Abstract
A Fatou-Bieberbach domain is a domain in C2 which is a biholomorphic image of C2 and is not all of C2. The existence of such domains has been known for a long time. There are several papers dealing with Fatou-Bieberbach domains [BF, BS, DE, E, FS, Gl, K, N1, N2, RR1, RR2, S]. Nevertheless, they remain quite misterious objects and there are many open questions about them. One of them is the following [RR1, Q.11, p.79)]: If Ω is a FatouBieberbach domain and L is a complex line is it possible that (a) L ∩Ω is connected (b) L∩Ω has finitely many components (c) L∩Ω is a circular disc. Let∆ be the open unit disc in C and let P = ∆×∆. Suppose one wants to obtain a Fatou-Bieberbach domain Ω whose intersection with the z-axis {(z, 0) : z ∈ C} is approximately the unit disc. One would try to find Ω such that
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