Targets and holes

Giulietti, P.; Koltai, P.; Vaienti, S.

doi:10.1090/proc/15384

Targets and holes
HTML articles powered by AMS MathViewer

by P. Giulietti, P. Koltai and S. Vaienti PDF

Proc. Amer. Math. Soc. 149 (2021), 3293-3306 Request permission

Abstract:

We address the extreme value problem of a one-dimensional dynamical system approaching a fixed target while constrained to avoid a fixed set, which can be thought of as a small hole. The presence of the latter influences the extremal index which depends explicitly on the escape rate.

References

Hale Aytaç, Jorge Milhazes Freitas, and Sandro Vaienti, Laws of rare events for deterministic and random dynamical systems, Trans. Amer. Math. Soc. 367 (2015), no. 11, 8229–8278. MR 3391915, DOI 10.1090/S0002-9947-2014-06300-9
Wael Bahsoun and Sandro Vaienti, Escape rates formulae and metastability for randomly perturbed maps, Nonlinearity 26 (2013), no. 5, 1415–1438. MR 3056132, DOI 10.1088/0951-7715/26/5/1415
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
Henk Bruin, Mark F. Demers, and Mike Todd, Hitting and escaping statistics: mixing, targets and holes, Adv. Math. 328 (2018), 1263–1298. MR 3771153, DOI 10.1016/j.aim.2017.12.020
Théophile Caby, Davide Faranda, Giorgio Mantica, Sandro Vaienti, and Pascal Yiou, Generalized dimensions, large deviations and the distribution of rare events, Phys. D 400 (2019), 132143, 15. MR 4008032, DOI 10.1016/j.physd.2019.06.009
Th. Caby, D. Faranda, S. Vaienti, and P. Yiou, On the computation of the extremal index for time series, J. Stat. Phys. 179 (2020), no. 5-6, 1666–1697. MR 4123370, DOI 10.1007/s10955-019-02423-z
Th. Caby, D. Faranda, S. Vaienti, and P. Yiou, Extreme value distributions of observation recurrences, Nonlinearity 34 (2021), no. 1, DOI 10.1088/1361-6544/abaff1.
Mark F. Demers, Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1139–1171. MR 2158400, DOI 10.1017/S0143385704000963
Mark F. Demers, Dispersing billiards with small holes, Ergodic theory, open dynamics, and coherent structures, Springer Proc. Math. Stat., vol. 70, Springer, New York, 2014, pp. 137–170. MR 3213499, DOI 10.1007/978-1-4939-0419-8_{8}
Mark F. Demers and Lai-Sang Young, Escape rates and conditionally invariant measures, Nonlinearity 19 (2006), no. 2, 377–397. MR 2199394, DOI 10.1088/0951-7715/19/2/008
Mark F. Demers and Mike Todd, Slow and fast escape for open intermittent maps, Comm. Math. Phys. 351 (2017), no. 2, 775–835. MR 3613520, DOI 10.1007/s00220-017-2829-6
D. Faranda, H. Ghoudi, P. Guiraud, and S. Vaienti, Extreme value theory for synchronization of coupled map lattices, Nonlinearity 31 (2018), no. 7, 3326–3358. MR 3816758, DOI 10.1088/1361-6544/aabc8e
Valerio Lucarini, Davide Faranda, Giorgio Turchetti, and Sandro Vaienti, Extreme value theory for singular measures, Chaos 22 (2012), no. 2, 023135, 17. MR 3388552, DOI 10.1063/1.4718935
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, 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, Speed of convergence for laws of rare events and escape rates, Stochastic Process. Appl. 125 (2015), no. 4, 1653–1687. MR 3310360, DOI 10.1016/j.spa.2014.11.011
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, DOI 10.1214/16-AIHP757
S. Galatolo and P. Giulietti, A linear response for dynamical systems with additive noise, Nonlinearity 32 (2019), no. 6, 2269–2301. MR 3951965, DOI 10.1088/1361-6544/ab0c2e
Crispin Gardiner, Stochastic methods, 4th ed., Springer Series in Synergetics, Springer-Verlag, Berlin, 2009. A handbook for the natural and social sciences. MR 2676235
Franz Hofbauer and Gerhard Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), no. 1, 119–140. MR 656227, DOI 10.1007/BF01215004
Gerhard Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts, vol. 42, Cambridge University Press, Cambridge, 1998. MR 1618769, DOI 10.1017/CBO9781107359987
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
Gerhard Keller and Carlangelo Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 1, 141–152. MR 1679080
Yuri Kifer and Ariel Rapaport, Geometric law for multiple returns until a hazard, Nonlinearity 32 (2019), no. 4, 1525–1545. MR 3925384, DOI 10.1088/1361-6544/aafbd4
Y. Kifer, Limit theorems for numbers of returnin arrays under $\phi$-mixing, Stoch. Dyn. (2020), DOI 10.1142/S0219493721400025.
Maxim Kirsebom, Philipp Kunde, and Tomas Persson, Shrinking targets and eventually always hitting points for interval maps, Nonlinearity 33 (2020), no. 2, 892–914. MR 4055367, DOI 10.1088/1361-6544/ab5160
M. R. Leadbetter, Georg Lindgren, and Holger Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 691492, DOI 10.1007/978-1-4612-5449-2
Carlangelo Liverani and Véronique Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 3, 385–412 (English, with English and French summaries). MR 1978986, DOI 10.1016/S0246-0203(02)00005-5
Valerio Lucarini, Davide Faranda, Ana Cristina Gomes Monteiro Moreira de Freitas, Jorge Miguel Milhazes de Freitas, Mark Holland, Tobias Kuna, Matthew Nicol, Mike Todd, and Sandro Vaienti, Extremes and recurrence in dynamical systems, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2016. MR 3558780, DOI 10.1002/9781118632321
Giulio Pianigiani and James A. Yorke, Expanding maps on sets which are almost invariant. Decay and chaos, Trans. Amer. Math. Soc. 252 (1979), 351–366. MR 534126, DOI 10.1090/S0002-9947-1979-0534126-2