Geodesic
Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers
Carlos Arnoldo Morales1 ; Maria José Pacifico1 ; Enrique R. Pujals1
1 Department of Mathematics, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro-RJ, Brazil
Annals of mathematics, Tome 160 (2004) no. 2, pp. 375-432.

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Inspired by Lorenz’ remarkable chaotic flow, we describe in this paper the structure of all $C^1$ 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 $C^1$ 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 $C^1$ robust attractor resembles a geometric Lorenz attractor.
DOI : 10.4007/annals.2004.160.375

Carlos Arnoldo Morales 1 ; Maria José Pacifico 1 ; Enrique R. Pujals 1

1 Department of Mathematics, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro-RJ, Brazil 
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Carlos Arnoldo Morales; Maria José Pacifico; Enrique R. Pujals. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Annals of mathematics, Tome 160 (2004) no. 2, pp. 375-432. doi : 10.4007/annals.2004.160.375. http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.375/

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