The Wolff-Denjoy theorem beyond the unit disc

Abstract: The Wolff-Denjoy theorem has been the motivation for a host of results that resemble the classical theorem for holomorphic self-maps of the unit disc. In this talk, we shall look at yet another result in this class. This result applies to a rather general class of bounded domains in one and higher dimensions, which may have rough boundaries and aren't necessarily contractible. While our techniques are motivated by the properties of holomorphic maps in several complex variables, the theory of such maps turns out to be incidental to these techniques. In fact, in this talk, we shall spend some time examining certain analogies between the Poincaré distance and the Hilbert distance on convex domains. This is relevant as there exists a Wolff--Denjoy-type theorem, by Beardon, in the latter setting. It is these analogies that give rise to the fundamental concept that underlies our result(s): namely, a weak notion of negative curvature for spaces equipped with the Kobayashi distance (of which the Poincaré distance is a special case). No knowledge of several complex variables will be assumed in this talk: indeed, most of the discussion will focus on basic complex analysis and on the properties of metric spaces and contractive maps. A large part of this talk will be based on joint work with Andrew Zimmer and Anwoy Maitra.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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