Dynamics of non-expansive maps on strictly convex Banach spaces
نویسندگان
چکیده
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 = (R, ‖ · ‖) 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 p where 1 < p < ∞ and p 6= 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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