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Extreme Value Laws in Dynamical Systems for Non-smooth Observations

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Abstract

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.

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Authors and Affiliations

  1. Centro de Matemática & Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    Ana Cristina Moreira Freitas

  2. Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Jorge Milhazes Freitas

  3. Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    Mike Todd

Authors
  1. Ana Cristina Moreira Freitas
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  2. Jorge Milhazes Freitas
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  3. Mike Todd
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Corresponding author

Correspondence to Jorge Milhazes Freitas.

Additional information

ACMF was partially supported by FCT grant SFRH/BPD/66174/2009. JMF was partially supported by FCT grant SFRH/BPD/66040/2009. MT was partially supported by FCT grant SFRH/BPD/26521/2006 and NSF grants DMS 0606343 and DMS 0908093. All three authors were supported by FCT through CMUP and PTDC/MAT/099493/2008.

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Freitas, A.C.M., Freitas, J.M. & Todd, M. Extreme Value Laws in Dynamical Systems for Non-smooth Observations. J Stat Phys 142, 108–126 (2011). https://doi.org/10.1007/s10955-010-0096-4

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  • DOI: https://doi.org/10.1007/s10955-010-0096-4

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