TAILIEUCHUNG - Đề tài " Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers "

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. . | Annals of Mathematics Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers By C. A. Morales M. J. Pacifico and E. R. Pujals Annals of Mathematics 160 2004 375 432 Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers By C. A. Morales M. J. Pacifico and E. R. Pujals 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. 1. Introduction A long-time goal in the theory of dynamical systems has been to describe and characterize systems exhibiting dynamical properties that are preserved under small perturbations. A cornerstone in this direction was the Stability Conjecture Palis-Smale 30 establishing that those systems that are identical up to a continuous change of coordinates of phase space to all nearby systems are characterized as the hyperbolic ones. Sufficient conditions for structural stability were proved by Robbin 36 for r 2 de Melo 6 and Robinson 38 for r 1 . Their necessity was reduced to showing that structural stability implies hyperbolicity Robinson 37 . And that was proved by Mane 23 in the discrete case for r 1 and Hayashi 13 in the framework of flows for r 1 . This has important consequences because there is a rich theory of hyperbolic systems describing their geometric and ergodic properties. In particular by Smale s spectral decomposition theorem 39 one has a description of the nonwandering set of a structural stable system as a finite number of disjoint .

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