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arXiv · 1207.2628

Bounds on the radius of the p-adic Mandelbrot set

Jacqueline Anderson
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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.

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Jacqueline Anderson. 2013-01-26. Bounds on the radius of the p-adic Mandelbrot set. https://doi.org/10.4064/aa158-3-5

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