Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Extreme Value Laws for sequences of intermittent maps


Authors: Ana Cristina Moreira Freitas, Jorge Milhazes Freitas and Sandro Vaienti
Journal: Proc. Amer. Math. Soc. 146 (2018), 2103-2116
MSC (2010): Primary 37A50, 60G70, 37B20, 37A25
DOI: https://doi.org/10.1090/proc/13892
Published electronically: January 29, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed point and prove the existence of Extreme Value Laws for such processes. We use an approach developed in an earlier work of the authors, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps.


References [Enhancements On Off] (What's this?)

  • [1] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, and Sandro Vaienti, Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 793-806. MR 3277171, https://doi.org/10.3934/dcds.2015.35.793
  • [2] Wael Bahsoun and Christopher Bose, Mixing rates and limit theorems for random intermittent maps, Nonlinearity 29 (2016), no. 4, 1417-1433. MR 3476513, https://doi.org/10.1088/0951-7715/29/4/1417
  • [3] Wael Bahsoun, Christopher Bose, and Yuejiao Duan, Decay of correlation for random intermittent maps, Nonlinearity 27 (2014), no. 7, 1543-1554. MR 3225871, https://doi.org/10.1088/0951-7715/27/7/1543
  • [4] Wael Bahsoun and Benoît Saussol, Linear response in the intermittent family: differentiation in a weighted $ C^0$-norm, Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 6657-6668. MR 3567814, https://doi.org/10.3934/dcds.2016089
  • [5] Viviane Baladi and Mike Todd, Linear response for intermittent maps, Comm. Math. Phys. 347 (2016), no. 3, 857-874. MR 3551256, https://doi.org/10.1007/s00220-016-2577-z
  • [6] Daniel Berend and Vitaly Bergelson, Ergodic and mixing sequences of transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 353-366. MR 776873, https://doi.org/10.1017/S0143385700002509
  • [7] P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 401-420. MR 1827111, https://doi.org/10.1017/S0143385701001201
  • [8] Jean-Pierre Conze and Albert Raugi, Limit theorems for sequential expanding dynamical systems on $ [0,1]$, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 89-121. MR 2331327, https://doi.org/10.1090/conm/430/08253
  • [9] Yuejiao Duan, ACIM for random intermittent maps: existence, uniqueness and stochastic stability, Dyn. Syst. 28 (2013), no. 1, 48-61. MR 3040766, https://doi.org/10.1080/14689367.2012.750646
  • [10] 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, https://doi.org/10.1007/s00440-009-0221-y
  • [11] 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, https://doi.org/10.1016/j.aim.2012.07.029
  • [12] Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Speed of convergence for laws of rare events and escape rates, Stochastic Process. Appl. 125 (2015), no. 4, 1653-1687. MR 3310360, https://doi.org/10.1016/j.spa.2014.11.011
  • [13] Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Sandro Vaienti, Extreme value laws for non stationary processes generated by sequential and random dynamical systems, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 3, 1341-1370 (English, with English and French summaries). MR 3689970
  • [14] Jorge Milhazes Freitas, Extremal behaviour of chaotic dynamics, Dyn. Syst. 28 (2013), no. 3, 302-332. MR 3170619, https://doi.org/10.1080/14689367.2013.806731
  • [15] Jorge Milhazes Freitas and Mike Todd, The statistical stability of equilibrium states for interval maps, Nonlinearity 22 (2009), no. 2, 259-281. MR 2475546, https://doi.org/10.1088/0951-7715/22/2/002
  • [16] Nicolai Haydn, Matthew Nicol, Andrew Török, and Sandro Vaienti, Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5293-5316. MR 3646763, https://doi.org/10.1090/tran/6812
  • [17] Mark Holland, Matthew Nicol, and Andrei Török, Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc. 364 (2012), no. 2, 661-688. MR 2846347, https://doi.org/10.1090/S0002-9947-2011-05271-2
  • [18] Jürg Hüsler, Asymptotic approximation of crossing probabilities of random sequences, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 2, 257-270. MR 701529, https://doi.org/10.1007/BF00538965
  • [19] Jürg Hüsler, Extreme values of nonstationary random sequences, J. Appl. Probab. 23 (1986), no. 4, 937-950. MR 867190
  • [20] Alexey Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity 29 (2016), no. 6, 1735-1754. MR 3502226, https://doi.org/10.1088/0951-7715/29/6/1735
  • [21] Juho Leppänen and Mikko Stenlund, Quasistatic dynamics with intermittency, Math. Phys. Anal. Geom. 19 (2016), no. 2, Art. 8, 23. MR 3506246, https://doi.org/10.1007/s11040-016-9212-2
  • [22] Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671-685. MR 1695915, https://doi.org/10.1017/S0143385799133856
  • [23] Matthew Nicol, Andrei Török, and Sandro Vaienti, Central limit theorems for sequential and random intermittent dynamical systems, To appear in Ergodic Theory and Dynamical Systems (arXiv:1510.03214), 2016.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37A50, 60G70, 37B20, 37A25

Retrieve articles in all journals with MSC (2010): 37A50, 60G70, 37B20, 37A25


Additional Information

Ana Cristina Moreira Freitas
Affiliation: Centro de Matemática & Faculdade de Economia da Universidade do Porto Rua Dr. Roberto Frias 4200-464 Porto Portugal
Email: amoreira@fep.up.pt

Jorge Milhazes Freitas
Affiliation: Centro de Matemática & Faculdade de Ciências da 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/proc/13892
Keywords: Non-stationarity, Extreme Value Theory, sequential dynamical systems, intermittent maps
Received by editor(s): May 23, 2016
Received by editor(s) in revised form: July 12, 2017
Published electronically: January 29, 2018
Communicated by: Yingfei Yi
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society