# Finiteness of homoclinic classes on sectional hyperbolic sets

## TL;DR

This paper proves that small perturbations of sectional hyperbolic sets on compact manifolds result in a finite number of homoclinic classes, enhancing understanding of their stability and structure.

## Contribution

It establishes the robust finiteness of homoclinic classes under perturbations in sectional hyperbolic sets, extending previous results on attractors and repellers.

## Key findings

- Homoclinic classes are finite under small perturbations.
- Attractors and repellers are special cases of homoclinic classes.
- Finiteness results are robust under perturbations.

## Abstract

We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain robustly finiteness of homoclinic classes on this scenary. Moreover, since attractor and repeller sets are particular cases of homoclinic classes, this result improve (A. M. L\'opez B, Finiteness and existence of attractors and repellers on sectional hyperbolic sets, Discrete and Continuous Dynamical Systems-A 37).