Dynamics of biholomorphic self-maps on bounded symmetric domains

Let g be a fixed-point free biholomorphic self-map of a bounded symmetric domain B. It is known that the sequence of iterates (gn) may not always converge locally uniformly on B even, for example, if B is an infinite dimensional Hilbert ball. However, g=ga∘T, for a linear isometry T, a=g(0) and a transvection ga, and we show that it is possible to determine the dynamics of ga. We prove that the sequence of iterates (gna) converges locally uniformly on B if, and only if, a is regular, in which case, the limit is a holomorphic map of B onto a boundary component (surprisingly though, generally not the boundary component of a∥a∥). We prove (gna) converges to a constant for all non-zero a if, and only if, B is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.