Periods of Nonexpansive Operators on Finite l1-Spaces

In [1] Akcoglu and Krengel showed that for every nonexpansive operator T with bounded orbits on a finitel1-space and for every x ∈ l1, there exists a p ∈ ℕ such that Tpnx converges. Using their result we show that there exists some p ∈ ℕ such that Tpnx converges for all x and we provide upper bounds of p as a function of the dimension of the space. In a special case we characterize the set of all p which are the period of some nonexpansive operator. Our main tool is the study of the order of automorphisms of finite sublattices of ℝl.