Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T19:16:09.386Z Has data issue: false hasContentIssue false
  • English
  • Français

On the statistical stability of Lorenz attractors with a $C^{1+\unicode[STIX]{x1D6FC}}$ stable foliation

Published online by Cambridge University Press:  16 April 2018

WAEL BAHSOUN  and
MARKS RUZIBOEV
WAEL BAHSOUN
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK email [email protected], [email protected]
MARKS RUZIBOEV
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK email [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove statistical stability for a family of Lorenz attractors with a $C^{1+\unicode[STIX]{x1D6FC}}$ stable foliation.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3169 - 3184
Copyright
© Cambridge University Press, 2018 

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

Afraimovich, V. S., Bykov, V. V. and Sil’nikov, L. P.. The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR 234(2) (1977), 336–339.Google Scholar
Alves, J. F.. Strong statistical stability of non-uniformly expanding maps. Nonlinearity 17(4) (2004), 1193–1215.Google Scholar
Alves, J. F. and Soufi, M.. Statistical stability of geometric Lorenz attractors. Fund. Math. 224(3) (2014), 219–231.Google Scholar
Alves, J. F. and Viana, M.. Statistical stability for robust classes of maps with non-uniform expansion. Ergod. Th. & Dynam. Sys. 22(1) (2002), 1–32.Google Scholar
Araújo, V. and Melbourne, I.. Exponential decay of correlations for nonuniformly hyperbolic flows with a C 1+𝛼 stable foliation, including the classical Lorenz attractor. Ann. Henri Poincaré 17(11) (2016), 2975–3004.Google Scholar
Araújo, V. and Melbourne, I.. Existence and smoothness of the stable foliation for sectional hyperbolic attractors. Bull. Lond. Math. Soc. 49(2) (2017), 351–367.Google Scholar
Araújo, V., Melbourne, I. and Varandas, P.. Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps. Comm. Math. Phys. 340(3) (2015), 901–938.Google Scholar
Araújo, V. and Pacífico, M. J.. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 53. Springer, Berlin, 2010.Google Scholar
Araújo, V., Pacífico, M. J., Pujals, E. R. and Viana, M.. Singular-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc. 361(5) (2009), 2431–2485.Google Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16) . World Scientific, Singapore, 2000.Google Scholar
Bortolotti, R. T.. Physical measures for certain partially hyperbolic attractors on 3-manifolds. Ergod. Th. & Dynam. Sys. https://doi.org/10.1017/etds.2017.24. Published online 8 May 2017.Google Scholar
Boyarsky, A. and Góra, P.. Laws of Chaos, Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser, Boston, 1997.Google Scholar
Galatolo, S. and Lucena, R.. Spectral gap and quantitative statistical stability for systems with contracting fibres and Lorenz like maps. Preprint, 2017, arXiv:1507.08191.Google Scholar
Gukenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 59–72.Google Scholar
Holland, M. and Melbourne, I.. Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. (2) 76(2) (2007), 345–364.Google Scholar
Keller, G.. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4) (1982), 313–333.Google Scholar
Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3) (1985), 461–478.Google Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1) (1999), 141–152.Google Scholar
Lorenz, E. D.. Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1963), 130–141.Google Scholar
Luzzatto, S., Melbourne, I. and Paccaut, F.. The Lorenz attractor is mixing. Comm. Math. Phys. 260 (2005), 393–401.Google Scholar
Morales, C. A., Pacífico, M. J. and Pujals, E. R.. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. of Math. (2) 160(2) (2004), 375–432.Google Scholar
Tucker, W.. The Lorenz attractor exists. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 1197–1202.Google Scholar
Tucker, W.. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2 (2002), 53–117.Google Scholar
Young, L.-S.. What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108 (2002), 733–754.Google Scholar