J an 2 00 9 Chapter 1 Lorenz-like chaotic attractors revised

We describe some recent results on the dynamics of singularhyperbolic (Lorenz-like) attractorsΛ introduced in [25]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exi sts a positive Lyapunov exponent at every orbit; (3) attractors in this class are exp ansive 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 me asure whose support is the whole attractor and which is the equilibrium state wit h respect to the centerunstable Jacobian; (6) the hitting time associated to a geom etric Lorenz attractor satisfies a logarithm law; (7) the rate of large deviations fo r the physical measure on the ergodic basin of a geometric Lorenz attractor is exponen tial.