Dynamics on the Morse boundary

Abstract

Let X be a proper geodesic metric space and let G be a group of isometries of X which acts geometrically. Cordes constructed the Morse boundary of X which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in G by their fixed points on the Morse boundary $$\partial _MX$$ . The dynamics on the Morse boundary is very similar to that of a $$\delta $$ -hyperbolic space. In particular, we show that the action of G on $$\partial _MX$$ is minimal if G is not virtually cyclic. We also get a uniform convergence result on the Morse boundary which gives us a weak north-south dynamics for a Morse isometry. This generalizes the work of Murray in the case of the contracting boundary of a CAT(0) space.