On the existence of cycle times for some non-expansive maps

We consider functions F : R n ! R n which are homogeneous and nonexpansive in thè 1 norm. We refer to these as topical functions. We study the existence of the cycle time vector (F) = lim k!1 F k (~ x)=k, which, if it exists, is independent of ~ x 2 R n. For a restricted class of topical functions, the cycle time is known to be implicated in the existence of xed points and this provides the motivation for the present paper. We give a characterisation of topical functions which extends an earlier result of Crandall and Tartar. We show that the sequence F k (~ x)=k converges weakly, in the sense that its images under the functions t always converge. We show that under suitable conditions, weak convergence may be realised by the convergence of components of the vector sequence F k (0)=k. We show further that when n = 2, itself exists. When n = 3, it may not, and we give a family of examples which show the extent of the departure from convergence. We discuss the problem of characterising those topical functions for which does exist.