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Singular-hyperbolic attractors are chaotic

Vitor Araujo·Maria Jose Pacifico·Enrique Pujals·Marcelo Viana
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Abstract

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.

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Vitor Araujo, Maria Jose Pacifico, Enrique Pujals, Marcelo Viana. 2007-03-22. Singular-hyperbolic attractors are chaotic. https://doi.org/10.1090/s0002-9947-08-04595-9

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