Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards
Annales Henri Lebesgue, Volume 4 (2021) , pp. 407-451.

Metadata

KeywordsSharp mixing rates, nonuniform hyperbolicity, billiards, multidimensional intermittent maps, operator renewal theory

Abstract

Gouëzel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.


References

[AFLV11] Alves, José. F.; Freitas, Jorge M.; Luzzatto, Stefano; Vaienti, Sandro From rates of mixing to recurrence times via large deviations, Adv. Math., Volume 228 (2011) no. 2, pp. 1203-1236 | Article | MR 2822221 | Zbl 1236.37003

[ALP05] Alves, José. F.; Luzzatto, Stefano; Pinheiro, Vilton Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 6, pp. 817-839 | Article | Numdam | MR 2172861 | Zbl 1134.37326

[AM19a] Antoniou, Marios; Melbourne, Ian Rate of convergence in the weak invariance principle for deterministic systems, Commun. Math. Phys., Volume 369 (2019) no. 3, pp. 1147-1165 | Article | MR 3975864 | Zbl 1427.37022

[AM19b] Araújo, Vítor; Melbourne, Ian Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation, Adv. Math., Volume 349 (2019), pp. 212-245 | Article | MR 3938851 | Zbl 1442.37019

[AP08] Alves, José. F.; Pinheiro, Vilton Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., Volume 131 (2008) no. 3, pp. 505-534 | Article | MR 2386574 | Zbl 1144.82324

[BCD11] Bálint, Péter; Chernov, Nikolai I.; Dolgopyat, Dmitry Limit theorems for dispersing billiards with cusps, Commun. Math. Phys., Volume 308 (2011) no. 2, pp. 479-510 | Article | MR 2851150 | Zbl 1241.37005

[BG06] Bálint, Péter; Gouëzel, Sébastien Limit theorems in the stadium billiard, Commun. Math. Phys., Volume 263 (2006) no. 2, pp. 461-512 | Article | MR 2207652 | Zbl 1170.37314

[Ble92] Bleher, Pavel M. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J. Stat. Phys., Volume 66 (1992) no. 1-2, pp. 315-373 | Article | MR 1149489 | Zbl 0925.82147

[BLvS03] Bruin, Henk; Luzzatto, Stefano; van Strien, Sebastian Decay of correlations in one-dimensional dynamics, Ann. Sci. Éc. Norm. Supér., Volume 36 (2003) no. 4, pp. 621-646 | Article | Numdam | MR 2013929 | Zbl 1039.37021

[BT18] Bruin, Henk; Terhesiu, Dalia Upper and lower bounds for the correlation function via inducing with general return times, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 1, pp. 34-62 | Article | MR 3742537 | Zbl 1388.37012

[Bun73] Bunimovič, Leonid A. The ergodic properties of billiards that are nearly scattering, Dokl. Akad. Nauk SSSR, Volume 211 (1973), pp. 1024-1026 | MR 330413

[Bun79] Bunimovič, Leonid A. On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys., Volume 65 (1979) no. 3, pp. 295-312 | Article | MR 530154

[CDKM19] Cuny, Christophe; Dedecker, Jerôme; Korepanov, Alexey; Merlevède, Florence Rates in almost sure invariance principle for slowly mixing dynamical systems, Ergodic Theory Dyn. Syst., Volume 40 (2019) no. 9, pp. 2317-2348 | Article | MR 4130806 | Zbl 1448.37008

[CFK + 19] Chevyrev, Ilya; Friz, Peter K.; Korepanov, Alexey; Melbourne, Ian; Zhang, Huilin Multiscale systems, homogenization, and rough paths, Probability and Analysis in Interacting Physical Systems: In Honor of S.R.S. Varadhan (Friz, Peter, ed.) (Springer Proceedings in Mathematics & Statistics), Volume 283, Springer, 2019, pp. 17-42 | Article | MR 3968507 | Zbl 1443.60060

[Che99] Chernov, Nikolai I. Decay of correlations and dispersing billiards, J. Stat. Phys., Volume 94 (1999) no. 3-4, pp. 513-556 | Article | MR 1675363 | Zbl 1047.37503

[CM06] Chernov, Nikolai I.; Markarian, Roberto Chaotic billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, 2006 | MR 2229799 | Zbl 1101.37001

[CM07] Chernov, Nikolai I.; Markarian, Roberto Dispersing billiards with cusps: slow decay of correlations, Commun. Math. Phys., Volume 270 (2007) no. 3, pp. 727-758 | Article | MR 2276463 | Zbl 1113.37020

