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

TL;DR

This paper proves that certain locally connected Smale spaces with a lifted local product structure are topologically conjugate to hyperbolic infra-nilmanifold automorphisms, generalizing known results about Anosov diffeomorphisms with pinched spectrum.

Contribution

It establishes a topological conjugacy between specific Smale spaces and hyperbolic infra-nilmanifold automorphisms, extending classical theorems to a broader class.

Findings

Smale spaces with lifted local product structures are conjugate to infra-nilmanifold automorphisms.

Generalization of Brin and Manning's theorem to Smale spaces.

Locally connected codimension one Smale spaces are conjugate to hyperbolic torus automorphisms.

Abstract

We show that if (X, f) is a locally connected Smale space 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.