Dynamics of non-expansive maps on strictly convex normed spaces

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 = (a"e (n) , ayen center dot ayen) 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 a"" (p) (n) where 1 < p < a and p not equal 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.