[CM15] Cuny, Christophe; Merlevède, Florence Strong invariance principles with rate for “reverse” martingales and applications, J. Theor. Probab., Volume 28 (2015) no. 1, pp. 137-183 | Article | MR 3320963 | Zbl 1353.60031

[CZ05a] Chernov, Nikolai I.; Zhang, Hong-kun Billiards with polynomial mixing rates, Nonlinearity, Volume 18 (2005) no. 4, pp. 1527-1553 | Article | MR 2150341 | Zbl 1143.37314

[CZ05b] Chernov, Nikolai I.; Zhang, Hong-kun A family of chaotic billiards with variable mixing rates, Stoch. Dyn., Volume 5 (2005) no. 4, pp. 535-553 | Article | MR 2185504 | Zbl 1111.37029

[CZ08] Chernov, Nikolai I.; Zhang, Hong-kun Improved estimates for correlations in billiards, Commun. Math. Phys., Volume 77 (2008) no. 2, pp. 305-321 | MR 2358286 | Zbl 1143.37024

[DM15] Dedecker, Jerôme; Merlevède, Florence Moment bounds for dependent sequences in smooth Banach spaces, Stochastic Processes Appl., Volume 125 (2015) no. 9, pp. 3401-3429 | Article | MR 3357614 | Zbl 1318.60028

[GM13] Gottwald, Georg A.; Melbourne, Ian Homogenization for deterministic maps and multiplicative noise, Proc. A, R. Soc. Lond., Volume 469 (2013), 20130201 | MR 3064670 | Zbl 1371.34084

[GM14] Gouëzel, Sébastien; Melbourne, Ian Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., Volume 19 (2014), 93, 30 pages | MR 3272326 | Zbl 1351.37041

[Gou04] Gouëzel, Sébastien Sharp polynomial estimates for the decay of correlations, Isr. J. Math., Volume 139 (2004), pp. 29-65 | Article | MR 2041223 | Zbl 1070.37003

[Gou05] Gouëzel, Sébastien Berry–Esseen theorem and local limit theorem for non uniformly expanding maps, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 41 (2005) no. 6, pp. 997-1024 | Article | Numdam | MR 2172207 | Zbl 1134.60323

[Gou07] Gouëzel, Sébastien Statistical properties of a skew product with a curve of neutral points, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 1, pp. 123-151 | Article | MR 2297091 | Zbl 1143.37022

[Gou10] Gouëzel, Sébastien Characterization of weak convergence of Birkhoff sums for Gibbs–Markov maps, Isr. J. Math., Volume 180 (2010), pp. 1-41 | Article | MR 2735054 | Zbl 1213.37017

[Hu04] Hu, Huyi Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dyn. Syst., Volume 24 (2004) no. 2, pp. 495-524 | Article | MR 2054191 | Zbl 1071.37026

[HV09] Hu, Huyi; Vaienti, Sandro Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dyn. Syst., Volume 29 (2009) no. 4, pp. 1185-1215 | Article | MR 2529645 | Zbl 1201.37058

[HV19] Hu, Huyi; Vaienti, Sandro Lower bounds for the decay of correlations in non-uniformly expanding maps, Ergodic Theory Dyn. Syst., Volume 39 (2019) no. 7, pp. 1936-1970 | Article | MR 3961681 | Zbl 1423.37032

[JZ18] Jung, Paul; Zhang, Hong-Kun Stable laws for chaotic billiards with cusps at flat points, Ann. Henri Poincaré, Volume 19 (2018) no. 12, pp. 3815-3853 | Article | MR 3877517 | Zbl 1407.37058

[Kel85] Keller, Gerhard Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 69 (1985), pp. 461-478 | Article | MR 787608 | Zbl 0574.28014

[KKM19] Korepanov, Alexey; Kosloff, Zemer; Melbourne, Ian Explicit coupling argument for nonuniformly hyperbolic transformations, Proc. R. Soc. Edinb., Sect. A, Math., Volume 149 (2019) no. 1, pp. 101-130 | Article | Zbl 1430.37026

[KM16] Kelly, David; Melbourne, Ian Smooth approximation of stochastic differential equations, Ann. Probab., Volume 44 (2016) no. 1, pp. 479-520 | MR 3456344 | Zbl 1372.60082

[Kor18] Korepanov, Alexey Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle, Commun. Math. Phys., Volume 359 (2018) no. 3, pp. 1123-1138 | Article | MR 3784542 | Zbl 1396.37038

[LSV99] Liverani, Carlangelo; Saussol, Benoît; Vaienti, Sandro A probabilistic approach to intermittency, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 3, pp. 671-685 | Article | MR 1695915 | Zbl 0988.37035

