Le Conte de la Mesure sur les Complexes Cubiques CAT(0)

We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite dimensional case, we prove that there must be an orbit of cardinality $2^N$, where $N$ is bounded by the dimension. This is a slight extension of the author's previous Tits' Alternative.