Non-autonomous basins of attraction and their boundaries
A Fatou-Bierberbach (FB) domain is a proper subdomain of \(\mathbb{C}^2\) biholomorphic to \(\mathbb{C}^2\). A classical example is given by an attracting basin of an attractive fixed-point of an automorphism of \(\mathbb{C}^2\). Note that such a domain may be dense in \(\mathbb{C}^2\). This article deals with the regularity of the boundary of some FB domains. The authors show that there exists a FB domain whose boundary has positive 4-dimensional Hausdorff measure. It is obtained as an attracting basin associated to a sequence of nonresonant automorphisms of \(\mathbb{C}^2\) having a common attractive fixed-point. Recall that such a basin is not necessarily biholomorphic to \(\mathbb{C}^2\), it may be a ``short \(\mathbb{C}^2\)'' [\textit{J. E. Fornaess}, Miyajima, Kimio (ed.) et al., Complex analysis in several variables -- Memorial conference of Kiyoshi Oka's centennial birthday. Papers from the conference, Kyoto, Japan, October 30--November 5, 2001 and Nara, Japan, November 6--8, 2001. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 42, 95--108 (2004; Zbl 1071.32007)]. The nonresonant condition ensures that the attracting basin is actually biholomorphic to \(\mathbb{C}^2\). The authors also show that there exists a FB domain whose boundary has an upper-box dimension 4 near any of its points. Such domains are obtained as ``calibrated basins'' associated to a sequence of automorphisms.
scientific article; zbMATH DE number 2183025
scientific article