4 N ov 2 00 5 SINGULAR - HYPERBOLIC ATTRACTORS ARE CHAOTIC
نویسندگان
چکیده
We prove that a singular-hyperbolic (or Lorenz-like) attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. In particular these results show that both the flow defined by the Lorenz equations and the geometric Lorenz flows are expansive and have physical measures which are u-Gibbs states.
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