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Extreme Value Laws for Dynamical Systems with Countable Extremal Sets

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Abstract

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set \(\mathcal M\), i.e.,the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set \(\mathcal M\) may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in Freitas (Adv Math 231(5): 2626–2665, 2012, Stoch Process Appl 125(4): 1653–1687, 2015).

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Notes

  1. We note that the sets \(I_j, I_{j,n}\) are not symmetric with respect to the centre \(\xi _j\) but by changing the metric we can still identify the sets \(I_j, I_{j,n}\) as balls around \(\xi _j\).

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Acknowledgements

DA was partially supported by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. ACMF and JMF were partially supported by FCT projects FAPESP/19805/2014 and PTDC/MAT-CAL/3884/2014. FBR was supported by BREUDS, a Brazilian-European partnership of the FP7-PEOPLE-2012-IRSES program, with project Number 318999, which is supported by an FP7 International International Research Staff Exchange Scheme (IRSES) grant of the European Union. All authors were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. JMF would like to thank Mike Todd for careful reading and useful suggestions.

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Authors and Affiliations

  1. Instytut Matematyczny Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-656, Warszawa, Polska

    Davide Azevedo

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

    Ana Cristina Moreira Freitas

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

    Jorge Milhazes Freitas

  4. Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500 - Prédio 43-111 - Agronomia, Caixa Postal 15080, Porto Alegre, RS, 91509-900, Brasil

    Fagner B. Rodrigues

Authors
  1. Davide Azevedo
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  2. Ana Cristina Moreira Freitas
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  3. Jorge Milhazes Freitas
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  4. Fagner B. Rodrigues
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Correspondence to Jorge Milhazes Freitas.

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Azevedo, D., Freitas, A.C.M., Freitas, J.M. et al. Extreme Value Laws for Dynamical Systems with Countable Extremal Sets. J Stat Phys 167, 1244–1261 (2017). https://doi.org/10.1007/s10955-017-1767-1

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  • DOI: https://doi.org/10.1007/s10955-017-1767-1

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