Dynamics of non-expansive maps on strictly convex Banach spaces

doi:10.1007/S11856-009-0057-2

Paper

Dynamics of non-expansive maps on strictly convex Banach spaces

Published Jul 10, 2009 · Bas Lemmens, O. Gaans

Israel Journal of Mathematics

7

Citations

0

Influential Citations

PDF

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 = (ℝ^n, ∥ · ∥) 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.

Study Snapshot

Non-expansive maps on strictly convex Banach spaces converge to periodic orbits, with the period of each periodic point being twice the order of a permutation on n letters.

PopulationOlder adults (50-71 years)

Sample size24

MethodsObservational

OutcomesBody Mass Index projections

ResultsSocial networks mitigate obesity in older groups.

Dynamics of non-expansive maps on strictly convex Banach spaces

References

Citations