This article is cited in 31 papers

Semigroups of analytic functions in analysis and applications

V. V. Goryainov

Volzhsky Institute of Humanities (branch) of the Volgograd State University

Abstract: This survey considers problems of analysis and certain related areas in which semigroups of analytic functions with respect to the operation of composition appear naturally. The main attention is devoted to holomorphic maps of a disk (or a half-plane) into itself. The role of fixed points is highlighted, both in the description of the structure of semigroups and in applications. Interconnections of the problem of fractional iteration with certain problems in the theory of random branching processes are pointed out, as well as with certain questions of non-commutative probability. The role of the infinitesimal description of semigroups of conformal maps in the development of the parametric method in the theory of univalent functions is shown.
Bibliography: 94 titles.

Keywords: one-parameter semigroup, infinitesimal generator, evolution family, evolution equation, fractional iterates, Koenigs function, fixed points.

UDC: 517.54

MSC: Primary 30-02; Secondary 30-03, 30C20, 30C35, 30C45, 30C50, 30C75, 30D05, 39B12, 39B32, 60E05, 60E10, 60J80

Received: 24.09.2012

DOI: 10.4213/rm9500

 English version: Russian Mathematical Surveys, 2012, 67:6, 975–1021

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