Control Theory for Semigroups over Local Fields

Let $G$ be a 1-connected, almost-simple Lie group over a local field and $\mathcal{S}$ a subsemigroup of $G$ with non-empty interior. The action of the regular hyperbolic elements in the interior of $\mathcal{S}$ 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 $D_{w}$ and $W(\mathcal{S})$ the subgroup of the Weyl group such that the control set $D_{w}$ coincides with the invariant control set $D_{1}$ is a Weyl subgroup of $W$. We conclude by showing that the control sets are parameterized by the lateral classes $W(S)\backslash W$.