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

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 $({\Psi }_{\alpha }{)}_{\alpha }$ of polynomial automorphisms of ${\mathbb{C}}^3$. We show that the first dynamical degree of ${\Psi }_{\alpha }$ is $>1$, that ${\Psi }_{\alpha }$ preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family $({\Psi }_{\alpha }{)}_{\alpha }$, in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter $\alpha$.