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Dynamics of non-expansive maps on strictly convex Banach spaces

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Abstract

This paper concerns the dynamics of non-expansive maps on strictly convex finite dimensional normed spaces. By using results of Edelstein and Lyubich, we show that if X = (ℝn, ∥ · ∥) is strictly convex and X has no 1-complemented Euclidean plane, then every bounded orbit of a non-expansive map f: X → X, converges to a periodic orbit. By putting extra assumptions on the derivatives of the norm, we also show that the period of each periodic point of a non-expansive map f: X → X is the order, or, twice the order of a permutation on n letters. This last result generalizes a theorem of Sine, who proved it for ℓ n p where 1 < p < ∞ and p ≠ 2. To obtain the results we analyze the ranges of non-expansive projections, the geometry of 1-complemented subspaces, and linear isometries on 1-complemented subspaces.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    Bas Lemmens

  2. Mathematical Institute, Leiden University, P.O. Box 9512, 2300, RA Leiden, The Netherlands

    Onno van Gaans

Authors
  1. Bas Lemmens
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  2. Onno van Gaans
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Correspondence to Bas Lemmens.

Additional information

B. Lemmens acknowledges the support by Marie Curie Intra European Fellowship (MEIF-CT-2005-515391) of the European Commission.

O. van Gaans acknowlegdes the support by “Vidi subsidie” (639.032.510) of the Netherlands Organisation for Scientific Research (N.W.O.).

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Lemmens, B., van Gaans, O. Dynamics of non-expansive maps on strictly convex Banach spaces. Isr. J. Math. 171, 425–442 (2009). https://doi.org/10.1007/s11856-009-0057-2

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  • DOI: https://doi.org/10.1007/s11856-009-0057-2

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