[Mar04] Markarian, Roberto Billiards with polynomial decay of correlations, Ergodic Theory Dyn. Syst., Volume 24 (2004) no. 1, pp. 177-197 | Article | MR 2041267 | Zbl 1115.37037

[Mel18] Melbourne, Ian Superpolynomial and polynomial mixing for semiflows and flows, Nonlinearity, Volume 31 (2018) no. 10, p. R268-R316 | Article | MR 3851312 | Zbl 1429.37013

[MN05] Melbourne, Ian; Nicol, Matthew Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., Volume 260 (2005) no. 1, pp. 131-146 | Article | MR 2175992 | Zbl 1084.37024

[MN08] Melbourne, Ian; Nicol, Matthew Large deviations for nonuniformly hyperbolic systems, Trans. Am. Math. Soc., Volume 360 (2008) no. 12, pp. 6661-6676 | Article | MR 2434305 | Zbl 1151.37031

[MT14] Melbourne, Ian; Terhesiu, Dalia Decay of correlations for nonuniformly expanding systems with general return times, Ergodic Theory Dyn. Syst., Volume 34 (2014) no. 3, pp. 893-918 | Article | Zbl 1332.37025

[MV16] Melbourne, Ian; Varandas, Paulo A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion, Stoch. Dyn., Volume 16 (2016) no. 3, 1660012, 13 pages | MR 3471365 | Zbl 1362.37073

[MV20] Melbourne, Ian; Varandas, Paulo Convergence to a Lévy process in the Skorohod 1 and 2 topologies for nonuniformly hyperbolic systems, including billiards with cusps, Commun. Math. Phys., Volume 375 (2020) no. 1, pp. 653-678 | Article | Zbl 1440.37013

[PM80] Pomeau, Yves; Manneville, Paul Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., Volume 74 (1980) no. 2, pp. 189-197 | Article | MR 576270

[Pol85] Pollicott, Mark On the rate of mixing of Axiom A flows, Invent. Math., Volume 81 (1985), pp. 413-426 | Article | MR 807065 | Zbl 0591.58025

[Sar02] Sarig, Omri Subexponential decay of correlations, Invent. Math., Volume 150 (2002) no. 3, pp. 629-653 | Article | MR 1946554 | Zbl 1042.37005

[Sar06] Sarig, Omri Continuous phase transitions for dynamical systems, Commun. Math. Phys., Volume 267 (2006) no. 3, pp. 631-667 | Article | MR 2249785 | Zbl 1123.37012

[Sau00] Saussol, Benoît Absolutely continuous invariant measures for multidimensional expanding maps, Isr. J. Math., Volume 116 (2000), pp. 223-248 | Article | MR 1759406 | Zbl 0982.37003

[SV07] Szász, Domokos; Varjú, Tamás Limit laws and recurrence for the planar Lorentz process with infinite horizon, J. Stat. Phys., Volume 129 (2007) no. 1, pp. 59-80 | Article | MR 2349520 | Zbl 1128.82011

[Tha80] Thaler, Maximilian Estimates of the invariant densities of endomorphisms with indifferent fixed point, Isr. J. Math., Volume 37 (1980), pp. 303-314 | Article | MR 599464 | Zbl 0447.28016

[VZ16] Vaienti, Sandro; Zhang, Hong-Kun Optimal bounds on correlation decay rates for nonuniform hyperbolic systems (2016) (https://arxiv.org/abs/1605.01793)

[You98] Young, Lai-Sang Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., Volume 147 (1998) no. 3, pp. 585-650 | Article | MR 1637655 | Zbl 0945.37009

[You99] Young, Lai-Sang Recurrence times and rates of mixing, Isr. J. Math., Volume 110 (1999), pp. 153-188 | Article | MR 1750438 | Zbl 0983.37005

[Zha17a] Zhang, Hong-Kun Decay of correlations for billiards with flat points I: channel effects, Dynamical systems, ergodic theory, and probability: in memory of Kolya Chernov (Contemporary Mathematics), Volume 698, American Mathematical Society, 2017, pp. 239-286 | MR 3716097 | Zbl 1398.37015

[Zha17b] Zhang, Hong-Kun Decay of correlations for billiards with flat points. II: Cusps effect, Dynamical systems, ergodic theory, and probability: in memory of Kolya Chernov (Contemporary Mathematics), Volume 698, American Mathematical Society, 2017, pp. 287-316 | MR 3716098 | Zbl 1398.37035

[Zwe98] Zweimüller, Roland Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, Volume 11 (1998) no. 5, pp. 1263-1276 | Article | MR 1644385 | Zbl 0922.58039