Bounds on the radius of the $p$-adic Mandelbrot set

Jacqueline Anderson¹

¹ Mathematics Department Box 1917 Brown University Providence, RI 02912, U.S.A.

Acta Arithmetica, Tome 158 (2013) no. 3, pp. 253-269

Résumé

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.

Détail

DOI : 10.4064/aa158-3-5

Keywords: d d dots mathbb degree polynomial say post critically bounded pcb its critical points have bounded orbit under iteration known geq pcb critical points have p adic absolute value equal similar result frac leq explore one parameter family cubic polynomials mathbb illustrate p adic mandelbrot set quite complicated contrast simple well understood geq