Bounds on the radius of the p-adic Mandelbrot set

• Autorzy: Jacqueline Anderson
• Języki publikacji: EN

Abstrakty

EN

Let $f(z) = z^d + a_{d-1}z^{d-1} + ... + a_1z ∈ ℂ_p[z]$ be a degree d polynomial. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p ≥ d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for 1/2d ≤ p < d. We also explore a one-parameter family of cubic polynomials over ℚ₂ to illustrate that the p-adic Mandelbrot set can be quite complicated when p < d, in contrast with the simple and well-understood p ≥ d case.

Kategorie tematyczne

• 11S82: Non-Archimedean dynamical systems \SeeMainly{}
• 37P05: Polynomial and rational maps

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 158
• Numer: 3
• Strony: 253-269

Daty

wydano

2013

Twórcy

autor

Jacqueline Anderson

• Mathematics Department, Box 1917, Brown University, Providence, RI 02912, U.S.A.

Identyfikatory

DOI

10.4064/aa158-3-5