Geodesic rays, the “Lion-Man” game, and the fixed point property

Abstract

This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main result holds in the setting of CAT(0) spaces that are additionally Gromov hyperbolic. IntroductionGiven a metric space X, the existence of either a geodesic ray in X or, more generally, of a curve that approximates a geodesic ray (e.g., a directional curve or a quasi-geodesic ray, see Section 3 for precise definitions), is connected to some intrinsic topological and geometric properties of the space. Such properties include the presence of upper or lower curvature bounds in the sense of Alexandrov [5,12,24], the Gromov hyperbolicity condition [5], the betwenness property [23], or, for normed spaces, the reflexivity [34]. On the other hand, the existence of different types of curves can be used in the analysis of several problems among which we mention the characterization of the fixed point property either for continuous [20,19,24] or nonexpansive mappings [34,12,27], and some pursuit-evasion games [2,23].Geometric properties of a set stand behind the existence of fixed points for continuous or nonexpansive mappings defined on the set in question. This fact has prompted a very fruitful research direction because, among other reasons, it leads to challenging questions regarding the geometry of Banach spaces (see, e.g., the monographs [3,13]). Klee [20] was a pioneer in the use of topological rays as a tool to characterize compactness of convex subsets of a locally convex metrizable linear topological space by means of the fixed point property for continuous mappings (see also [10]). Counterparts of this result for geodesic metric spaces were proved in [24,25]. The notion of directional curves was introduced in [34] to characterize the approximate fixed point property for nonexpansive mappings in convex subsets of a class of metric spaces which includes, in particular, Banach spaces or complete Busemann convex geodesic spaces. Further results in this direction were proved in [19,27]. Namely, the absence of geodesic rays from a closed and convex set is equivalent to its fixed point property for continuous mappings in complete R-trees, while for the more general setting of complete CAT(κ) spaces with κ < 0, it is equivalent to its fixed point property for nonexpansive mappings.Persuit-evasion problems are not only interesting because their analysis requires the use of tools from different areas of mathematics, but also because of their application in other disciplines such as robotics or the modeling of animal behavior. In a recent survey, Chung et al.[8] classify persuit-evasion games based on the environment where the game is played, the information available to the players, the restrictions imposed on the players' motion, and the way of defining capture.The game that we consider in this work was proposed in [1, 2] (see also [4]) and is a discrete variant of the cla...