Dynamics of a Family of Polynomial Automorphisms of C-3, a Phase Transition

摘要

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 C-3. We show that the first dynamical degree psi(alpha) of 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.