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

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

2008-09-12

arxiv.org/abs/0809.2116v1

Mathematics

Complex Variables

6 pages, comments welcome

Scientific paper

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.

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