The Robust Singular

Robust transitivity is the strongest form of mixing for an attractor: every open set eventually visits every other open set, and this property survives all small perturbations of the system. In three dimensions, all known robustly transitive attractors with singularities are sectional-hyperbolic — meaning every two-dimensional section through the attractor expands uniformly. The Lorenz attractor is the prototype. The natural conjecture: robust transitivity with singularities requires sectional hyperbolicity.

Arbieto, Britto, Morales, and Rego (arXiv:2603.16476) construct counterexamples. For every dimension n ≥ 5, they build a C¹ vector field with a robustly transitive singular attractor that is not sectional-hyperbolic. The attractor is singular-hyperbolic — a weaker condition requiring only that the singularity dominates the contraction — but the uniform expansion on two-dimensional sections fails.

The construction works by exploiting what higher dimensions allow. In three dimensions, the singular point forces a rigid geometric structure on the flow around it — there’s essentially only one way to organize the expanding and contracting directions. In five or more dimensions, there’s enough room for the expanding directions to twist relative to each other in ways that preserve topological mixing but break the uniform expansion condition. The robustness comes from the singular hyperbolicity; the failure of sectional hyperbolicity comes from the dimensional freedom.

The structural insight: the tight connection between robust mixing and uniform expansion is a low-dimensional accident. In three dimensions, the geometric constraints are so severe that sectional hyperbolicity is essentially forced. In higher dimensions, the attractor has enough geometric freedom to sustain robust mixing through a weaker mechanism. The mixing is topological — it doesn’t need every section to expand, just enough structure to prevent the attractor from splitting.

Arbieto, Britto, Morales, & Rego, “Robust transitivity without sectional-hyperbolicity,” arXiv:2603.16476 (2026).