Holomorphic invariance on bounded symmetric domains

We establish domains of invariance for holomorphic self-maps of a bounded s ymmetric domain of arbitrary dimension, generalising results of Wolff and J ulia in the open unit disc Delta of C and well known results on the Hilbert ball. The holomorphically invariant domains are analogues of the Poincare balls and their limiting horocycles in Delta. The results appear to be new even in the finite dimensional (non-Hilbert space) case.