Intransitive sectional-Anosov flows on 3-manifolds

TL;DR

This paper constructs examples of intransitive sectional-Anosov flows with dense periodic orbits and a specified number of equilibria on 3-manifolds, characterized by their union of homoclinic classes.

Contribution

It demonstrates the existence of Venice masks with any finite number of equilibria on certain 3-manifolds, expanding understanding of sectional-Anosov flows.

Findings

Existence of Venice masks with n equilibria for each positive integer n

Maximal invariant set as a finite union of homoclinic classes

Intersections of homoclinic classes contained in closures of unstable manifolds

Abstract

For each $n\in\mathbb{Z}^+$, we show the existence of Venice masks (i.e. intransitive sectional-Anosov flows with dense periodic orbits) containing $n$ equilibria on certain compact 3-manifolds. These examples are characterized because of the maximal invariant set is a finite union of homoclinic classes. Here, the intersection between two different homoclinic classes is contained in the closure of the union of unstable manifolds of the singularities.