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Transitive Anosov flows on non-compact manifolds

Thomas Barthelmé, Lingfeng Lu

TL;DR

This work investigates transitivity of Anosov flows on non-compact 3-manifolds, proving that lifted flows to certain abelian covers are transitive precisely when the base flow is $Y$-full, and leveraging virtual RFRS properties to guarantee the existence of such covers in many cases. It extends the equivalence between transitivity and density of periodic orbits from maximal abelian covers to general abelian covers, and shows that for $\mathbb{R}$-covered flows lifts to regular covers are either transitive or wandering, with a complete obstruction classification. The authors also construct non-compact transitive Anosov flows where every periodic orbit is unique in its free homotopy class, providing a flow analogue of Barbot–Fenley’s suspension characterization in a non-compact setting. Collectively, the results broaden the understanding of transitivity and suspension-type characterizations of Anosov dynamics beyond compact manifolds, and supply concrete methods to realize transitive lifts on infinite covers.

Abstract

In this article we study topological transitivity of Anosov flows on non-compact 3-manifolds. We provide homological conditions under which the lifts of a transitive Anosov flow to certain infinite covers of the manifold remain transitive. With some deep results in 3-manifold topology, we then deduce that such cover can be obtained for any non-graph manifold admitting a transitive Anosov flow. Moreover, for a large class of Anosov flows known as $\mathbb{R}$-covered, which are always transitive, we show that their lifts to any regular covers are either transitive or consist exclusively of wandering orbits. Finally, we construct a family of transitive Anosov flows on non-compact manifolds that satisfy a homotopical characterization of suspension flows, answering the flow version of a question concerning the existence of transitive Anosov diffeomorphisms on non-compact manifolds.

Transitive Anosov flows on non-compact manifolds

TL;DR

This work investigates transitivity of Anosov flows on non-compact 3-manifolds, proving that lifted flows to certain abelian covers are transitive precisely when the base flow is -full, and leveraging virtual RFRS properties to guarantee the existence of such covers in many cases. It extends the equivalence between transitivity and density of periodic orbits from maximal abelian covers to general abelian covers, and shows that for -covered flows lifts to regular covers are either transitive or wandering, with a complete obstruction classification. The authors also construct non-compact transitive Anosov flows where every periodic orbit is unique in its free homotopy class, providing a flow analogue of Barbot–Fenley’s suspension characterization in a non-compact setting. Collectively, the results broaden the understanding of transitivity and suspension-type characterizations of Anosov dynamics beyond compact manifolds, and supply concrete methods to realize transitive lifts on infinite covers.

Abstract

In this article we study topological transitivity of Anosov flows on non-compact 3-manifolds. We provide homological conditions under which the lifts of a transitive Anosov flow to certain infinite covers of the manifold remain transitive. With some deep results in 3-manifold topology, we then deduce that such cover can be obtained for any non-graph manifold admitting a transitive Anosov flow. Moreover, for a large class of Anosov flows known as -covered, which are always transitive, we show that their lifts to any regular covers are either transitive or consist exclusively of wandering orbits. Finally, we construct a family of transitive Anosov flows on non-compact manifolds that satisfy a homotopical characterization of suspension flows, answering the flow version of a question concerning the existence of transitive Anosov diffeomorphisms on non-compact manifolds.

Paper Structure

This paper contains 10 sections, 18 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Transitivity on abelian covers: A homological condition
  3. Transitivity on regular covers: the $\mathbb{R}$-covered case
  4. Transitive Anosov flows and the homotopical characterization of suspension flows
  5. Anosov-like actions and structure of the orbit space
  6. Homological conditions for transitivity
  7. Y-full flows
  8. Manifolds with RFRS fundamental groups and their covers
  9. $\mathbb{R}$-covered flows and regular covers.
  10. Non-compact Anosov flows and homotopical characterization of suspensions

Key Result

Theorem A

Let $M$ be a compact 3-manifold that admits a transitive Anosov flow. If $M$ is not a graph manifoldA graph manifold is a $P^2$-irreducible (irreducible and contains no two-sided $\mathbb{R} \mathrm{P}^2$'s) $3$-manifold such that its JSJ decomposition consists only of Seifert fibered pieces, see e.

Figures (3)

  • Figure 1: Two solid leaves making a perfect fit (a); and a lozenge (b)
  • Figure 2: A local leaf configuration of a Smale chain
  • Figure 3: Induced foliations on the transverse torus (a); and the projections in the orbit space (b).

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1.1: Barbot, Fenley
  • Theorem D
  • Remark 1.2
  • Definition 2.1
  • Theorem 2.2: BM_book, Theorem 2.7.2
  • Definition 2.3
  • Definition 2.4
  • ...and 33 more