Singular-hyperbolic attractors are chaotic

Araujo, V.; Pacifico, M.; Pujals, E.; Viana, M.

doi:10.1090/S0002-9947-08-04595-9

Singular-hyperbolic attractors are chaotic
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by V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana PDF

Trans. Amer. Math. Soc. 361 (2009), 2431-2485

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

V. S. Afraĭmovič, V. V. Bykov, and L. P. Sil′nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 336–339 (Russian). MR 0462175
Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
José Ferreira Alves and Vítor Araújo, Random perturbations of nonuniformly expanding maps, Astérisque 286 (2003), xvii, 25–62 (English, with English and French summaries). Geometric methods in dynamics. I. MR 2052296
D. V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst., 90:1–235, 1967.
D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
Aubin Arroyo and Enrique R. Pujals, Dynamical properties of singular-hyperbolic attractors, Discrete Contin. Dyn. Syst. 19 (2007), no. 1, 67–87. MR 2318274, DOI 10.3934/dcds.2007.19.67
S. Bautista and C. A. Morales. Existence of periodic orbits for singular-hyperbolic attractors. Preprint, IMPA Serie A 288/2004, 2004.
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180–193. MR 341451, DOI 10.1016/0022-0396(72)90013-7
C. M. Carballo, C. A. Morales, and M. J. Pacifico, Maximal transitive sets with singularities for generic $C^1$ vector fields, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 287–303. MR 1817090, DOI 10.1007/BF01241631
W. Colmenárez. SRB measures for singular hyperbolic attractors. Ph.D. thesis, UFRJ, Rio de Janeiro, 2002.
Wilmer J. Colmenárez Rodriquez, Nonuniform hyperbolicity for singular hyperbolic attractors, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4131–4140. MR 2159702, DOI 10.1090/S0002-9947-04-03706-7
A. Fathi, M.-R. Herman, and J.-C. Yoccoz, A proof of Pesin’s stable manifold theorem, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 177–215. MR 730270, DOI 10.1007/BFb0061417
J. Guckenheimer. A strange, strange attractor. In The Hopf bifurcation theorem and its applications, pages 368–381. Springer-Verlag, 1976.
John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), no. 1, 69–77. MR 422932, DOI 10.1007/BF01608556
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
Franz Hofbauer and Gerhard Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), no. 1, 119–140. MR 656227, DOI 10.1007/BF01215004
Gerhard Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 3, 461–478. MR 787608, DOI 10.1007/BF00532744
H. B. Keynes and M. Sears, $\scr F$-expansive transformation groups, General Topology Appl. 10 (1979), no. 1, 67–85. MR 519714, DOI 10.1016/0016-660X(79)90029-1
Yuri Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, vol. 16, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1015933, DOI 10.1007/978-1-4615-8181-9
M. Komuro, Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems (Kyoto, 1984) World Sci. Publishing, Singapore, 1984, pp. 4–26. MR 797594
R. Labarca and M. J. Pacífico, Stability of singularity horseshoes, Topology 25 (1986), no. 3, 337–352. MR 842429, DOI 10.1016/0040-9383(86)90048-0
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
E. N. Lorenz. Deterministic nonperiodic flow. J. Atmosph. Sci., 20:130–141, 1963.
Stefano Luzzatto, Ian Melbourne, and Frederic Paccaut, The Lorenz attractor is mixing, Comm. Math. Phys. 260 (2005), no. 2, 393–401. MR 2177324, DOI 10.1007/s00220-005-1411-9
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
Roger J. Metzger, Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 2, 247–276 (English, with English and French summaries). MR 1753089, DOI 10.1016/S0294-1449(00)00111-6
C. A. Morales and M. J. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001), no. 2, 359–378. MR 1819802, DOI 10.1088/0951-7715/14/2/310
Carlos Arnoldo Morales, Maria José Pacífico, and Enrique Ramiro Pujals, On $C^1$ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 1, 81–86 (English, with English and French summaries). MR 1649489, DOI 10.1016/S0764-4442(97)82717-6
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Strange attractors across the boundary of hyperbolic systems, Comm. Math. Phys. 211 (2000), no. 3, 527–558. MR 1773807, DOI 10.1007/s002200050825
C. A. Morales and E. R. Pujals, Singular strange attractors on the boundary of Morse-Smale systems, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 6, 693–717 (English, with English and French summaries). MR 1476293, DOI 10.1016/S0012-9593(97)89936-3
Carlos Arnoldo Morales, A note on periodic orbits for singular-hyperbolic flows, Discrete Contin. Dyn. Syst. 11 (2004), no. 2-3, 615–619. MR 2083434, DOI 10.3934/dcds.2004.11.615
C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1575–1600. MR 2018613, DOI 10.1017/S0143385702001621
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3393–3401. MR 1610761, DOI 10.1090/S0002-9939-99-04936-9
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 (2004), no. 2, 375–432. MR 2123928, DOI 10.4007/annals.2004.160.375
C. A. Morales, M. J. Pacifico, and B. San Martin, Expanding Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal. 36 (2005), no. 6, 1836–1861. MR 2178223, DOI 10.1137/S0036141002415785
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541, DOI 10.1007/978-1-4612-5703-5
Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
Ya. Pesin. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR. Izv., 10:1261–1302, 1976.
Ya. B. Pesin and Ya. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 417–438 (1983). MR 721733, DOI 10.1017/S014338570000170X
Charles Pugh and Michael Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), no. 1, 1–54. MR 983869, DOI 10.1090/S0002-9947-1989-0983869-1
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Warwick Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1197–1202 (English, with English and French summaries). MR 1701385, DOI 10.1016/S0764-4442(99)80439-X
Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117. MR 1870856, DOI 10.1007/s002080010018
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$^ \textrm {o}$ Colóquio Brasileiro de Matemática [21th Brazilian Mathematics Colloquium].
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
Sherman Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc. 246 (1978), 493–500. MR 515555, DOI 10.1090/S0002-9947-1978-0515555-9