On the essential hyperbolicity of sectional-Anosov flows

TL;DR

This paper demonstrates that sectional-Anosov flows on compact 3-manifolds have a finite set of hyperbolic attractors and singularities with dense basins, advancing understanding of their hyperbolic structure and dynamical properties.

Contribution

It establishes the essential hyperbolicity of sectional-Anosov flows on 3-manifolds and links topological features to dynamical behavior.

Findings

Finite collection of hyperbolic attractors and singularities

Basins of these attractors are dense in the manifold

Characterization of sensitivity to initial conditions

Abstract

We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving \cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.