Irreducibility of Anosov diffeomorphisms

Determine whether every Anosov diffeomorphism f: M -> M on a compact manifold M yields an irreducible dynamical system (M, f), i.e., whether all Anosov diffeomorphisms are irreducible.

Background

Anosov diffeomorphisms form a fundamental class within Smale spaces. The paper proves that every irreducible Smale space satisfies the Garden of Eden theorem, which in particular covers all Anosov diffeomorphisms on tori because these are known to be irreducible.

Beyond tori, it remains unsettled whether irreducibility holds for all Anosov diffeomorphisms. The authors note classical equivalences for Anosov systems—irreducibility, topological mixing, non-wandering, and density of periodic points—and recall that all known manifolds admitting Anosov diffeomorphisms are infra-nilmanifolds, for which irreducibility is known by Manning. The general case, however, is open.