CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X. (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α,β∈∂X with α≠β, (1) lim sup{d∂X(gα,gβ)∣g∈G}>0 and (2) lim inf{d∂X(gα,gβ)∣g∈G}=0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.