Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Singular-hyperbolic attractors are chaotic


Authors: V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana
Journal: Trans. Amer. Math. Soc. 361 (2009), 2431-2485
MSC (2000): Primary 37C10; Secondary 37C40, 37D30
DOI: https://doi.org/10.1090/S0002-9947-08-04595-9
Published electronically: December 3, 2008
MathSciNet review: 2471925
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a singular-hyperbolic attractor of a $ 3$-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $ u$-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction.

This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.

In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are $ u$-Gibbs states.


References [Enhancements On Off] (What's this?)

  • 1. V. S. Afraimovich, V. V. Bykov, and L. P. Shil'nikov.
    On the appearance and structure of the Lorenz attractor.
    Dokl. Acad. Sci. USSR, 234:336-339, 1977. MR 0462175 (57:2150)
  • 2. J. F. Alves, C. Bonatti, and M. Viana.
    SRB measures for partially hyperbolic systems whose central direction is mostly expanding.
    Invent. Math., 140(2):351-398, 2000. MR 1757000 (2001j:37063b)
  • 3. Jose F. Alves and Vitor Araujo.
    Random perturbations of nonuniformly expanding maps.
    Astérisque, 286:25-62, 2003. MR 2052296 (2005e:37058)
  • 4. D. V. Anosov.
    Geodesic flows on closed Riemannian manifolds of negative curvature.
    Proc. Steklov Math. Inst., 90:1-235, 1967.
  • 5. D. V. Anosov and Ja. G. Sinaĭ.
    Certain smooth ergodic systems.
    Uspehi Mat. Nauk, 22(5 (137)):107-172, 1967. MR 0224771 (37:370)
  • 6. A. Arroyo and E. R. Pujals.
    Dynamical properties of singular hyperbolic attractors.
    Preprint IMPA Serie A, 292/2004. MR 2318274
  • 7. S. Bautista and C. A. Morales.
    Existence of periodic orbits for singular-hyperbolic attractors.
    Preprint, IMPA Serie A 288/2004, 2004.
  • 8. R. Bowen.
    Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math.
    Springer-Verlag, 1975. MR 0442989 (56:1364)
  • 9. R. Bowen and D. Ruelle.
    The ergodic theory of Axiom A flows.
    Invent. Math., 29:181-202, 1975. MR 0380889 (52:1786)
  • 10. R. Bowen and P. Walters.
    Expansive one-parameter flows.
    J. Differential Equations, 12:180-193, 1972. MR 0341451 (49:6202)
  • 11. C. M. Carballo, C. A. Morales, and M. J. Pacifico.
    Maximal transitive sets with singularities for generic $ C\sp 1$ vector fields.
    Bol. Soc. Brasil. Mat. (N.S.), 31(3):287-303, 2000. MR 1817090 (2002b:37075)
  • 12. W. Colmenárez.
    SRB measures for singular hyperbolic attractors.
    Ph.D. thesis, UFRJ, Rio de Janeiro, 2002.
  • 13. Wilmer J. Colmenárez Rodriquez.
    Nonuniform hyperbolicity for singular hyperbolic attractors.
    Trans. Amer. Math. Soc., 357(10):4131-4140 (electronic), 2005. MR 2159702 (2006d:37048)
  • 14. A. Fathi, M.-R. Herman, and J.-C. Yoccoz.
    A proof of Pesin's stable manifold theorem.
    In Geometric dynamics (Rio de Janeiro, 1981), volume 1007 of Lecture Notes in Math., pages 177-215. Springer, Berlin, 1983. MR 730270 (85j:58122)
  • 15. J. Guckenheimer.
    A strange, strange attractor.
    In The Hopf bifurcation theorem and its applications, pages 368-381. Springer-Verlag, 1976.
  • 16. J. Guckenheimer and R. F. Williams.
    Structural stability of Lorenz attractors.
    Publ. Math. IHES, 50:59-72, 1979. MR 556582 (82b:58055a)
  • 17. M. Hénon.
    A two dimensional mapping with a strange attractor.
    Comm. Math. Phys., 50:69-77, 1976. MR 0422932 (54:10917)
  • 18. M. Hirsch, C. Pugh, and M. Shub.
    Invariant manifolds, volume 583 of Lect. Notes in Math.
    Springer-Verlag, New York, 1977. MR 0501173 (58:18595)
  • 19. F. Hofbauer and G. Keller.
    Ergodic properties of invariant measures for piecewise monotonic transformations.
    Math. Z., 180:119-140, 1982. MR 656227 (83h:28028)
  • 20. G Keller.
    Generalized bounded variation and applications to piecewise monotonic transformations.
    Z. Wahrsch. Verw. Gebiete, 69(3):461-478, 1985. MR 787608 (86i:28024)
  • 21. H. B. Keynes and M. Sears.
    $ \mathcal{F}$-expansive transformation groups.
    General Topology Appl., 10(1):67-85, 1979. MR 519714 (80f:54039)
  • 22. Yuri Kifer.
    Random perturbations of dynamical systems, volume 16 of Progress in Probability and Statistics.
    Birkhäuser Boston Inc., Boston, MA, 1988. MR 1015933 (91e:58159)
  • 23. M. Komuro.
    Expansive properties of Lorenz attractors.
