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Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

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An Erratum to this article was published on 28 December 2015

Abstract

We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.

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Notes

  1. We remark that to have the left hand of item (1) well defined we can assume (without changing \(\mu ^{1}\)) that \(\mu _{\gamma }^{2}\) is defined in some way, for example \(\mu _{\gamma }^{2}=m\) (the one dimensional Lebesgue measure on the leaf) for each leaf where the density of \(\mu _{x}^{2}\) is null.

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Acknowledgments

S.G. wishes to thank IMPA, PUC, and UFRJ (Rio de Janeiro), where a part of this work has been done, for their warm hospitality. All authors wish to thank Carlangelo Liverani for illuminating discussions about convergence to equilibrium and decay of correlations.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N, Ondina, Salvador, BA, 40170-110, Brazil

    Vitor Araújo

  2. Dipartimento di Matematica Applicata, Universita di Pisa, Via Buonarroti 1, Pisa, Italy

    Stefano Galatolo

  3. Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, Rio de Janeiro, 21945-970, Brazil

    Maria José Pacifico

Authors
  1. Vitor Araújo
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  2. Stefano Galatolo
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  3. Maria José Pacifico
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Correspondence to Maria José Pacifico.

Additional information

Vitor Araújo and Maria José Pacifico were partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ, Balzan Research Project of J. Palis.

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Araújo, V., Galatolo, S. & Pacifico, M.J. Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors. Math. Z. 276, 1001–1048 (2014). https://doi.org/10.1007/s00209-013-1231-0

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