On closed manifolds admitting an Anosov diffeomorphism but no expanding map

TL;DR

This paper constructs explicit 12-dimensional nilmanifolds with Anosov diffeomorphisms but no expanding maps, demonstrating the minimal dimension for such examples within infra-nilmanifolds and exploring their algebraic properties.

Contribution

It provides the first explicit low-dimensional examples of such manifolds and establishes their minimality in the class of infra-nilmanifolds, advancing understanding of Anosov dynamics.

Findings

Constructed explicit 12-dimensional nilmanifolds with Anosov diffeomorphisms but no expanding maps

Proved these examples have the smallest possible dimension among infra-nilmanifolds

Developed a method to construct positive gradings from eigenvalues using Galois group actions

Abstract

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.