Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions

Abstract

Let $K$ be a closed, normal cone with nonempty interior $\inta(K)$
in a Banach space $X$. Let $\Sigma = \{x\in\inta(K) : q(x) = 1\}$
where $q \colon \inta(K)\rightarrow (0,\infty)$ is continuous and
homogeneous of degree $1$ and it is usually assumed that $\Sigma$
is bounded in norm. In this framework there is a complete metric
$d$, {\it Hilbert's projective metric}, defined on $\Sigma$ and a
complete metric $\overline d$, {\it Thompson's metric}, defined on
${\rm \int}(K)$. We study primarily maps $f\colon \Sigma\rightarrow\Sigma$
which are nonexpansive with respect to $d$, but also maps $g
\colon {\rm \int}(K)\rightarrow {\rm \int}(K)$ which are nonexpansive with respect
to $\overline{d}$. 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 $\partial\Sigma := \overline\Sigma
\setminus \Sigma$. If $x \in \Sigma$, $f \colon \Sigma\rightarrow
\Sigma$ is nonexpansive with respect to Hilbert's projective
metric, $f$ has no fixed points on $\Sigma$ and $f$ satisfies
certain mild compactness assumptions, we prove that $\omega
(x;f)$, the omega limit set of $x$ under $f$ in the norm topology,
is contained in $\partial\Sigma$; and there exists
$\eta\in\partial\Sigma$, $\eta$ independent of $x$, such that $(1
- t) y + t\eta \in\partial K$ for $0 \leq t \leq 1$ and all $y\in
\omega (x;f)$. This generalizes results of Beardon and of
Karlsson and Noskov. We give some evidence for the conjecture
that $\text{\rm co}(\omega(x;f))$, the convex hull of $\omega(x;f)$,
is contained in $\partial K$.