Iterates of holomorphic and KD-nonexpansive mappings in convex domains in Cn

✍ Scribed by Tadeusz Kuczumow; Adam Stachura

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 408 KB
• Volume: 81
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(90)90005-8

✦ Synopsis

Throughout this paper D will be a bounded domain in C" and k, will be its Kobayashi distance.
In [ 1 ] Abate introduced the following notion of horospheres in D. For q,~ D, XE~D, and R>O the small horosphere Ezo(.~, R) and the big horosphere F,,(x, R) of center x, pole z0 and radius R are defined by E;,(x, R) = {ZE D: lim sup [ko(z, w) -k,(z,, w)] < i log R}, II' -5 F,,(x, R) = {z E D: lim inf [k,(z, w) -k,(z,, w)] < flog R}. H"T Similarly as the ellipsoids in the open unit ball B c C": E,(x, R)= ZEB: I(z, x) -(1 -r)l z + 112 -(2, x)x11 z r2 0, and r = R/(R + 1 ), horospheres are useful tools in investigations of holomorphic mappings [l, 10, 11, 16, 23, 251. THEOREM A [l]. Let D c @" be a convex domain and f: D + D be a holomorphir map without fixed points. Then there exists x E aD such that for everyz,ED, R>O, andiEN fi(E;& R)) = F;,(.v, R). Since for strongly pseudoconvex C2 domains D we have F,,(x, R) n 8D = (x} it gives [ 1 ] 90 OOOI-8708/90$7.50