Remarks on Automorphisms of $\mathbb{C}^* \times \mathbb{C}^*$ and their basins

TL;DR

This paper investigates the basins of attraction for certain automorphisms of a7^2, showing that under a conjecture, no basins exist for specific directions, limiting the construction of Fatou-Bieberbach domains.

Contribution

It links a conjecture about automorphisms of a7^* a7^* to the non-existence of certain basins, advancing understanding of complex dynamics in a7^2.

Findings

No basins for non-degenerate characteristic directions under the conjecture

Limits on constructing Fatou-Bieberbach domains avoiding both axes

Conditional results depending on a known conjecture

Abstract

We study basins of attraction of automorphisms of $\CC^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of $\CC^*\times\CC^*$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method.