Singular-hyperbolic attractors are chaotic
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 -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.
Cite
@article{arxiv.math/0511352,
title = {Singular-hyperbolic attractors are chaotic},
author = {Vitor Araujo and Maria Jose Pacifico and Enrique Pujals and Marcelo Viana},
journal= {arXiv preprint arXiv:math/0511352},
year = {2009}
}
Comments
55 pages, extra figures (now a total of 16), major rearrangement of sections and corrected proofs, improved introduction