Abstract
We study partially hyperbolic attractors ofC 2 diffeomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Hölder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems.
Similar content being viewed by others
References
J. F. Alves,SRB measures for nonhyperbolic systems with multidimensional expansion, PhD thesis, IMPA, 1997.
C. Bonatti and M. Viana,SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics115 (2000), 157–193.
M. Brin and Ya. Pesin,Partially hyperbolic dynamical systems, Izvestiya Akademii Nauk SSSR1 (1974), 177–212.
M. Carvalho,Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems13 (1993), 21–44.
A. A. Castro,Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting central direction, PhD thesis, IMPA, 1998.
D. Dolgopyat,On dynamics of mostly contracting diffeomorphisms, Preprint, 1998.
M. Hirsch, C. Pugh and M. Shub,Invariant manifold, Lecture Notes in Mathematics583, Springer-Verlag, Berlin, 1977.
C. Livernani,Decay of correlations, Annals of Mathematics142 (1995), 239–301.
R. Mañe,Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987.
Ya. Pesin and Ya. Sinai,Gibbs measures for partially hyperbolic attractors, Ergodic Theory and Dynamical Systems2 (1982), 417–438.
C. Pugh and M. Shub,Ergodic attractors, Transactions of the American Mathematical Society312 (1989), 1–54.
M. Shub,Global Stability of Dynamica Systems, Springer-Verlag, Berlin, 1987.
M. Viana,Stochastic dynamics of deterministic systems, Lecture Notes from the XXI Brazilian Mathematical Colloqium, IMPA, Rio de Janeiro, 1997.
L.-S. Young,Statistical properties of dynamicals systems with some hyperbolicity, Annals of Mathematics147 (1998), 585–650.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by IMPA and CNPq-01/2000.
Rights and permissions
About this article
Cite this article
Armando, A., Júnior, C. Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Isr. J. Math. 130, 29–75 (2002). https://doi.org/10.1007/BF02764070
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02764070