Dynamics of non cohomologically hyperbolic automorphisms of $\mathbb{C}^3$

TL;DR

This paper investigates the complex dynamics of certain non cohomologically hyperbolic automorphisms of C^3, constructing invariant currents and measures, and analyzing their properties and supports.

Contribution

It introduces a method to extend these automorphisms to algebraically stable maps on a compactification and constructs canonical invariant currents and measures.

Findings

Constructed algebraically stable extensions on a compactification.

Defined and studied properties of invariant currents and their intersections.

Proved the support of the invariant measure is compact and pluripolar.

Abstract

We study the dynamics of a family of non cohomologically hyperbolic automorphisms $f$ of $\mathbb{C}^3$. We construct a compactification $X$ of $\mathbb{C}^3$ where their extensions are algebraically stable. We finally construct canonical invariant closed positive $(1,1)$-currents for $f^*$, $f_*$ and we study several of their properties. Moreover, we study the well defined current $T_f \wedge T_{f^{-1}}$ and the dynamics of $f$ on its support. Then we construct an invariant positive measure $T_f \wedge T_{f^{-1}}\wedge \phi_{\infty}$, where $\phi_{\infty}$ is a function defined on the support of $T_f \wedge T_{f^{-1}}$. We prove that the support of this measure is compact and pluripolar. We prove also that this measure is canonical, in some sense that will be precised.