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Minimally critical regular endomorphisms of $\mathbb{A}^N$
Published online by Cambridge University Press: 21 October 2021
Abstract
We study the dynamics of the map
$f:\mathbb {A}^N\to \mathbb {A}^N$
defined by
for $A\in \operatorname {SL}_N$ , $\mathbf {b}\in \mathbb {A}^N$ , and $d\geq 2$ , a class which specializes to the unicritical polynomials when $N=1$ . In the case $k=\mathbb {C}$ we obtain lower bounds on the sum of Lyapunov exponents of f, and a statement which generalizes the compactness of the Mandelbrot set. Over $\overline {\mathbb {Q}}$ we obtain estimates on the critical height of f, and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.
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- © The Author(s), 2021. Published by Cambridge University Press
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