Iteration of self-maps on a product of Hilbert balls

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