Let $D$ be a Hermitian symmetric space of tube type, $S$ its Silov boundary and $G$ the neutral component of the group of bi-holomorphic diffeomorphisms of $D$. Our main interest is in studying the action of $G$ on $S^3=S\times S\times S$. Sections 1 and 2 are part of a joint work with B. Ørsted (see [4]). In Section 1, as a pedagogical introduction, we study the case where $D$ is the unit disc and $S$ is the circle. This is a fairly elementary and explicit case, where one can easily get a flavour of the more general results. In Section...