Iteration of holomorphic maps on Lie balls

We investigate iterations of fixed-point free holomorphic self-maps on a Lie ball of any dimension, where a Lie ball is a bounded symmetric domain and the open unit ball of a spin factor which can be infinite dimensional. We describe the invariant domains of a holomorphic self-map f on a Lie ball D when f   is fixed-point free and compact, and show that each limit function of the iterates (fn)(fn) has values in a one-dimensional disc on the boundary of D  . We show that the Möbius transformation gaga induced by a nonzero element a in D may fail the Denjoy–Wolff-type theorem, even in finite dimension. We determine those which satisfy the theorem.