Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T17:02:51.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical profile of a class of rank-one attractors

Published online by Cambridge University Press:  08 May 2012

QIUDONG WANG  and
LAI-SANG YOUNG
QIUDONG WANG
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA (email: [email protected])
LAI-SANG YOUNG
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1221 - 1264
Copyright
Copyright © 2012 Cambridge University Press 

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

[BC1]Benedicks, M. and Carleson, L.. On iterations of $1-ax^2$ on $(-1, 1)$. Ann. of Math. (2) 122 (1985), 1–25.Google Scholar
[BC2]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. of Math. (2) 133 (1991), 73–169.Google Scholar
[BV]Benedicks, M. and Viana, M.. Solution of the basin problem for Hénon attractors. Invent. Math. 143 (2001), 375–434.Google Scholar
[BY1]Benedicks, M. and Young, L.-S.. Sinai–Ruelle–Bowen measures for certain Hénon maps. Invent. Math. 112 (1993), 541–576.Google Scholar
[BY2]Benedicks, M. and Young, L.-S.. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261 (2000), 13–56.Google Scholar
[B1]Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[B2]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[C]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21 (2001), 401–420.CrossRefGoogle Scholar
[CE]Collet, P. and Eckmann, J. P.. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13–46.Google Scholar
[DSV]Dolgopyat, D., Szász, D. and Varjú, T.. Recurrence properties of Lorentz gas. Duke Math. J. 142 (2008), 241–281.Google Scholar
[GWY]Guckenheimer, J., Weschelberger, M. and Young, L.-S.. Chaotic attractors of relaxation oscillators. Nonlinearity 19 (2006), 701–720.Google Scholar
[HPS]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.Google Scholar
[HK]Hunt, B. and Kaloshin, V.. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms, part I. Ann. of Math. (2) 165 (2007), 89–170.Google Scholar
[Kl]Kaloshin, V.. Generic diffeomorphisms with superexponential growth of number of periodic points. Comm. Math. Phys. 211(1) (2000), 253–271.CrossRefGoogle Scholar
[Kt]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
[L]Ledrappier, F.. Propriétés ergodiques des mesures de Sinai. Publ. Math. Inst. Hautes Études Sci. 59 (1984), 163–188.CrossRefGoogle Scholar
[LY]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. of Math. (2) 122 (1985), 509–574.Google Scholar
[LWY]Lu, K., Wang, Q. and Young, L.-S.. Strange attractors for periodically forced parabolic equations. Preprint, 2007, Memoirs of the AMS to appear.Google Scholar
[MN]Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005), 131–146.Google Scholar
[M]Misiurewicz, M.. Absolutely continuous invariant measures for certain maps of an interval. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 17–51.CrossRefGoogle Scholar
[MV]Mora, L. and Viana, M.. Abundance of strange attractors. Acta. Math. 171 (1993), 1–71.Google Scholar
[P]Pesin, Y. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32(4) (1977), 55–114.Google Scholar
[PS]Pugh, C. and Shub, M.. Ergodic attractors. Trans. Amer. Math. Soc. 312 (1989), 1–54.Google Scholar
[RbY]Rey-Bellet, L. and Young, L.-S.. Large deviations in nonuniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28(2) (2008), 587–612.Google Scholar
[R]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27–58.Google Scholar
[S]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.Google Scholar
[SV]Szász, D. and Varjú, T.. Local limit theorem for the Lorentz process and its recurrence in the plane. Ergod. Th. & Dynam. Sys. 24 (2004), 257–278.Google Scholar
[T]Takahasi, H.. On the basin problem for Hénon-like attractors. J. Math. Kyoto Univ. 46(2) (2006), 303–348.Google Scholar
[TTY]Thieullen, P., Tresser, C. and Young, L.-S.. Positive exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64 (1994), 121–172.Google Scholar
[V]Viana, M.. Strange attractors in higher dimensions. Bull. Braz. Math. Soc. 24 (1993), 13–62.CrossRefGoogle Scholar
[Wa]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1981.Google Scholar
[WO]Wang, Q. and Ott, W.. Dissipative homoclinic loops and rank one chaos. Comm. Pure Appl. Math. 64(11) (2011), 1439–1496.Google Scholar
[WY1]Wang, Q. and Young, L.-S.. Strange attractors with one direction of instability. Comm. Math. Phys. 218 (2001), 1–97.Google Scholar
[WY2]Wang, Q. and Young, L.-S.. Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480.Google Scholar
[WY3]Wang, Q. and Young, L.-S.. Non-uniformly expanding 1D maps. Comm. Math. Phys. 264(1) (2006), 255–282.Google Scholar
[WY4]Wang, Q. and Young, L.-S.. From invariant curves to strange attractors. Comm. Math. Phys. 225 (2002), 275–304.Google Scholar
[WY5]Wang, Q. and Young, L.-S.. Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Comm. Math. Phys. 240 (2002), 509–529.Google Scholar
[Y1]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.Google Scholar
[Y2]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar
[Y3]Young, L.-S.. Ergodic theory of differentiable dynamical systems. Real and Complex Dynamical Systems. Eds. Branner, B. and Hjorth, P.. Kluwer Academic Publishers, Dordrecht, 1995, pp. 293–336.Google Scholar