BOUNDS ON THE RADIUS OF THE P-ADIC MANDELBROT SET

Abstract. Let f(z) = zd + ad−1zd−1 + · · · + a1z ∈ Cp[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 12d ≤ 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. 1