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On the Existence of Cycle Times for Some Nonexpansive Maps
Gunawardena, Jeremy ; Keane, Mike
Keyword(s): cycle time, dynamical system, min-maz function, nonexpansive map, supremum norm
Abstract: We consider functions F: Rn → Rn which are homogenous and nonexpansive in the norm. We refer to these as topical functions. We study the existence of cycle time vector X(F) = limk → ∞ Fk (x)/k, which if it exists, is independent of x ε Rn. For a restricted class of topical functions, the cycle time is known to be implicated in the existence of fixed 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 Fk(x)/k converges weakly, in the sense that its images under the functions t(x1, ··· , xn) = max(x1, ··· , xn) and b(x1, ··· , xn) = min(x1, ··· , xn) always converge. We show that under suitable conditions, weak convergence may be realised by the convergence by components of the vector sequence F(0)/k. We show further that when n=2, X itself exists. When n = 3, it may not, as we give a family of examples which show the extent of the departure from convergence. We discuss the problem characterising those topical functions for which X does not exist.
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