The dynamics of contractions

Abstract

We show that, providing a metric space $X$ has a boundary that is in some sense similar to the boundary of hyperbolic space, the iterates of a contraction $f:X\to X$ converge locally uniformly to a point in, or on the boundary of, $X$. This generalises the Denjoy–Wolff theorem for analytic self-maps of the unit disc in the complex plane, and also shows that if $D$ is a bounded strictly convex subdomain of ${\Bbb R}^n$, then any contraction of $D$ with respect to the Hilbert metric of $D$ converges to a point in the closure of $D$.