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Journal of Lie Theory 11 (2001), No. 2, 559--604
Copyright Heldermann Verlag 2001

On Asymptotic Behavior and Rectangular Band Structures in SL(2, R)

Brigitte E. Breckner
Babes-Bolyai University, Faculty of Mathematics, Str. M. Kogalniceanu 1, 3400 Cluj-Napoca, Romania

Wolfgang A. F. Ruppert
Institut fuer Mathematik, Universitaet fuer Bodenkultur, Peter-Jordan-Str. 82, 1190 Wien, Austria

We associate with every subsemigroup of Sl(2, R), not contained in a single Borel group, an "asymptotic object", a rectangular band which is defined on a closed subset of a torus surface. Using this concept we show that the horizon set (in the sense of J. D. Lawson [J. Lie Theory 4 (1994) 17--29]) of a connected open subsemigroup of Sl(2, R) is always convex, in fact the interior of a three dimensional Lie semialgebra. Other applications include the classification of all exponential subsemigroups of Sl(2, R) and the asymptotics of semigroups of integer matrices in Sl(2, R).

Keywords: Asymptotic objects, asymptotic property, subsemigroups of Sl(2, R), Lie semigroups, Lie semialgebras and their classification, compression semigroups, diamond product, rectangular domain, umbrella sets, control theory in Lie groups, asymptotics of integer

MSC: 22E15; 22E67, 22E46, 22A15, 22A25

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