sense of the [*Pontryagin duality*]{}. In this terminology, the subgroup $ \Z_\mu $ of $ \Z $ in Theoren \[Intro YorTh\] is exactly the dual group of $ H_\mu^s $. In the non-abelian cases, however, we have a typical example given in Example \[ex: sym group\] where $ H_\mu(\bP^o) $ is strictly included in $ H_\mu^s $ for any extremal point $ \bP^o $.

Now we recall Tsirelson’s stochastic differential equation (SDE in short). Let $ (t_k : \, k \in -\N) $ be a decreasing sequence of the interval $ (0,1] $ such that $ t_k \to 0 $ as $ k \to -\infty $. Define $ A(t,X) $, which is called [*Tsirelson’s drift*]{}, as $$A(t,X) = \sum_{k \in -\N} \left\{ \frac{X_{t_{k-1}} - X_{t_{k-2}}}{t_{k-1} - t_{k-2}} \right\} 
1_{ [t_{k-1},t_{k}) }(t) , \quad 0 < t < 1 .$$ Then Tsirelson’s SDE is given by $$X_t = B_t + \int_0^t A(s,X) ds , \quad 0 \le t \le 1 
\label{Tsirel SDE}$$ where $ (B_t) $ is a one-dimensional Brownian motion. If $ ((X_t),(B_t)) $ is a solution of the equation , then the sequences $ (\eta_k) $ and $ (\xi_k) $ defined by $$\eta_k = \frac{X_{t_{k}} - X_{t_{k-1}}}{t_{k} - t_{k-1}} 