Laws of rare events for deterministic and random dynamical systems
HTML articles powered by AMS MathViewer
- by Hale Aytaç, Jorge Milhazes Freitas and Sandro Vaienti PDF
- Trans. Amer. Math. Soc. 367 (2015), 8229-8278 Request permission
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.References
- José F. Alves, Jorge M. Freitas, Stefano Luzzatto, and Sandro Vaienti, From rates of mixing to recurrence times via large deviations, Adv. Math. 228 (2011), no. 2, 1203–1236. MR 2822221, DOI 10.1016/j.aim.2011.06.014
- José Ferreira Alves and Vítor Araújo, Random perturbations of nonuniformly expanding maps, Astérisque 286 (2003), xvii, 25–62 (English, with English and French summaries). Geometric methods in dynamics. I. MR 2052296
- Viviane Baladi, Michael Benedicks, and Véronique Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1, 77–126 (English, with English and French summaries). MR 1886006, DOI 10.1016/S0012-9593(01)01083-7
- Viviane Baladi, Michael Benedicks, and Véronique Maume-Deschamps, Corrigendum: “Almost sure rates of mixing for i.i.d. unimodal maps” [Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1, 77–126; MR1886006 (2003d:37027)], Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 319–322 (English, with English and French summaries). MR 1980314, DOI 10.1016/S0012-9593(03)00011-9
- Abraham Boyarsky and PawełGóra, Laws of chaos, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. Invariant measures and dynamical systems in one dimension. MR 1461536, DOI 10.1007/978-1-4612-2024-4
- Jean-René Chazottes, Zaqueu Coelho, and Pierre Collet, Poisson processes for subsystems of finite type in symbolic dynamics, Stoch. Dyn. 9 (2009), no. 3, 393–422. MR 2566908, DOI 10.1142/S0219493709002713
- Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
- Andrew Ferguson and Mark Pollicott, Escape rates for Gibbs measures, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 961–988. MR 2995652, DOI 10.1017/S0143385711000058
- Ana Cristina Moreira Freitas and Jorge Milhazes Freitas, On the link between dependence and independence in extreme value theory for dynamical systems, Statist. Probab. Lett. 78 (2008), no. 9, 1088–1093. MR 2422964, DOI 10.1016/j.spl.2007.11.002
- Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields 147 (2010), no. 3-4, 675–710. MR 2639719, DOI 10.1007/s00440-009-0221-y
- Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys. 142 (2011), no. 1, 108–126. MR 2749711, DOI 10.1007/s10955-010-0096-4
- Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, The extremal index, hitting time statistics and periodicity, Adv. Math. 231 (2012), no. 5, 2626–2665. MR 2970462, DOI 10.1016/j.aim.2012.07.029
- Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics, Comm. Math. Phys. 321 (2013), no. 2, 483–527. MR 3063917, DOI 10.1007/s00220-013-1695-0
- Jorge Milhazes Freitas, Extremal behaviour of chaotic dynamics, Dyn. Syst. 28 (2013), no. 3, 302–332. MR 3170619, DOI 10.1080/14689367.2013.806731
- N. Haydn, Y. Lacroix, and S. Vaienti, Hitting and return times in ergodic dynamical systems, Ann. Probab. 33 (2005), no. 5, 2043–2050. MR 2165587, DOI 10.1214/009117905000000242
- Nicolai Haydn and Sandro Vaienti, The compound Poisson distribution and return times in dynamical systems, Probab. Theory Related Fields 144 (2009), no. 3-4, 517–542. MR 2496441, DOI 10.1007/s00440-008-0153-y
- Masaki Hirata, Poisson law for Axiom A diffeomorphisms, Ergodic Theory Dynam. Systems 13 (1993), no. 3, 533–556. MR 1245828, DOI 10.1017/S0143385700007513
- Olav Kallenberg, Random measures, 4th ed., Akademie-Verlag, Berlin; Academic Press, Inc., London, 1986. MR 854102
- Gerhard Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94 (1982), no. 4, 313–333. MR 685377, DOI 10.1007/BF01667385
- Gerhard Keller, Rare events, exponential hitting times and extremal indices via spectral perturbation, Dyn. Syst. 27 (2012), no. 1, 11–27. MR 2903242, DOI 10.1080/14689367.2011.653329
- Gerhard Keller and Carlangelo Liverani, Rare events, escape rates and quasistationarity: some exact formulae, J. Stat. Phys. 135 (2009), no. 3, 519–534. MR 2535206, DOI 10.1007/s10955-009-9747-8
- Yuri Kifer, Ergodic theory of random transformations, Progress in Probability and Statistics, vol. 10, Birkhäuser Boston, Inc., Boston, MA, 1986. MR 884892, DOI 10.1007/978-1-4684-9175-3
- Yuri Kifer and Pei-Dong Liu, Random dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 379–499. MR 2186245, DOI 10.1016/S1874-575X(06)80030-5
- Yuri Kifer and Ariel Rapaport, Poisson and compound Poisson approximations in a nonconventional setup, Probability and Related Fields, DOI 10.1007/s00440-013-0541-9.
- M. R. Leadbetter, On extreme values in stationary sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 (1973/74), 289–303. MR 362465, DOI 10.1007/BF00532947
- Philippe Marie and Jérôme Rousseau, Recurrence for random dynamical systems, Discrete Contin. Dyn. Syst. 30 (2011), no. 1, 1–16. MR 2773129, DOI 10.3934/dcds.2011.30.1
- Marek Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), no. 1, 69–80. MR 728198, DOI 10.4064/sm-76-1-69-80
- Benoît Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math. 116 (2000), 223–248. MR 1759406, DOI 10.1007/BF02773219
- Benoit Saussol, An introduction to quantitative Poincaré recurrence in dynamical systems, Rev. Math. Phys. 21 (2009), no. 8, 949–979. MR 2568049, DOI 10.1142/S0129055X09003785
- Marcelo Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium 1997, IMPA, 1997.
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
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
- MR Author ID: 754460
- 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
- MR Author ID: 176525
- Email: vaienti@cpt.univ-mrs.fr
- 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 - © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3391915