A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shift

Abstract

We construct an example announced in the title. It answers in a strong way a well-known open problem in topological dynamics. In fact our construction is an existence theorem. It is based on a Borsuk-Ulam type theorem whose proof heavily relies on p-adic completions of G-complexes and the equivariant Sullivan conjecture.