in the crucial set of φ The explicit connections in the case of quadratic rational maps and cubic polynomials suggest the possibility of a general theory linking the crucial set to the moduli space Md for all d.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
