Lower and upper bounds for ω-limit sets of nonexpansive maps

Abstract If D is a subset of R n and f: D → D is an l 1-norm nonexpansive map, then it is known that every bounded orbit of f approaches a periodic orbit. Moreover, the minimal period of each periodic point of f is bounded by n! 2m, where m = 2n − 1. In this paper we shall describe two different procedures to construct periodic orbits of l 1-norm nonexpansive maps. These constructions yield that a lower bound for the largest possible minimal period of a periodic point of an l 1-norm non-expansive map is given by 3·2n − 1, n ≥ 3. Ifn ≤ 5, we shall also improve the upper bound for the largest possible minimal period.

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