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

Volume 158 / 2013

Jacqueline Anderson Acta Arithmetica 158 (2013), 253-269 MSC: Primary 11S82; Secondary 37P05. DOI: 10.4064/aa158-3-5

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.

Authors

  • Jacqueline AndersonMathematics Department
    Box 1917
    Brown University
    Providence, RI 02912, U.S.A.
    e-mail

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