Fixed points, semigroups and rigidity of holomorphic mappings – David Shoikhet (ORT College Braude, Israele)

Venue

Sala Conferenze (Puteano, Centro De Giorgi).

Abstract

There is a long history associated with the problem of iterating nonexpansive and holomorphic mappings and finding their fixed points, with the modern results of K. Goebel, W.-A. Kirk, T. Kuczumow, S. Reich, W. Rudin and J.-P. Vigue’ being among the most important. Historically, complex dynamics and geometrical function theory have beenintensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty yearsthe theory of holomorphic mappings on complex spaces has been studied bymany mathematicians with many applications to nonlinear analysis, functionalanalysis, differential equations, classical and quantum mechanics. The laws ofdynamics are usually presented as equations of motion which are written in theabstract form of a dynamical system: dx/dt + f(x) = 0, where x is a variabledescribing the state of the system under study, and f is a vector-function of x. The study of such systems when f is a monotone or an accretive (generally non-linear) operator on the underlying space has recently been the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems. In this talk we give a brief description of the classical statements which combine the celebrated Julia Theorem of 1920, Carathéodory’s contribution in 1929 and Wolff’s boundary version of the Schwarz Lemma of 1926 with their modern interpretations for discrete and continuous semigroups of hyperbolically nonexpansive mappings in Hilbert spaces. We also present flow-invariance conditions for holomorphic and hyperbolically monotone mappings. Finally, we study the asymptotic behavior of one-parameter continuous semigroups (flows) of holomorphic mappings. We present angular characteristics of the flows trajectories at their Denjoy-Wolff points, as well as at their regular repelling points (whenever they exist). This enables us by using linearization models in the spirit of functional Schroeder’s and Abel’s equations and eigenvalue problems for composition operators to establish new rigidity properties of holomorphic generators which cover the famous Burns-Krantz Theorem.

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