Iteration of self-maps on a product of Hilbert balls

Let D=D1×⋯×DpD=D1×⋯×Dp be a product of Hilbert balls, with coordinate maps πj:D¯→D¯j on the closure D¯, for j=1,…,pj=1,…,p. Let f be a fixed-point free self-map on D  , which is nonexpansive in the Kobayashi distance, and compact for p⩾2p⩾2. We describe the horospheres invariant under f   and show that there exist a boundary point (ξ1,…,ξp)(ξ1,…,ξp) of D   and a nonempty set J⊂{1,…,p}J⊂{1,…,p} such that each limit function h   of the iterates (fn)(fn) satisfies ξj∈πj∘h(D)¯ for all j∈Jj∈J and πj∘h(⋅)=ξjπj∘h(⋅)=ξj whenever πj∘h(D)πj∘h(D) meets the boundary of DjDj. For a single Hilbert ball D1D1, either liminfn→∞‖f2n(0)‖<1 or (fn)(fn) converges locally uniformly to a constant map taking value at the boundary of D1D1.