ASYMPTOTIC BEHAVIOR AND DENJOY-WOLFF THEOREMS FOR HILBERT METRIC NONEXPANSIVE MAPS

Abstract

OF THE DISSERTATION Asymptotic Behavior and Denjoy-Wolff Theorems for Hilbert Metric Nonexpansive Maps by Brian C. Lins Dissertation Director: Roger D. Nussbaum We study the asymptotic behavior of fixed point free Hilbert metric nonexpansive maps on bounded convex domains. For such maps, we prove that the omega limit sets are contained in a convex subset of the boundary when the domain is either polyhedral or two dimensional. Similar results are obtained for several classes of positive operators defined on closed cones, including linear maps, affine linear maps, max-min operators, and reproduction-decimation operators. We discuss the relationship between these results and other Denjoy-Wolff type theorems. In particular, we investigate the interaction of nonexpansive maps with the horofunction boundary in the Hilbert geometry and in finite dimensional normed spaces.