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

Hitting times distribution and extreme value laws for semi-flows

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.

Citation: Maria José Pacifico, Fan Yang. Hitting times distribution and extreme value laws for semi-flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5861-5881. doi: 10.3934/dcds.2017255
References:
[1]

V. S. Afraimovic, V. V. Bykov, 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, 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, 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, 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, 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, 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, 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, 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, R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci., 50 (1979), 59-72.

[13]

C. Gupta, M. Holland, 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, 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, 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, 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.

[21]

P. Mattila, J. Marklof, Entry and return times for semi-flows, Nonlinearity, 30 (2017), 810-824, arXiv: 1605. 02715.

[22]

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

[23]

C. A. Morales, M. J. Pacifico, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci., 50 (1979), 59-72.

[13]

C. Gupta, M. Holland, 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, 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, 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, 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.

[21]

P. Mattila, J. Marklof, Entry and return times for semi-flows, Nonlinearity, 30 (2017), 810-824, arXiv: 1605. 02715.

[22]

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

[23]

C. A. Morales, M. J. Pacifico, 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, 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, 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.

[1]

V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73

[2]

Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources . Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565

[3]

Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689

[4]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237

[5]

Nicolai Haydn, Sandro Vaienti. The limiting distribution and error terms for return times of dynamical systems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 589-616. doi: 10.3934/dcds.2004.10.589

[6]

Yu Zhou. On the distribution of auto-correlation value of balanced Boolean functions. Advances in Mathematics of Communications, 2013, 7 (3) : 335-347. doi: 10.3934/amc.2013.7.335

[7]

Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251

[8]

Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling. Networks & Heterogeneous Media, 2013, 8 (2) : 433-463. doi: 10.3934/nhm.2013.8.433

[9]

Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks & Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749

[10]

Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329

[11]

Ivana Bochicchio, Claudio Giorgi, Elena Vuk. On the viscoelastic coupled suspension bridge. Evolution Equations & Control Theory, 2014, 3 (3) : 373-397. doi: 10.3934/eect.2014.3.373

[12]

P. J. McKenna. Oscillations in suspension bridges, vertical and torsional. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 785-791. doi: 10.3934/dcdss.2014.7.785

[13]

Elvise Berchio, Filippo Gazzola. The role of aerodynamic forces in a mathematical model for suspension bridges. Conference Publications, 2015, 2015 (special) : 112-121. doi: 10.3934/proc.2015.0112

[14]

Alberto Ferrero, Filippo Gazzola. A partially hinged rectangular plate as a model for suspension bridges. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5879-5908. doi: 10.3934/dcds.2015.35.5879

[15]

Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096

[16]

Jon Aaronson, Omri Sarig, Rita Solomyak. Tail-invariant measures for some suspension semiflows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 725-735. doi: 10.3934/dcds.2002.8.725

[17]

Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34.

[18]

Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185

[19]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008

[20]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]