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Growth of Volume in Fatou-Bieberbach Regions

1993; Kyoto University; Volume: 29; Issue: 1 Linguagem: Inglês

10.2977/prims/1195167547

1663-4926

Jean-Pierre Rosay, Walter Rudin,

Algebraic Geometry and Number Theory

Here J0 is the complex Jacobian of 0 [7; p. 11], hence J0 is holomorphic, |/0 1 is subharmonic and of course positive, and its integral over C is therefore infinite. Let B be the open unit ball of C. Thus rB is the ball of radius r, centered at 0. The preceding paragraph shows that vol (Qr\rB} must tend to oo as r— >oo, whenever Q is F.B. Theorem 1 of the present paper shows that this can happen arbitrarily slowly. We became interested in vol (Qr\rB) because we wanted to know (we still don't) whether there is an F. B. region in C that does not intersect the set {zw=Q} , a union of two intersecting complex lines. (This is a special case of a more general question : Which analytic varieties V can be avoided by F. B. regions? When F is a complex line in C then it can be done; see [3] or Example 9.7 in [6]. In this direction, Bedford and Smillie [1] proved an interesting result concerning algebraic varieties.) If there were such a region, it would be the range of a biholomorphic 0 of the form