We introduce new tools for studying boundary dynamics of a co-compact proper isometric action of a group G on a CAT(0) space X. In this situation, X has a G-equivariant compactification CX, with the residual set BX (also known as the visual boundary) carrying an additional metric structure, the Tits metric, which induces a far stronger, often non-compact, topology -- denote that by TX.

Results of Swenson and Kleiner imply that the Tits boundary TX has finite geometric dimension and contains isometrically embedded round spheres of this dimension, bounding flats in X.

An important set of questions due to Ballman and Buyalo, motivated by Ballman's rank rigidity theorem for Hadamard manifolds, is about the extent to which the dynamics of the action of G on X determines the geometry of X. Intuitively, one could view this as a question regarding the interaction between dynamics of G and the density/tightness of the family of maximal flats in X / round spheres in TX.

In order to capture limit effects of the group action on BX, we study an extension of the action of G on X to an action of its Stone-Cech compactification on CX, and produce maximal flats whose boundaries intersect every minimal G-invariant subset of CX.

As a quick application, we improve on the Ballmann-Buyalo and Sweson- Papasoglu bounds on the diameter of the Tits boundary of a higher rank group. More seriously, one can obtain a necessary and sufficient dynamical condition for G to be virtually-Abelian, and we formulate a new approach to Ballmann’s rank rigidity conjectures.