Mathematics – Dynamical Systems
Scientific paper
2005-11-14
Transactions of the American Mathematical Society, 361 (2009), 2431-2485
Mathematics
Dynamical Systems
55 pages, extra figures (now a total of 16), major rearrangement of sections and corrected proofs, improved introduction
Scientific paper
10.1090/S0002-9947-08-04595-9
We prove that a singular-hyperbolic 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. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.
Araujo Vitor
Pacifico Maria Jose
Pujals Enrique
Viana Marcelo
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