This article is cited in 19 papers

One-parameter semigroups of analytic functions, fixed points and the Koenigs function

V. V. Goryainov, O. S. Kudryavtseva

The Volzhsky Institute of Humanities

Abstract: Analogues of the Berkson-Porta formula for the infinitesimal generator of a one-parameter semigroup of holomorphic maps of the unit disc into itself are obtained in the case when, along with a Denjoy-Wolff point, there also exist other fixed points. With each one-parameter semigroup a so-called Koenigs function is associated, which is a solution, common for all elements of the one-parameter semigroup, of a certain functional equation (Schröder's equation in the case of an interior Denjoy-Wolff point and Abel's equation in the case of a boundary Denjoy-Wolff point). A parametric representation for classes of Koenigs functions that takes account of the Denjoy-Wolff point and other fixed points of the maps in the one-parameter semigroup is presented.
Bibliography: 19 titles.

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

UDC: 517.54

MSC: 30D05

Received: 09.03.2010

DOI: 10.4213/sm7708

 English version: Sbornik: Mathematics, 2011, 202:7, 971–1000

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