Annales Academić Scientiarum Fennicć
Mathematica
Volumen 35, 2010, 439-472

LINEARIZATION MODELS FOR PARABOLIC DYNAMICAL SYSTEMS VIA ABEL'S FUNCTIONAL EQUATION

Mark Elin, Dmitry Khavinson, Simeon Reich and David Shoikhet

ORT Braude College, Department of Mathematics
P.O. Box 78, 21982 Karmiel, Israel; mark_elin 'at' braude.ac.il

University of South Florida, Department of Mathematics
Tampa, FL 33620-5700, U.S.A.; dkhavins 'at' cas.usf.edu

Technion - Israel Institute of Technology, Department of Mathematics
32000 Haifa, Israel; sreich 'at' tx.technion.ac.il

ORT Braude College, Department of Mathematics
P.O. Box 78, 21982 Karmiel, Israel; davs 'at' braude.ac.il

Abstract. We study linearization models for continuous one-parameter semigroups of parabolic type. In particular, we introduce new limit schemes to obtain solutions of Abel's functional equation and to study asymptotic behavior of such semigroups. The crucial point is that these solutions are univalent functions convex in one direction. In a parallel direction, we find analytic conditions which determine certain geometric properties of those functions, such as the location of their images in either a half-plane or a strip, and their containing either a half-plane or a strip. In the context of semigroup theory these geometric questions may be interpreted as follows: is a given one-parameter continuous semigroup either an outer or an inner conjugate of a group of automorphisms? In other words, the problem is finding a fractional linear model of the semigroup which is defined by a group of automorphisms of the open unit disk. Our results enable us to establish some new important analytic and geometric characteristics of the asymptotic behavior of one-parameter continuous semigroups of holomorphic mappings, as well as to study the problem of existence of a backward flow invariant domain and its geometry.

2000 Mathematics Subject Classification: Primary 30C45, 30D05, 37F99, 47H20.

Key words: Abel's functional equation, asymptotic behavior, convex in one direction, generator, linearization model, semigroup of parapolic type.

Reference to this article: M. Elin, D. Khavinson, S. Reich and D. Shoikhet: Linearization models for parabolic dynamical systems via Abel's functional equation. Ann. Acad. Sci. Fenn. Math. 35 (2010), 439-472.

Full document as PDF file

doi:10.5186/aasfm.2010.3528