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Rare Events for Cantor Target Sets

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Abstract

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and its iterates.

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References

  1. Alves, J.F., Freitas, J.M., Luzzatto, S., Vaienti, S.: From rates of mixing to recurrence times via large deviations. Adv. Math. 228(2), 1203–1236 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Aytaç, H., Freitas, J.M., Vaienti, S.: Laws of rare events for deterministic and random dynamical systems. Trans. Am. Math. Soc. 367(11), 8229–8278 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Azevedo, D., Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B.: Clustering of extreme events created by multiple correlated maxima. Phys. D 315, 33–48 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Azevedo, D., Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B.: Extreme value laws for dynamical systems with countable extremal sets. J. Stat. Phys. 167(5), 1244–1261 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences, volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (994) Revised reprint of the 1979 original

  6. Boyarsky, A., Góra, P.: Laws of Chaos. Probability and its Applications. Birkhäuser Boston, Boston (1997). (Invariant measures and dynamical systems in one dimension)

    MATH  Google Scholar 

  7. Caby, T., Faranda, D., Mantica, G., Vaienti, S., Yiou, P.: Generalized dimensions, large deviations and the distribution of rare events. Phys. D 400, 132143, 15 (2019)

    MathSciNet  Google Scholar 

  8. Chazottes, J.-R., Coelho, Z., Collet, P.: Poisson processes for subsystems of finite type in symbolic dynamics. Stoch. Dyn. 9(3), 393–422 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21(2), 401–420 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Das, M., Ngai, S.-M.: Graph-directed iterated function systems with overlaps. Indiana Univ. Math. J. 53(1), 109–134 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Derzko, N.A., Pfeffer, A.M.: Bounds for the spectral radius of a matrix. Math. Comput. 19, 62–67 (1965)

    MathSciNet  MATH  Google Scholar 

  12. Edgar, G.: Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (2008)

    Google Scholar 

  13. Falconer, K.: Fractal Geometry, 2nd edn. Wiley, Hoboken (2003). (Mathematical foundations and applications)

    MATH  Google Scholar 

  14. Faranda, D., Alvarez-Castro, M.C., Messori, G., Rodrigues, D., Yiou, P.: The hammam effect or how a warm ocean enhances large scale atmospheric predictability. Nat. Commun. 10(1), 1316 (2019)

    ADS  Google Scholar 

  15. Faranda, D., Ghoudi, H., Guiraud, P., Vaienti, S.: Extreme value theory for synchronization of coupled map lattices. Nonlinearity 31(7), 3326–3358 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Faranda, D., Messori, G., Alvarez-Castro, M.C., Yiou, P.: Dynamical properties and extremes of northern hemisphere climate fields over the past 60 years. Nonlinear Process. Geophys. 24(4), 713–725 (2017)

    ADS  Google Scholar 

  17. Faranda, D., Sato, Y., Saint-Michel, B., Wiertel, C., Padilla, V., Dubrulle, B., Daviaud, F.: Stochastic chaos in a turbulent swirling flow. Phys. Rev. Lett. 119, 014502 (2017)

    ADS  Google Scholar 

  18. Ferguson, A., Pollicott, M.: Escape rates for gibbs measures. Ergod. Theory Dyn. Syst. 32, 961–988 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Freitas, A.C.M., Freitas, J.M., Todd, M.: Hitting time statistics and extreme value theory. Probab. Theory Relat. Fields 147(3–4), 675–710 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Freitas, A.C.M., Freitas, J.M., Todd, M.: Extreme value laws in dynamical systems for non-smooth observations. J. Stat. Phys. 142(1), 108–126 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Freitas, A.C.M., Freitas, J.M., Todd, M.: The extremal index, hitting time statistics and periodicity. Adv. Math. 231(5), 2626–2665 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Freitas, A.C.M., Freitas, J.M., Todd, M.: The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics. Commun. Math. Phys. 321(2), 483–527 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Freitas, A.C.M., Freitas, J.M., Todd, M., Vaienti, S.: Rare events for the Manneville–Pomeau map. Stoch. Process. Appl. 126(11), 3463–3479 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Holland, M., Nicol, M., Török, A.: Extreme value theory for non-uniformly expanding dynamical systems. Trans. Am. Math. Soc. 364(2), 661–688 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Hsing, T.: Extremal index estimation for a weakly dependent stationary sequence. Ann. Stat. 21(4), 2043–2071 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Keller, G.: Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst. 27(1), 11–27 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Keller, G., Liverani, C.: Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3), 519–534 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Lucarini, V., Bódai, T.: Transitions across melancholia states in a climate model: Reconciling the deterministic and stochastic points of view. Phys. Rev. Lett. 122, 158701 (2019)

