Fixed Point Theorems and Denjoy–wolff Theorems for Hilbert’s Projective Metric in Infinite Dimensions

Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ int(K) : q(x) = 1} where q: int(K) → (0,∞) is continuous and homogeneous of degree 1 and it is usually assumed that Σ is bounded in norm. In this framework there is a complete metric d, Hilbert’s projective metric, defined on Σ and a complete metric d, Thompson’s metric, defined on int(K). We study primarily maps f : Σ → Σ which are nonexpansive with respect to d, but also maps g: int(K) → int(K) which are nonexpansive with respect to d. We prove under essentially minimal compactness assumptions, fixed point theorems for f and g. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert’s projective metric near ∂Σ := Σ \ Σ. If x ∈ Σ, f : Σ → Σ is nonexpansive with respect to Hilbert’s projective metric, f has no fixed points on Σ and f satisfies certain mild compactness assumptions, we prove that ω(x; f), the omega limit set of x under f in the norm topology, is contained in ∂Σ; and there exists η ∈ ∂Σ, η independent of x, such that (1 − t)y + tη ∈ ∂K for 0 ≤ t ≤ 1 and all y ∈ ω(x; f). This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that co(ω(x; f)), the convex hull of ω(x; f), is contained in ∂K. 2000 Mathematics Subject Classification. 47H07, 47H09, 47H10.