Bounds on the radius of the p-adic Mandelbrot set
Number Theory
2017-03-29 v2 Dynamical Systems
Abstract
Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. 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 is greater than or equal to 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 primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.
Keywords
Cite
@article{arxiv.1207.2628,
title = {Bounds on the radius of the p-adic Mandelbrot set},
author = {Jacqueline Anderson},
journal= {arXiv preprint arXiv:1207.2628},
year = {2017}
}