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arXiv · 1606.09633

Dynamics of a family of polynomial automorphisms of $\mathbb{C}^3$, a phase transition

Julie Déserti·Martin Leguil
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Abstract

The polynomial automorphisms of the affine plane have been studied a lot: if $f$ is such an automorphism, then either $f$ preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals $1$, or $f$ preserves no rational curves, has a countable centralizer and its first dynamical degree is $>1$. In higher dimensions there is no such description. In this article we study a family $(Ψ_α)_α$ of polynomial automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of $Ψ_α$ is $>1$, that $Ψ_α$ preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family $(Ψ_α)_α$, in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter $α$.

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Julie Déserti, Martin Leguil. 2016-07-01. Dynamics of a family of polynomial automorphisms of $\mathbb{C}^3$, a phase transition. https://arxiv.org/abs/1606.09633

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