On the automorphism group of certain Short ${\mathbb{C}}^2$'s

Abstract

For a Hénon map of the form $H(x,y)=(y,p(y)-ax)$, where $p$ is a polynomial of degree at least two and $a\ne 0$, it is known that the sub-level sets of the Green's function $G_H^{+}$ associated with $H$ are Short ${\mathbb{C}}^2$'s. For a given $c>0$, we study the holomorphic automorphism group of such a Short ${\mathbb{C}}^2$, namely ${\Omega }_c=\{G_H^{+}<c\}$. The unbounded domain ${\Omega }_c\subset {\mathbb{C}}^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that ${\Omega }_c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, Aut$({\Omega }_c)$ cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short ${\mathbb{C}}^2$'s to be biholomorphic.

final version; accepted in International Mathematics Research Notices

References in corpus (3)

• Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
• Examples of non-autonomous basins of attraction-II
• Notes on Short ${\mathbb{C}}^k$'s