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Dominated splitting and zero volume for incompressible threeflows
 Nonlinearity
"... ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able ..."
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ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of BochiMañé (see [26, 13, 9]) and of Newhouse (see [30, 10]) for flows with singularities. That is we obtain for a residual subset of the C 1 incompressible flows on 3manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.
MULTIDIMENSIONAL ROVELLALIKE ATTRACTORS
, 909
"... Abstract. We present a multidimensional flow exhibiting a Rovellalike attractor: a transitive invariant set with a nonLorenzlike singularity accumulated by regular orbits and a multidimensional nonuniformly expanding invariant direction. Moreover, this attractor has a physical measure with full ..."
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Abstract. We present a multidimensional flow exhibiting a Rovellalike attractor: a transitive invariant set with a nonLorenzlike singularity accumulated by regular orbits and a multidimensional nonuniformly expanding invariant direction. Moreover, this attractor has a physical measure with full support but persists along certain submanifolds of the space of vector fields. As in the 3dimensional Rovellalike attractor, this example is not robust. The construction introduces a class of multidimensional dynamics, whose suspension provides the Rovellalike attractor, which are partially hyperbolic, and whose quotient over stable leaves is a multidimensional endomorphism to which BenedicksCarleson type arguments are applied to prove nonuniform expansion. 1.
Rapid mixing for the lorenz attractor and statistical limit laws for their time1 maps. arXiv preprint arXiv:1311.5017
, 2013
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WHAT IS NEW ON LORENZLIKE ATTRACTORS
, 804
"... Abstract. We describe some recent results on the dynamics of Lorenzlike attractors Λ: (1) there is an invariant foliation whose leaves are forward contracted by the flow; (2) there is a positive Lyapunov exponent at every orbit; (3) they are expansive and so sensitive with respect to initial data; ..."
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Abstract. We describe some recent results on the dynamics of Lorenzlike attractors Λ: (1) there is an invariant foliation whose leaves are forward contracted by the flow; (2) there is a positive Lyapunov exponent at every orbit; (3) they are expansive and so sensitive with respect to initial data; (4) they have zero volume if the flow is C 2; (5) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the centerunstable Jacobian. 1.
VOLUME OF PARTIALLY HYPERBOLIC HORSESHOELIKE ATTRACTORS
"... Abstract. We prove that any partially hyperbolic horseshoelike attractor of a C1generic diffeomorphism has zero volume. By modification of Poincare ́ cross section of the geometric model of Lorenz attractor, we build a C1diffeomorphism with a partially hyperbolic horseshoelike attractor of posi ..."
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Abstract. We prove that any partially hyperbolic horseshoelike attractor of a C1generic diffeomorphism has zero volume. By modification of Poincare ́ cross section of the geometric model of Lorenz attractor, we build a C1diffeomorphism with a partially hyperbolic horseshoelike attractor of positive volume. 1. introduction A hyperbolic set is a compact invariant set over which the tangent bundle splits into two invariant subbundles, one is contracting and the other one is expanding. The Lebesgue measure (volume) of hyperbolic sets is an interesting subject considered in many articles. The scenario was begun by the seminal works of Bowen in 70’s. Bowen has proved in [5] that a hyperbolic attractor of positive volume of a C2diffeomorphism does contains some stable and unstable manifolds. On the other hand, he showed in [4] the existence of a C1diffeomorphism admitting a totally disconnected hyperbolic set of positive volume. The issue of the volume and the interior of a hyperbolic sets followed by many authors. For instance, it is shown in [2] that a transitive hyperbolic set which attracts a set with positive volume necessarily attracts a neighborhood of itself. It is also proven in [2] that there are no
Lorentzlike chaotic attractors revisited
, 2009
"... We describe some recent results on the dynamics of singularhyperbolic (Lorenzlike) attractors Λ introduced in [25]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class a ..."
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We describe some recent results on the dynamics of singularhyperbolic (Lorenzlike) attractors Λ introduced in [25]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class are expansive and so sensitive with respect to initial data; (4) they have zero volume if the flow is C², or else the flow is globally hyperbolic; (5) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the centerunstable Jacobian; (6) the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; (7) the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential.