# Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers | Annals of Mathematics

Abstract Inspired by Lorenz’ remarkable chaotic flow, we describe in this paper the structure of all C1 robust transitive sets with singularities for flows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C1 robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any C1 robust attractor resembles a geometric Lorenz attractor.

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### 数学年刊（Annals of Mathematics）

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