A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces

Abstract

Abstract. We establish a Denjoy-Wolff theorem for compact holomorphic self-mappings of bounded and strictly convex domains in arbitrary complex Banach spaces.

Wolff–Denjoy theorems in geodesic spaces

“…In 2012, Budzyńska [7] (see also [8,10,11] for infinite-dimensional generalizations) obtained the Wolff-Denjoy theorem for nonexpansive maps on a strictly convex bounded domain D ⊂ C n with the Kobayashi distance. Budzyńska's arguments were sharpened by Abate and Raissy in [3].…”

Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

“…The classical Denjoy-Wolff theorem asserts that all orbits of a fixed point free holomorphic mapping f : D → D on the open unit disc D ⊆ C converge to a unique point η ∈ ∂D. Since the inception of the theorem by Denjoy [14] and Wolff [51,52] in the nineteen-twenties a variety of extensions have been obtained; see for example [1,8,9,10,24,46]. A detailed account of its history and an extensive list of references can be found in the recent survey articles [4, Appendices G and H], [26], and [45].…”

A Wolff Theorem for finite rank bounded symmetric domains

“…In a Hilbert space ellipsoids replace the internally tangent discs [18]. In strictly convex domains, in C n [1,2,7], or in Banach spaces [9,10], the internally tangent discs are replaced by horospheres defined in terms of the Kobayashi distance. Although these horospheres are defined for arbitrary Banach spaces [2,3,32], if the boundary of the ball is more complicated they are considerably less tractable, even in finite dimensions [17,4,16].…”