Robust transitivity without sectional-hyperbolicity

TL;DR

This paper constructs higher-dimensional vector fields with robustly transitive singular attractors that are not sectional-hyperbolic, yet remain singular-hyperbolic, advancing understanding of dynamical systems' hyperbolic properties.

Contribution

It provides the first examples of such systems in dimensions five and above, improving previous constructions and expanding the class of known dynamical behaviors.

Findings

Existence of robust transitive singular attractors not sectional-hyperbolic in dimensions ≥5

Attractors are singular-hyperbolic despite not being sectional-hyperbolic

Advances the understanding of hyperbolic structures in dynamical systems

Abstract

For any integer $n \geq 5$, we construct an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the first such examples improving some features of the constructions in [17, 32].