    In The theory of dynamical systems and its applications to nonlinear problems, pages 4-26. World Sci. Publishing, Kyoto, 1984. MR 797594 (86j:58082)
  • 24. R. Labarca and M.J. Pacifico.
    Stability of singular horseshoes.
    Topology, 25:337-352, 1986. MR 842429 (87h:58106)
  • 25. F. Ledrappier and L.-S. Young.
    The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin's entropy formula.
    Ann. of Math, 122:509-539, 1985. MR 819556 (87i:58101a)
  • 26. E. N. Lorenz.
    Deterministic nonperiodic flow.
    J. Atmosph. Sci., 20:130-141, 1963.
  • 27. Stefano Luzzatto, Ian Melbourne, and Frederic Paccaut.
    The Lorenz attractor is mixing.
    Comm. Math. Phys., 260(2):393-401, 2005. MR 2177324 (2006g:37056)
  • 28. R. Mañé.
    Ergodic theory and differentiable dynamics.
    Springer-Verlag, New York, 1987. MR 889254 (88c:58040)
  • 29. Roger J. Metzger.
    Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 17(2):247-276, 2000. MR 1753089 (2001d:37037)
  • 30. C. Morales and M. J. Pacifico.
    Mixing attractors for 3-flows.
    Nonlinearity, 14(2):359-378, 2001. MR 1819802 (2002a:37036)
  • 31. C. Morales, M. J. Pacifico, and E. Pujals.
    On $ {C}^1$ robust singular transitive sets for three-dimensional flows.
    C. R. Acad. Sci. Paris, 326, Série I:81-86, 1998. MR 1649489 (99j:58183)
  • 32. C. Morales, M. J. Pacifico, and E. Pujals.
    Strange attractors across the boundary of hyperbolic systems.
    Comm. Math. Phys., 211(3):527-558, 2000. MR 1773807 (2001g:37036)
  • 33. C. Morales and E. Pujals.
    Singular strange attractors on the boundary of Morse-Smale systems.
    Ann. Sci. École Norm. Sup., 30:693-717, 1997. MR 1476293 (98k:58137)
  • 34. C. A. Morales.
    A note on periodic orbits for singular-hyperbolic flows.
    Discrete Contin. Dyn. Syst., 11(2-3):615-619, 2004. MR 2083434 (2005e:37060)
  • 35. C. A. Morales and M. J. Pacifico.
    A dichotomy for three-dimensional vector fields.
    Ergodic Theory Dynam. Systems, 23(5):1575-1600, 2003. MR 2018613 (2005a:37030)
  • 36. C. A. Morales, M. J. Pacifico, and E. R. Pujals.
    Singular hyperbolic systems.
    Proc. Amer. Math. Soc., 127(11):3393-3401, 1999. MR 1610761 (2000c:37034)
  • 37. C. A. Morales, M. J. Pacifico, and E. R. Pujals.
    Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers.
    Ann. of Math. (2), 160(2):375-432, 2004. MR 2123928 (2005k:37054)
  • 38. C. A. Morales, M. J. Pacifico, and B. San Martin.
    Expanding Lorenz attractors through resonant double homoclinic loops.
    SIAM J. Math. Anal., 36(6):1836-1861 (electronic), 2005. MR 2178223 (2006g:37075)
  • 39. J. Palis and W. de Melo.
    Geometric Theory of Dynamical Systems.
    Springer-Verlag, 1982. MR 669541 (84a:58004)
  • 40. J. Palis and F. Takens.
    Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations.
    Cambridge University Press, 1993. MR 1237641 (94h:58129)
  • 41. Ya. Pesin.
    Families of invariant manifolds corresponding to non-zero characteristic exponents.
    Math. USSR. Izv., 10:1261-1302, 1976.
  • 42. Ya. Pesin and Ya. Sinai.
    Gibbs measures for partially hyperbolic attractors.
    Ergod. Th. & Dynam. Sys., 2:417-438, 1982. MR 721733 (85f:58071)
  • 43. C. Pugh and M. Shub.
    Ergodic attractors.
    Trans. Amer. Math. Soc., 312:1-54, 1989. MR 983869 (90h:58057)
  • 44. S. Smale.
    Differentiable dynamical systems.
    Bull. Am. Math. Soc., 73:747-817, 1967. MR 0228014 (37:3598)
  • 45. W. Tucker.
    The Lorenz attractor exists.
    C. R. Acad. Sci. Paris, 328, Série I:1197-1202, 1999. MR 1701385 (2001b:37051)
  • 46. Warwick Tucker.
    A rigorous ODE solver and Smale's 14th problem.
    Found. Comput. Math., 2(1):53-117, 2002. MR 1870856 (2003b:37055)
  • 47. Marcelo Viana.
    Stochastic dynamics of deterministic systems.
    Publicações Matemáticas do IMPA [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997.
    21 $ \sp {\rm o}$ Colóquio Brasileiro de Matemática [21th Brazilian Mathematics Colloquium].
  • 48. P. Walters.
    An introduction to ergodic theory.
    Springer-Verlag, 1982. MR 648108 (84e:28017)
  • 49. R. F. Williams.
    The structure of Lorenz attractors.
    Inst. Hautes Études Sci. Publ. Math., 50:73-99, 1979. MR 556583 (82b:58055b)
  • 50. S. Wong.
    Some metric properties of piecewise monotonic mappings of the unit interval.
    Trans. Amer. Math. Soc., 246:493-500, 1978. MR 515555 (80c:28014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37C10, 37C40, 37D30