    ADS  Google Scholar 

  29. Lucarini, V., Faranda, D., Freitas, A.C.M., Freitas, J.M., Holland, M., Kuna, T., Nicol, M., Vaienti, S.: Extremes and Recurrence in Dynamical Systems. A Wiley Series of Texts, Monographs and Tracts. Wiley, Hoboken (2016). (Pure and Applied Mathematics)

    MATH  Google Scholar 

  30. Lucarini, V., Faranda, D., Turchetti, G., Vaienti, S.: Extreme value theory for singular measures. Chaos 22(2), 023135 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Mantica, G., Perotti, L.: Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis. J. Phys. A 49(37), 374001, 21 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Mauldin, R.D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. Am. Math. Soc. 309(2), 811–829 (1988)

    MathSciNet  MATH  Google Scholar 

  33. McClure, M.: Intersections of self-similar sets. Fractals 16(2), 187–197 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Messori, G., Caballero, R., Bouchet, F., Faranda, D., Grotjahn, R., Harnik, N., Jewson, S., Pinto, J.G., Rivière, G., Woollings, T., Yiou, P.: An interdisciplinary approach to the study of extreme weather events: large-scale atmospheric controls and insights from dynamical systems theory and statistical mechanics. Bull. Am. Meteorol. Soc. 99(5), ES81–ES85 (2018)

    Google Scholar 

  35. Messori, G., Caballero, R., Faranda, D.: A dynamical systems approach to studying midlatitude weather extremes. Geophys. Res. Lett. 44(7), 3346–3354 (2017)

    ADS  Google Scholar 

  36. Newhouse, S.E.: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 50, 101–151 (1979)

    MathSciNet  MATH  Google Scholar 

  37. O’Brien, G.L.: Extreme values for stationary and Markov sequences. Ann. Probab. 15(1), 281–291 (1987)

    MathSciNet  MATH  Google Scholar 

  38. Palis, J., Takens, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge Studies in Advanced Mathematics, vol. 35. Cambridge University Press, Cambridge (1993). (Fractal dimensions and infinitely many attractors)

    MATH  Google Scholar 

  39. Saussol, B.: An introduction to quantitative Poincaré recurrence in dynamical systems. Rev. Math. Phys. 21(8), 949–979 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Saw, E.W., Kuzzay, D., Faranda, D., Guittonneau, A., Daviaud, F., Wiertel-Gasquet, C., Padilla, V., Dubrulle, B.: Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow. Nat. Commun. 7, 12466 (2016)

    ADS  Google Scholar 

  41. Shmerkin, P.: On Furstenberg’s intersection conjecture, self-similar measures, and the \(L^q\) norms of convolutions. Ann. Math. (2) 189(2), 319–391 (2019)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Ana Cristina Moreira Freitas

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

    Jorge Milhazes Freitas

  3. Fagner Bernardini Rodrigues 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, Brazil

    Fagner B. Rodrigues

  4. Centro de Matemática & Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, Porto, 4169-007, Portugal

    Jorge Valentim Soares

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  1. Ana Cristina Moreira Freitas
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  2. Jorge Milhazes Freitas
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  3. Fagner B. Rodrigues
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Correspondence to Jorge Milhazes Freitas.

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Communicated by C. Liverani.

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All authors were partially supported by FCT Projects FAPESP/19805/2014, PTDC/MAT-CAL/3884/2014 and PTDC/MAT-PUR/28177/2017, with national funds, and by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. JS was also supported by FCT scholarship with Reference Number PD/BD/128061/2016. We thank Vuksan Mijović, Mike Todd and Romain Aimino for helpful comments and suggestions. We also would like to thank Davide Faranda, Giorgio Mantica and Sandro Vaienti, in particular, for the help in providing physical motivation for our results.

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Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B. et al. Rare Events for Cantor Target Sets. Commun. Math. Phys. 378, 75–115 (2020). https://doi.org/10.1007/s00220-020-03794-1

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