Sectional Anosov flows: Existence of Venice masks with two singularities

TL;DR

This paper proves the existence of Venice masks, a special type of sectional Anosov flows with two equilibria, on certain 3-manifolds, expanding the known examples beyond those with one or three singularities.

Contribution

It introduces new examples of Venice masks with two singularities, differing from previously known cases, and analyzes their homoclinic class intersections.

Findings

Existence of Venice masks with two equilibria on specific 3-manifolds.

Different intersection properties of homoclinic classes in these examples.

Extension of known classifications of sectional Anosov flows.

Abstract

We show the existence of Venice masks (i.e. nontransitive sectional Anosov flows with dense periodic orbits), containing two equilibria on certain compact 3-manifolds. Indeed, the only known examples of venice masks have one or three singularities, and they are characterized by having two properties: are the union non disjoint of two homoclinic classes and whose intersection is the closure of the unstable manifold of a singularity. Thus, we present two type of examples containing two equilibria in which the homoclinic classes composing their maximal invariant set intersect in a very different way.