Retrieve articles in all journals with MSC (2000): 37C10, 37C40, 37D30


Additional Information

V. Araujo
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970, Rio de Janeiro, RJ-Brazil – and – Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: vitor.araujo@im.ufrj.br, vdaraujo@fc.up.pt

M. J. Pacifico
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, Brazil
Email: pacifico@im.ufrj.br, pacifico@impa.br

E. R. Pujals
Affiliation: IMPA, Estrada D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: enrique@impa.br

M. Viana
Affiliation: IMPA, Estrada D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: viana@impa.br

DOI: https://doi.org/10.1090/S0002-9947-08-04595-9
Keywords: Singular-hyperbolic attractor, Lorenz-like flow, physical measure, expansive flow, equilibrium state
Received by editor(s): July 5, 2006
Received by editor(s) in revised form: March 27, 2007
Published electronically: December 3, 2008
Additional Notes: The first author was partially supported by CMUP-FCT (Portugal), CNPq (Brazil) and grants BPD/16082/2004 and POCI/MAT/61237/2004 (FCT-Portugal) while enjoying a post-doctorate leave from CMUP at PUC-Rio and IMPA
The second, third and fourth authors were partially supported by PRONEX, CNPq and FAPERJ-Brazil
Article copyright: © Copyright 2008 by the authors

American Mathematical Society