On robust expansiveness for sectional hyperbolic attracting sets

TL;DR

This paper establishes that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive under certain conditions, extending previous results and linking robustness of transitivity, hyperbolicity, and chaos.

Contribution

It proves the robustness of expansiveness for sectional-hyperbolic attracting sets and characterizes robust transitivity and chaos in low-dimensional flows.

Findings

Sectional-hyperbolic attracting sets are robustly expansive under strong dissipative conditions.

A robustly transitive attractor is sectional-hyperbolic if and only if it is robustly expansive.

In 3-flows, attracting sets are singular-hyperbolic if and only if they are robustly chaotic.

Abstract

We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions).