2007/03/09 by Marcelo Firer, Firer, Marcelo, Daniel Miranda +1
Mathematics · #12J25 #93C99 #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC) #math.MG #math.OC #msc:12J25 #msc:93C99
paper · pdf · doi:10.48550/arxiv.math/0703250
31 pages; 1 figure
arxiv created 2007/08/27 · arxiv updated 2009/12/01
Let G be a 1-connected, almost-simple Lie group over a local field and S a subsemigroup of G with non-empty interior. The action of the regular hyperbolic elements in the interior of S on the flag manifold G/P and on the associated Euclidean building allows us to prove that the invariant control set exists and is unique. We also provide a characterization of the set of transitivity of the control sets: its elements are the fixed points of type w for a regular hyperbolic isometry, where w is an element of the Weyl group of G. Thus, for each w in W there is a control set Dw and W(S) the subgroup of the Weyl group such that the control set Dw coincides with the invariant control set D1 is a Weyl subgroup of W. We conclude by showing that the control sets are parameterized by the lateral classes W(S)\backslash W.