Iteration of self-maps on a product of Hilbert balls

作者：Chu Cho Ho^*; Rigby Michael

来源：Journal of Mathematical Analysis and Applications, 2014, 411(2): 773-786.

DOI：10.1016/j.jmaa.2013.10.009

• Iteration
• Hilbert ball
• Polydisc
• Kobayashi distance
• JB*-triple

Let D = D-1 x ... x D-p be a product of Hilbert balls, with coordinate maps pi(J) :(D) over bar -> (D) over bar (j): on the closure (D) over bar, 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 (xi(1),..., xi(p)) of D and a nonempty set J c {1,..., p} such that each limit function h of the iterates (f(n)) satisfies xi(j) is an element of pi(j) o h(D) for all j E J and pi(j) o h(.) = xi(j) whenever pi(J) o h(D) meets the boundary of D-j. For a single Hilbert ball D-1, either lim inf(n ->infinity) parallel to f(2n) (0)parallel to < 1 or (f(n)) converges locally uniformly to a constant map taking value at the boundary of D-1.