Lattice Isomorphisms and Iterates of Nonexpansive Maps

It is easy to see that the I, norm and the sup norm 11. Ilm (Ilxll, = max{Ix, I: 1 I i 5 n)) on I?’ are polyhedral. If E is a finite dimensional Banach space with a polyhedral norm 11. )I, D is a compact subset of E and f: D + D is a nonexpansive map, Weller [2] has shown that for each x E D, there again exists an integer px such that (1.1) holds. The original arguments did not give upper bounds for the integer px, x E D. However, subsequent work has proved (see [3-l 11) that there exists an integer N, depending only on the integer m in equation (1.2), such that px I N for all x E D. In general, the problem of finding optimal upper bounds for the integers px appears to be a very difficult combinatorialgeometrical question: see [3-9, 121 for a more complete discussion. The original motivation for the study of II-nonexpansive maps in [l] was to understand nonlinear analogues of diffusion on finite state spaces. However, as discussed in [S], these results, for the case of the sup norm and other polyhedral norms, also have applications to certain cone mappings and, hence, to a variety of examples and applications in [12-141. These ideas also apply to certain autonomous differential equations x’(f) = f(x(t)) (see [S]) and