Minimally critical regular endomorphisms of A^N

TL;DR

This paper investigates a class of endomorphisms of affine space that generalize unicritical polynomials, providing bounds on Lyapunov exponents, estimates on critical height, and rigidity results across different fields.

Contribution

It introduces a new class of endomorphisms called minimally critical regular endomorphisms and extends classical results to higher dimensions and various fields.

Findings

Lower bounds on sum of Lyapunov exponents over complex numbers

Generalization of Mandelbrot set compactness to higher dimensions

Rigidity results for post-critically finite morphisms

Abstract

We study the dynamics of a class of endomorphisms of A^N which restricts, when N = 1, to the class of unicritical polynomials. Over the complex numbers, we obtain lower bounds on the sum of Lyapunov exponents, and a statement which generalizes the compactness of the Mandelbrot set. Over the algebraic numbers, we obtain estimates on the critical height, and over general algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.