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

Acta Arithmetica (2013)

- Volume: 158, Issue: 3, page 253-269
- ISSN: 0065-1036

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topJacqueline Anderson. "Bounds on the radius of the p-adic Mandelbrot set." Acta Arithmetica 158.3 (2013): 253-269. <http://eudml.org/doc/279226>.

@article{JacquelineAnderson2013,

abstract = {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.},

author = {Jacqueline Anderson},

journal = {Acta Arithmetica},

keywords = {-adic Mandelbrot set; non-Archimedean dynamical systems},

language = {eng},

number = {3},

pages = {253-269},

title = {Bounds on the radius of the p-adic Mandelbrot set},

url = {http://eudml.org/doc/279226},

volume = {158},

year = {2013},

}

TY - JOUR

AU - Jacqueline Anderson

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

JO - Acta Arithmetica

PY - 2013

VL - 158

IS - 3

SP - 253

EP - 269

AB - 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.

LA - eng

KW - -adic Mandelbrot set; non-Archimedean dynamical systems

UR - http://eudml.org/doc/279226

ER -

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