# On the intersection of homoclinic classes in intransitive sectional-Anosov flows

## TL;DR

This paper investigates the structure of homoclinic classes and omega-limit sets in sectional-Anosov flows, revealing conditions under which these sets decompose into hyperbolic and singular components, especially in nontransitive flows.

## Contribution

It establishes new structural results about the intersection of homoclinic classes and the nature of omega-limit sets in sectional-Anosov flows supported on 3-manifolds.

## Key findings

- Omega-limit set of non-recurrent points in certain flows is a closed orbit.
- Intersections of homoclinic classes decompose into singular points, hyperbolic sets, and regular points with specific limit set properties.

## Abstract

We show that if X is a Venice mask (i.e. nontransitive sectional-Anosov flow with dense periodic orbits) supported on a compact 3-manifold, then the omega-limit set of every non-recurrent point in the unstable manifold of some singularity is a closed orbit. In addition, we prove that the intersection of two different homoclinic classes in the maximal invariant set of a sectional-Anosov flow can be decomposed as the disjoint union of, singular points, a non-singular hyperbolic set, and regular points whose alpha-limit set and omega-limit set is formed by singular points or hyperbolic sets.