Semigroups over Local fields

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)\W.