Contributions to the Theory of Asymptotically Sectional Hyperbolic Flows

TL;DR

This paper advances the understanding of asymptotically sectional-hyperbolic flows by proving key properties such as sectional-hyperbolicity of star ASH attractors, entropy flexibility, and bounds on periodic orbit growth.

Contribution

It proves that star ASH attractors are sectional-hyperbolic, establishes entropy flexibility, and provides bounds on periodic orbit growth in ASH attractors.

Findings

Star ASH attractors are sectional-hyperbolic.

All ASH attractors have entropy flexibility.

Lower bounds for periodic orbit growth in ASH attractors.

Abstract

In this paper, we make several contributions to the theory of asymptotically sectional-hyperbolic (ASH) flows. First, we prove that every star ASH attractor for a $C^2$ vector field is, in fact, sectional-hyperbolic (SH). Second, we establish that all ASH attractors exhibit the property of entropy flexibility. Additionally, we show that any ASH attractor for three-dimensional vector fields is entropy-expansive and admits periodic orbits. Finally, we provide a lower bound for the growth rate of periodic orbits in an ASH attractor.