Acta Arithmetica (2013)

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

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## Abstract

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Let $f\left(z\right)={z}^{d}+{a}_{d-1}{z}^{d-1}+...+{a}_{1}z\in {ℂ}_{p}\left[z\right]$ 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.

## How to cite

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Jacqueline 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},
url = {http://eudml.org/doc/279226},
volume = {158},
year = {2013},
}

TY - JOUR
AU - Jacqueline Anderson
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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