On Brolin's theorem over the quaternions

Cinzia Bisi; Antonino De Martino

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arXiv · 2003.09899

On Brolin's theorem over the quaternions

Cinzia Bisi·Antonino De Martino

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Abstract

In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.

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Cinzia Bisi, Antonino De Martino. 2020-03-22. On Brolin's theorem over the quaternions. https://arxiv.org/abs/2003.09899

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