On convolution powers on semidirect products

LetK be a compact group of linear operators of thed-dimensional spaceR ^d andG _K,d denote the semidirect productK byR ^d . It is shown that if an adapted probability measureμ onG _K,d is not scattered (i.e. for some compactF we havex ₀ ∈ R ^d (gF)>0), then there exists a nonzero vectorx ₀ ∈R ^d such thatk ₁(x ₀)=k ₂(x ₀) holds for all (k ₁,x ₁) and (k ₂,x ₂) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of thed-dimensional Euclidian space is scattered. I thank the Foundation for Research Development for financial support.