Asymptotic behavior and rectangular band structures in SL\((2,\mathbb{R})\) (Q2730499)

In this article the authors study in some detail subsemigroups of Sl\((2,\mathbb R)\) not contained in a Borel subgroup. They attach to each such subsemigroup an ``asymptotic object'', an idempotent semigroup called a rectangular band defined on a closed subset of a toral surface; this idempotent semigroup plays a key role in their investigations. They also introduce ``umbrellas'', limiting directions at infinity (in the Lie algebra) of a subsemigroup. A principal result is that for a connected open subsemigroup of Sl\((2,\mathbb R)\) the umbrella is convex, and is in fact the interior of a three-dimensional Lie semialgebra. Other applications include the classification of exponential subsemigroups and the asymptotic behavior of semigroups of integer matrices.