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Bounds on the radius of the p-adic Mandelbrot set

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Anderson J.

Acta Arithmetica

2013

tom 158

nr 3

253-269

Let $f(z) = z^d + a_{d-1}z^{d-1} + ... + a_1z ∈ ℂ_p[z]$ 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 ≥ 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 1/2d ≤ 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.

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1 Acta Arithmetica

1 Anderson J.

1 2013

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Bounds on the radius of the p-adic Mandelbrot set