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arXiv · 1308.1734

Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows

A. Arbieto·A. Rojas·B. Santiago
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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 Poincaré 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.

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A. Arbieto, A. Rojas, B. Santiago. 2013-08-08. Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows. https://arxiv.org/abs/1308.1734

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