Locally connected Smale spaces, pinched spectrum, and infra-nilmanifolds

Abstract

We show that if (X,f) is a locally connected Smale space (e.g., a basic set of an Axiom-A diffeomorphism) such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X,f) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.