  For connected reductive groups G over a finite extension F of Q_p and L the
maximal unramified extension of F we study the sets H_{\mu, N}(G) of elements b
in G(L) with given Hodge points of (b\sigma), (b\sigma)^2, ..., (b\sigma)^N. We
explain the relationship to stratifications of some moduli scheme of abelian
varieties defined by Goren and Oort respectively Andreatta and Goren. We show
that for sufficiently large N the Newton point is constant on the sets H_{\mu,
N}(G) and compute such N for certain classes of groups.
