Annales Academię Scientiarum Fennicę

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

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