Building Anosov flows on 3-manifolds

TL;DR

This paper introduces a method to construct various types of Anosov flows on 3-manifolds by gluing neighborhoods of hyperbolic sets, leading to multiple novel examples and applications in dynamical systems.

Contribution

It provides a new technique for building Anosov flows on 3-manifolds and demonstrates several applications, including the existence of manifolds with multiple distinct Anosov flows.

Findings

Constructed 3-manifolds with both transitive and non-transitive Anosov flows.

Built manifolds supporting multiple distinct Anosov vector fields.

Created transitive attractors with prescribed entrance foliations.

Abstract

We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector field admitting infinitely many pairwise non-isotopic trans- verse tori.