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Laws of rare events for deterministic and random dynamical systems


Authors: Hale Aytaç, Jorge Milhazes Freitas and Sandro Vaienti
Journal: Trans. Amer. Math. Soc. 367 (2015), 8229-8278
MSC (2010): Primary 37A50, 60G70, 37B20, 60G10, 37A25, 37H99
DOI: https://doi.org/10.1090/S0002-9947-2014-06300-9
Published electronically: November 10, 2014
MathSciNet review: 3391915
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Abstract: The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non-periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against $ L^1$ observables will play a central role in our investigations.


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Additional Information

Hale Aytaç
Affiliation: Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Address at time of publication: Universidade Federal da Bahia, Instituto de Matemática, Av. Adhemar de Barros, S/N, Ondina, 40170-110 Salvador-BA, Brazil
Email: aytach@fc.up.pt

Jorge Milhazes Freitas
Affiliation: Centro de Matemática & Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: jmfreita@fc.up.pt

Sandro Vaienti
Affiliation: Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France – and – Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France
Email: vaienti@cpt.univ-mrs.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06300-9
Keywords: Random dynamical systems, extreme values, hitting times statistics, extremal index
Received by editor(s): January 31, 2013
Received by editor(s) in revised form: September 13, 2013
Published electronically: November 10, 2014
Additional Notes: The first author was partially supported by FCT (Portugal) grant SFRH/BD/33371/2008
The first and second authors were supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT and CMUP under the project PEst-C/MAT/UI0144/2011.
The second author was partially supported by FCT grant SFRH/BPD/66040/2009 and by FCT project PTDC/MAT/099493/2008
The third author was supported by the CNRS-PEPS Mathematical Methods of Climate Theory and by the ANR-Project Perturbations; part of this work was done while he was visiting the Centro de Modelamiento Matemático, UMI2807, in Santiago de Chile with a CNRS support (délégation).
All three authors were supported by FCT project PTDC/MAT/120346/2010, which is financed by national and European Community structural funds through the programs FEDER and COMPETE
Article copyright: © Copyright 2014 American Mathematical Society