Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows

TL;DR

This paper demonstrates that generic three-dimensional flows exhibit either infinitely many sinks or hyperbolic/singular-hyperbolic attractors with full measure basins, and explores their structural properties and approximations.

Contribution

It establishes the generic dichotomy for three-dimensional flows and characterizes the accumulation points of sinks, advancing understanding of flow dynamics and attractor structures.

Findings

Generic flows have either infinitely many sinks or hyperbolic/singular-hyperbolic attractors.

The set of sink accumulation points lacks dominated splitting in the orientable case.

Any three-dimensional flow can be approximated by flows with homoclinic tangencies or singular-Axiom A flows.

Abstract

We prove that every $C^1$ generic three-dimensional flow has either infinitely many sinks, or, infinitely many hyperbolic or singular-hyperbolic attractors whose basins form a full Lebesgue measure set. We also prove in the orientable case that the set of accumulation points of the sinks of a $C^1$ generic three-dimensional flow has no dominated splitting with respect to the linear Poincar\'e flow. As a corollary we obtain that every three-dimensional flow can be $C^1$ approximated by flows with homoclinic tangencies or by singular-Axiom A flows.