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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Singular-hyperbolic attractors are chaotic


Authors: V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana
Journal: Trans. Amer. Math. Soc. 361 (2009), 2431-2485
MSC (2000): Primary 37C10; Secondary 37C40, 37D30
Published electronically: December 3, 2008
MathSciNet review: 2471925
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Abstract: We prove that a singular-hyperbolic attractor of a $ 3$-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, 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 is 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.

In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are $ u$-Gibbs states.


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Additional Information

V. Araujo
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970, Rio de Janeiro, RJ-Brazil – and – Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: vitor.araujo@im.ufrj.br, vdaraujo@fc.up.pt

M. J. Pacifico
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, Brazil
Email: pacifico@im.ufrj.br, pacifico@impa.br

E. R. Pujals
Affiliation: IMPA, Estrada D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: enrique@impa.br

M. Viana
Affiliation: IMPA, Estrada D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: viana@impa.br

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04595-9
PII: S 0002-9947(08)04595-9
Keywords: Singular-hyperbolic attractor, Lorenz-like flow, physical measure, expansive flow, equilibrium state
Received by editor(s): July 5, 2006
Received by editor(s) in revised form: March 27, 2007
Published electronically: December 3, 2008
Additional Notes: The first author was partially supported by CMUP-FCT (Portugal), CNPq (Brazil) and grants BPD/16082/2004 and POCI/MAT/61237/2004 (FCT-Portugal) while enjoying a post-doctorate leave from CMUP at PUC-Rio and IMPA
The second, third and fourth authors were partially supported by PRONEX, CNPq and FAPERJ-Brazil
Article copyright: © Copyright 2008 by the authors