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November  2017, 37(11): 5861-5881. doi: 10.3934/dcds.2017255

Hitting times distribution and extreme value laws for semi-flows

Maria José Pacifico and  Fan Yang ,

Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil

* Corresponding author: Fan Yang

Received  September 2016 Revised  June 2017 Published  July 2017

Fund Project: Maria José Pacifico is partially supported by CNPq, FAPERJ.
Fan Yang is partially supported by CAPES.

For flows whose return map on a cross section has sufficient mixing property, we show that the hitting time distribution of the flow to balls is exponential in limit. We also establish a link between the extreme value distribution of the flow and its hitting time distribution, generalizing a previous work by Freitas et al in the discrete time case. Finally we show that for maps that can be modeled by Young's tower with polynomial tail, the extreme value laws hold.

Keywords: Hitting times distribution, extreme value laws, suspension flow.
Mathematics Subject Classification: Primary: 37A05, 37A50, 37C10; Secondary: 60F99.
Citation: Maria José Pacifico, Fan Yang. Hitting times distribution and extreme value laws for semi-flows. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5861-5881. doi: 10.3934/dcds.2017255
References:
[1]

V. S. Afraimovic, V. V. Bykov and L. P. Silnikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk., 234 (1977), 336-339.  doi: 10.2307/2152750.

[2]

V. Araújo and M. J. Pacifico, Three-Dimensional Flows, volume 53 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, 2010.

[3]

V. Araújo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.

[4]

J. R. Chazottes and P. Collet, Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 49-80.  doi: 10.1017/S0143385711000897.

[5]

P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergod. Th. & Dynam. Sys., 21 (2001), 401-420.  doi: 10.1017/S0143385701001201.

[6]

A. C. M. Freitas and J. M. Freitas, On the link between dependence and independence in extreme value theory for dynamical systems, Stat. Probab. Lett., 78 (2008), 1088-1093.  doi: 10.1016/j.spl.2007.11.002.

[7]

A. C. M. Freitas, J. M. Freitas and M. Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields, 147 (2010), 675-710.  doi: 10.1007/s00440-009-0221-y.

[8]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys., 142 (2011), 108-126.  doi: 10.1007/s10955-010-0096-4.

[9]

J. M. Freitas, N. Haydn and M. Nicol, Convergence of rare event point processes to the Poisson process for planar billiards, Nonlinearity, 27 (2014), 1669-1687.  doi: 10.1088/0951-7715/27/7/1669.

[10]

S. Galatolo, I. Nisoli and M. J. Pacifico, Decay of correlations and logarithm laws for Rovella attractors, preprint, arXiv: 1701.08743.

[11]

S. Galatolo and M. J. Pacifico, Lorenz like flows: Exponential decay of correlations for the poincaré map, logarithm law, quantitative recurrence, Ergodic Theory and Dynamical Systems, 30 (2010), 1703-1737.  doi: 10.1017/S0143385709000856.

[12]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci., 50 (1979), 59-72. 

[13]

C. Gupta, M. Holland and M. Nicol, Extreme value theory and return time statistics for dispersing billard maps and flows, Lozi maps and Lorenz-like maps, Ergod. Th. & Dynam. Sys., 31 (2011), 1363-1390.  doi: 10.1017/S014338571000057X.

[14]

N. Haydn and K. Wassilewska, Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems, Discrete Contin. Dyn. Sys., 36 (2016), 2585-2611.  doi: 10.3934/dcds.2016.36.2585.

[15]

M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergod. Th. & Dynam. Sys., 13 (1993), 533-556.  doi: 10.1017/S0143385700007513.

[16]

M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications, Comm. Math. Phys., 206 (1999), 33-55.  doi: 10.1007/s002200050697.

[17]

M. Holland, M. Nicol and A. Török, Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc., 364 (2012), 661-688.  doi: 10.1090/S0002-9947-2011-05271-2.

[18]

M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics, Springer-Verlag, New York, 1983.

[19]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36.  doi: 10.1007/978-0-387-21830-4_2.

[20]

V. Lucarini, D. Faranda, A. C. M. Freitas, J. M. Freitas, M. Holland, T. Kuna, M. Nicol and S. Vaienti, Extremes and Recurrence in Dynamical Systems, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, Hoboken, NJ, 2016. doi: 10.1002/9781118632321.

[21]

P. Mattila, J. Marklof, Entry and return times for semi-flows, Nonlinearity, 30 (2017), 810-824, arXiv: 1605.02715. doi: 10.1088/1361-6544/aa518b.

[22]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1$^{st}$ ed. Cambridge: Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.

[23]

C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Am. Math. Soc., 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.

[24]

F. Péne and B. Saussol, Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing, Ergod. Th. & Dynam. Sys., 36 (2016), 2602-2626.  doi: 10.1017/etds.2015.28.

[25]

B. Pitskel, Poisson law for Markov chains, Ergod. Th. & Dynam. Sys., 11 (1991), 501-513.  doi: 10.1017/S0143385700006301.

[26]

J. Rousseau, Recurrence rates for observations of flows, Ergod. Th. & Dynam. Sys., 32 (2012), 1727-1751.  doi: 10.1017/S014338571100037X.

[27]

J. Rousseau, B. Saussol and P. Varandas, Exponential law for random subshifts of finite type, Stochastic Processes and their Applications, 124 (2014), 3260-3276.  doi: 10.1016/j.spa.2014.04.016.

