Some properties of singular hyperbolic attractors.

*(English. Russian original)*Zbl 1174.37008
Sb. Math. 200, No. 1, 35-76 (2009); translation from Mat. Sb. 200, No. 1, 37-80 (2009).

The concept of a singular hyperbolic attractor was introduced in 1998 by C. A. Morales, M. J. Pacífico, and E. R. Pujals [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 81–86 (1998; Zbl 0918.58036)] and D. V. Turaev and L. P. Shil’nikov [Sb. Math. 189, No. 2, 137–160 (1998); translation from Mat. Sb. 189, No. 2, 291–314 (1998; Zbl 0927.37017)]. This definition was motivated by the celebrated Lorenz system and the notion of a hyperbolic attractor. Yet another definition in terms of an invariant system of cones in the dual space has been suggested later by the author [Sb. Math. 196, No. 4, 561–594 (2005); translation from Mat. Sb. 196, No. 4, 99–134 (2005; Zbl 1101.37022)].

The purpose of this nicely written paper is to explore properties of singular hyperbolic attractors which are then used to establish existence of invariant measures of Sinaĭ-Bowen-Ruelle type for singular hyperbolic flows. One of the principal goals is to prove existence of strong unstable spaces and manifolds on a sufficiently representative set. A detailed analysis of Sinaĭ-Bowen-Ruelle invariant measures will be provided in forthcoming papers.

The purpose of this nicely written paper is to explore properties of singular hyperbolic attractors which are then used to establish existence of invariant measures of Sinaĭ-Bowen-Ruelle type for singular hyperbolic flows. One of the principal goals is to prove existence of strong unstable spaces and manifolds on a sufficiently representative set. A detailed analysis of Sinaĭ-Bowen-Ruelle invariant measures will be provided in forthcoming papers.

Reviewer: Yuri V. Rogovchenko (Kalmar)

##### MSC:

37D10 | Invariant manifold theory for dynamical systems |

37C10 | Dynamics induced by flows and semiflows |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |

37C75 | Stability theory for smooth dynamical systems |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37D30 | Partially hyperbolic systems and dominated splittings |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |