Tori with hyperbolic dynamics in 3-manifolds

TL;DR

This paper classifies 3-manifolds that admit invariant hyperbolic tori under diffeomorphisms, revealing only specific manifolds support such structures, which impacts the understanding of partially hyperbolic dynamics.

Contribution

It provides a classification of 3-manifolds with Anosov tori, showing only three types admit such hyperbolic invariant tori, advancing the understanding of hyperbolic dynamics in 3-manifolds.

Findings

Only three irreducible 3-manifolds admit Anosov tori.

Implications for partially hyperbolic dynamics and invariant foliations.

First example of a non-dynamically coherent partially hyperbolic diffeomorphism with 1D center.

Abstract

Let M be a closed orientable irreducible 3-manifold, and let f be a diffeomorphism over M. We call an embedded 2-torus T an Anosov torus if it is invariant and the induced action of f over \pi_1(T) is hyperbolic. We prove that only few irreducible 3-manifolds admit Anosov tori: (1) the 3-torus, (2) the mapping torus of -id, and (3) the mapping torus of hyperbolic automorphisms of the 2-torus. This has consequences for instance in the context of partially hyperbolic dynamics of 3-manifolds: if there is an invariant center-unstable foliation, then it cannot have compact leaves [19]. This has lead to the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle [19].