[28]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., 147 (1998), 585-650.  doi: 10.2307/120960.

[29]

L.-S. Young, Recurrence time and rate of mixing, Israel J. of Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.

[30]

L. Zhang, Borel-Cantelli lemmas and extreme value theory for geometric Lorenz models, Nonlinearity, 29 (2016), 232-255.  doi: 10.1088/0951-7715/29/1/232.

show all references

References:
[1]

V. S. Afraimovic, V. V. Bykov and L. P. Silnikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk., 234 (1977), 336-339.  doi: 10.2307/2152750.

[2]

V. Araújo and M. J. Pacifico, Three-Dimensional Flows, volume 53 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, 2010.

[3]

V. Araújo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.

[4]

J. R. Chazottes and P. Collet, Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 33 (2013), 49-80.  doi: 10.1017/S0143385711000897.

[5]

P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergod. Th. & Dynam. Sys., 21 (2001), 401-420.  doi: 10.1017/S0143385701001201.

[6]

A. C. M. Freitas and J. M. Freitas, On the link between dependence and independence in extreme value theory for dynamical systems, Stat. Probab. Lett., 78 (2008), 1088-1093.  doi: 10.1016/j.spl.2007.11.002.

[7]

A. C. M. Freitas, J. M. Freitas and M. Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields, 147 (2010), 675-710.  doi: 10.1007/s00440-009-0221-y.

[8]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys., 142 (2011), 108-126.  doi: 10.1007/s10955-010-0096-4.

[9]

J. M. Freitas, N. Haydn and M. Nicol, Convergence of rare event point processes to the Poisson process for planar billiards, Nonlinearity, 27 (2014), 1669-1687.  doi: 10.1088/0951-7715/27/7/1669.

[10]

S. Galatolo, I. Nisoli and M. J. Pacifico, Decay of correlations and logarithm laws for Rovella attractors, preprint, arXiv: 1701.08743.

[11]

S. Galatolo and M. J. Pacifico, Lorenz like flows: Exponential decay of correlations for the poincaré map, logarithm law, quantitative recurrence, Ergodic Theory and Dynamical Systems, 30 (2010), 1703-1737.  doi: 10.1017/S0143385709000856.

[12]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci., 50 (1979), 59-72. 

[13]

C. Gupta, M. Holland and M. Nicol, Extreme value theory and return time statistics for dispersing billard maps and flows, Lozi maps and Lorenz-like maps, Ergod. Th. & Dynam. Sys., 31 (2011), 1363-1390.  doi: 10.1017/S014338571000057X.

[14]

N. Haydn and K. Wassilewska, Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems, Discrete Contin. Dyn. Sys., 36 (2016), 2585-2611.  doi: 10.3934/dcds.2016.36.2585.

[15]

M. Hirata, Poisson law for Axiom A diffeomorphisms, Ergod. Th. & Dynam. Sys., 13 (1993), 533-556.  doi: 10.1017/S0143385700007513.

[16]

M. Hirata, B. Saussol and S. Vaienti, Statistics of return times: A general framework and new applications, Comm. Math. Phys., 206 (1999), 33-55.  doi: 10.1007/s002200050697.

[17]

M. Holland, M. Nicol and A. Török, Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc., 364 (2012), 661-688.  doi: 10.1090/S0002-9947-2011-05271-2.

[18]

M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics, Springer-Verlag, New York, 1983.

[19]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36.  doi: 10.1007/978-0-387-21830-4_2.

[20]

V. Lucarini, D. Faranda, A. C. M. Freitas, J. M. Freitas, M. Holland, T. Kuna, M. Nicol and S. Vaienti, Extremes and Recurrence in Dynamical Systems, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, Hoboken, NJ, 2016. doi: 10.1002/9781118632321.

[21]

P. Mattila, J. Marklof, Entry and return times for semi-flows, Nonlinearity, 30 (2017), 810-824, arXiv: 1605.02715. doi: 10.1088/1361-6544/aa518b.

[22]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1$^{st}$ ed. Cambridge: Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.

[23]

C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Am. Math. Soc., 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.

[24]

F. Péne and B. Saussol, Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing, Ergod. Th. & Dynam. Sys., 36 (2016), 2602-2626.  doi: 10.1017/etds.2015.28.

[25]

B. Pitskel, Poisson law for Markov chains, Ergod. Th. & Dynam. Sys., 11 (1991), 501-513.  doi: 10.1017/S0143385700006301.

[26]

J. Rousseau, Recurrence rates for observations of flows, Ergod. Th. & Dynam. Sys., 32 (2012), 1727-1751.  doi: 10.1017/S014338571100037X.

[27]

J. Rousseau, B. Saussol and P. Varandas, Exponential law for random subshifts of finite type, Stochastic Processes and their Applications, 124 (2014), 3260-3276.  doi: 10.1016/j.spa.2014.04.016.

[28]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., 147 (1998), 585-650.  doi: 10.2307/120960.

[29]

L.-S. Young, Recurrence time and rate of mixing, Israel J. of Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.

[30]

L. Zhang, Borel-Cantelli lemmas and extreme value theory for geometric Lorenz models, Nonlinearity, 29 (2016), 232-255.  doi: 10.1088/0951-7715/29/1/232.

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