Holomorphic invariance on bounded symmetric domains

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