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

TL;DR

The paper constructs a specific example of a free Z-action by isometries on a compact metric space that cannot be embedded into a cubical shift, addressing a longstanding open problem in topological dynamics.

Contribution

It provides an existence proof of a non-embeddable free Z-action on a compact metric space, using advanced topological and algebraic techniques.

Findings

Constructed a specific example of a non-embeddable free Z-action

Demonstrated the limitations of embedding certain dynamical systems into cubical shifts

Connected the problem to a Borsuk-Ulam type theorem and p-adic completions

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.