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Abstract

Let $f(z) = z^d + a_{d-1}z^{d-1} + \dots + a_1z \in \mathbb {C}_p[z]$ 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 $p\geq d$ 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 $\frac 12d \leq p < d $. We also explore a one-parameter family of cubic polynomials over $\mathbb {Q}_2$ to illustrate that the $p$-adic Mandelbrot set can be quite complicated when $p < d$, in contrast with the simple and well-understood $p \geq d$ case.