Periodic points of nonexpansive maps

Summary

Introduction

If M is a topological space and f : M → M is a continuous map, it is frequently of interest to understand, for x ∈ M, the behaviour of iterates f^k(x) as k approaches infinity. In this generality little can be said; but if f is “nonexpansive with respect to a metric ρ on M”, i.e. if ρ(f(x), f ≤ ρ(x, y) for all x, y ∈ M, then a variety of useful theorems, some of recent vintage, can be brought into play. Surprisingly, this simple and powerful observation has often been missed in studying specific classes of nonlinear operators.

In this note we shall state some theorems about nonexpansive maps, describe some conjectures and open questions, and indicate some specific classes of maps for which these observations prove useful.

Examples of Maps of Interest

We begin by recalling some terminology and basic facts. Let S denote a compact Hausdorff space, C(S) = X, the Banach space of continuous functions on S in the sup norm ∥x∥_∞ = sup{|x(s) s ∈ S}, K the cone of nonnegative functions on S and K the interior of S. If S is the set of integers i with 1 ≤ i ≤ n, we identify C(S) with Rⁿ in the obvious way and write K:= Kⁿ, so Kⁿ:= {x ∈ Rⁿ|x_i ≥ 0 for 1 ≤ i ≤ n}. We shall denote by u ∈ C(S) the constant map defined by u(s) = 1 for all s ∈ S. If S = {i ∈ Z| 1 ≤ i ≤ n}, we have u ∈ Rⁿ and u_i= 1 for 1 ≤ i ≤ n.