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Rapid Mixing for the Lorenz Attractor and Statistical Limit Laws for Their Time-1 Maps

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Abstract

We prove that every geometric Lorenz attractor satisfying a strong dissipativity condition has superpolynomial decay of correlations with respect to the unique Sinai–Ruelle–Bowen measure. Moreover, we prove the central limit theorem and almost sure invariance principle for the time-1 map of the flow of such attractors. In particular, our results apply to the classical Lorenz attractor.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil

    V. Araújo & P. Varandas

  2. Institute of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

    I. Melbourne

Authors
  1. V. Araújo
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  2. I. Melbourne
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  3. P. Varandas
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Corresponding author

Correspondence to V. Araújo.

Additional information

Communicated by M. Lyubich

I.M. was partially supported by a Santander Staff Mobility Award at the University of Surrey, by a European Advanced Grant StochExtHomog (ERC AdG 320977) and by CNPq (Brazil) through PVE Grant No. 313759/2014-6. V.A. and P.V. were partially supported by CNPq, PRONEX-Dyn.Syst. and FAPESB (Brazil). This research has been supported in part by EU Marie-Curie IRSES Brazilian–European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS). We are grateful to Oliver Butterley for very helpful discussions regarding the regularity of the strong stable foliation and to the anonymous referees for pointing out several details in our argument that had to be addressed, greatly improving the final text of this work.

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Araújo, V., Melbourne, I. & Varandas, P. Rapid Mixing for the Lorenz Attractor and Statistical Limit Laws for Their Time-1 Maps. Commun. Math. Phys. 340, 901–938 (2015). https://doi.org/10.1007/s00220-015-2471-0

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