Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards
[Bornes polynomiales précises pour la décroissance des corrélations d’applications non-uniformément hyperboliques en plusieurs dimensions et de billards]
Bruin, Henk1 ; Melbourne, Ian2 ; Terhesiu, Dalia3
1 Faculty of Mathematics, University of Vienna, 1090 Vienna, (Austria)
2 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, (UK)
3 Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, (The Netherlands)
Annales Henri Lebesgue, Tome 4 (2021), pp. 407-451.

Gouëzel et Sarig ont introduit la théorie du renouvellement d’opérateurs pour démontrer des résultats précis sur la décroissance des corrélations dans certaines classes d’applications non-uniformément dilatantes. Dans cet article, nous appliquons cette méthode à des billards plans dispersifs et des applications intermittentes non-markoviennes en plusieurs dimensions.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/ahl.76
Classification : 37A25, 37D25, 37D50, 60K05
Mots clés : Sharp mixing rates, nonuniform hyperbolicity, billiards, multidimensional intermittent maps, operator renewal theory
Bruin, Henk 1 ; Melbourne, Ian 2 ; Terhesiu, Dalia 3

1 Faculty of Mathematics, University of Vienna, 1090 Vienna, (Austria)
2 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, (UK)
3 Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, (The Netherlands) 
@article{AHL_2021__4__407_0,
     author = {Bruin, Henk and Melbourne, Ian and Terhesiu, Dalia},
     title = {Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards},
     journal = {Annales Henri Lebesgue},
     pages = {407--451},
     publisher = {\'ENS Rennes},
     volume = {4},
     year = {2021},
     doi = {10.5802/ahl.76},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ahl.76/}
}
TY  - JOUR
AU  - Bruin, Henk
AU  - Melbourne, Ian
AU  - Terhesiu, Dalia
TI  - Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards
JO  - Annales Henri Lebesgue
PY  - 2021
SP  - 407
EP  - 451
VL  - 4
PB  - ÉNS Rennes
UR  - http://archive.numdam.org/articles/10.5802/ahl.76/
DO  - 10.5802/ahl.76
LA  - en
ID  - AHL_2021__4__407_0
ER  - 
%0 Journal Article
%A Bruin, Henk
%A Melbourne, Ian
%A Terhesiu, Dalia
%T Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards
%J Annales Henri Lebesgue
%D 2021
%P 407-451
%V 4
%I ÉNS Rennes
%U http://archive.numdam.org/articles/10.5802/ahl.76/
%R 10.5802/ahl.76
%G en
%F AHL_2021__4__407_0
Bruin, Henk; Melbourne, Ian; Terhesiu, Dalia. Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards. Annales Henri Lebesgue, Tome 4 (2021), pp. 407-451. doi : 10.5802/ahl.76. http://archive.numdam.org/articles/10.5802/ahl.76/

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 et al., eds.) (Springer Proceedings in Mathematics & Statistics), Volume 283, Springer, 2019, pp. 17-42 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

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

[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 | DOI | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

Cité par Sources :