Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:26:47.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of moments for Axiom A and non-uniformly hyperbolic flows

Published online by Cambridge University Press:  10 June 2011

IAN MELBOURNE  and
ANDREI TÖRÖK
IAN MELBOURNE
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: [email protected])
ANDREI TÖRÖK
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA (email: [email protected]) Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania
Get access Rights & Permissions [Opens in a new window]

Abstract

In the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for non-uniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded. Non-uniformly hyperbolic systems covered by our result include Hénon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau–Manneville intermittency maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 1091 - 1100
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Afraĭmovič, V. S., Bykov, V. V. and Silnikov, L. P.. The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR 234 (1977), 336–339.Google Scholar
[2]Armstead, D. N., Hunt, B. R. and Ott, E.. Anomalous diffusion in infinite horizon billiards. Phys. Rev. E 67 (2003), 021110.CrossRefGoogle ScholarPubMed
[3]Bálint, P. and Melbourne, I.. Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. J. Stat. Phys. 133 (2008), 435–447.CrossRefGoogle Scholar
[4]Bálint, P. and Melbourne, I.. Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas. Preprint, November 2010.Google Scholar
[5]Benedicks, M. and Young, L.-S.. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261 (2000), 13–56.Google Scholar
[6]Billingsley, P.. The Lindeberg–Lévy theorem for martingales. Proc. Amer. Math. Soc. 12 (1961), 788–792.Google Scholar
[7]Bowen, R.. Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.Google Scholar
[8]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[9]Bunimovič, L. A.. The ergodic properties of billiards that are nearly scattering. Dokl. Akad. Nauk SSSR 211 (1973), 1024–1026.Google Scholar
[10]Burkholder, D. L.. Distribution function inequalities for martingales. Ann. Probab. 1 (1973), 19–42.CrossRefGoogle Scholar
[11]Chernov, N.. Decay of correlations and dispersing billiards. J. Stat. Phys. 94 (1999), 513–556.CrossRefGoogle Scholar
[12]Chernov, N. and Markarian, R.. Dispersing billiards with cusps: slow decay of correlations. Comm. Math. Phys. 270 (2007), 727–758.CrossRefGoogle Scholar
[13]Chernov, N. and Zhang, H.-K.. A family of chaotic billiards with variable mixing rates. Stoch. Dyn. 5 (2005), 535–553.CrossRefGoogle Scholar
[14]Chernov, N. I. and Zhang, H.-K.. Billiards with polynomial mixing rates. Nonlinearity 18 (2005), 1527–1553.CrossRefGoogle Scholar
[15]Chernov, N. I. and Zhang, H.-K.. Improved estimates for correlations in billiards. Comm. Math. Phys. 77 (2008), 305–321.Google Scholar
[16]Courbage, M., Edelman, M., Saberi Fathi, S. M. and Zaslavsky, G. M.. Problem of transport in billiards with infinite horizon. Phys. Rev. E 77 (2008), 036203.CrossRefGoogle ScholarPubMed
[17]Durrett, R.. Probability: Theory and Examples, 3rd edn. Duxbury Press, Belmont, CA, 1996.Google Scholar
[18]Field, M. J., Melbourne, I. and Török, A.. Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergod. Th. & Dynam. Sys. 23 (2003), 87–110.CrossRefGoogle Scholar
[19]Gordin, M. I.. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969), 1174–1176.Google Scholar
[20]Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004), 29–65.CrossRefGoogle Scholar
[21]Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 59–72.CrossRefGoogle Scholar
[22]Holland, M. and Melbourne, I.. Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. 76 (2007), 345–364.CrossRefGoogle Scholar
[23]Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.CrossRefGoogle Scholar
[24]Lorenz, E. D.. Deterministic nonperiodic flow. J. Atmospheric Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[25]McMullen, C.. Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173 (2008), 365–425.CrossRefGoogle Scholar
[26]Melbourne, I.. Decay of correlations for slowly mixing flows. Proc. Lond. Math. Soc. 98 (2009), 163–190.CrossRefGoogle Scholar
[27]Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005), 131–146.CrossRefGoogle Scholar
[28]Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008), 6661–6676.Google Scholar
[29]Melbourne, I. and Török, A.. Statistical limit theorems for suspension flows. Israel J. Math. 144 (2004), 191–209.CrossRefGoogle Scholar
[30]Merlevède, F., Peligrad, M. and Utev, S.. Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 (2006), 1–36 (electronic).CrossRefGoogle Scholar
[31]Parry, W. and Pollicott, M.. Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics (Astérisque, 187–188). Société Mathématique de France, Montrouge, 1990.Google Scholar
[32]Pollicott, M.. On the rate of mixing of Axiom A flows. Invent. Math. 81 (1985), 413–426.CrossRefGoogle Scholar
[33]Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), 189–197.CrossRefGoogle Scholar
[34]Ratner, M.. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973), 181–197.CrossRefGoogle Scholar
[35]Rio, E.. Théorie asymptotique des processus aléatoires faiblement dépendants (Mathématiques & Applications (Berlin) [Mathematics & Applications], 31). Springer, Berlin, 2000.Google Scholar
[36]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison Wesley, Reading, MA, 1978.Google Scholar
[37]Sarig, O. M.. Subexponential decay of correlations. Invent. Math. 150 (2002), 629–653.CrossRefGoogle Scholar
[38]Serfling, R. J.. Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 (1970), 1227–1234.CrossRefGoogle Scholar
[39]Sinaĭ, Y. G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), 21–70.CrossRefGoogle Scholar
[40]Szász, D. and Varjú, T.. Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129 (2007), 59–80.CrossRefGoogle Scholar
[41]Tucker, W.. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2 (2002), 53–117.CrossRefGoogle Scholar
[42]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.CrossRefGoogle Scholar
[43]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar
[44]Zweimüller, R.. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998), 1263–1276.CrossRefGoogle Scholar
[45]Zweimüller, R.. Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 20 (2000), 1519–1549.CrossRefGoogle Scholar