Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T13:13:01.506Z Has data issue: false hasContentIssue false
  • English
  • Français

A sectional-Anosov connecting lemma

Published online by Cambridge University Press:  21 July 2009

S. BAUTISTA  and
C. MORALES
S. BAUTISTA
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogota, Colombia (email: sbautistad@unal.edu.co)
C. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: morales@impa.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

The Anosov flows on compact manifolds M satisfy the following property: if p,q are points such that for all positive ϵ there is a trajectory from a point ϵ-close to p to a point ϵ-close to q, then there is a point whose α-limit set is that of p and whose ω-limit set is that of q. Here we give a version of this property for sectional-Anosov flows, namely, vector fields inwardly transverse to the boundary whose maximal invariant set is sectional-hyperbolic. Indeed, if in addition M is three-dimensional and p has non-singular α-limit set, then there is a point whose α-limit set is that of p and whose ω-limit set is either a singularity or that of q.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 339 - 359
Copyright
Copyright © Cambridge University Press 2009

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]Afraimovich, V. S., Bykov, V. V. and Shilnikov, L. P.. On attracting structurally unstable limit sets of Lorenz attractor type. Trudy Moskov. Mat. Obshch. 44 (1982), 150–212 (Russian).Google Scholar
[2]Araujo, V. and Pacifico, M. J.. Three dimensional flows. 26 ∘ Colóquio Brasileiro de Matemática (Publicações Matemáticas do IMPA). Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007.Google Scholar
[3]Bautista, S. and Morales, C.. Existence of periodic orbits for sectional-hyperbolic sets. Moscow Math. J. 6(2) (2006), 265–297, 406.CrossRefGoogle Scholar
[4]Bautista, S. and Morales, C.. Characterizing omega-limit sets which are closed orbits. J. Differential Equations 245(3) (2008), 637–652.CrossRefGoogle Scholar
[5]Bautista, S., Morales, C. and Pacifico, M. J.. On the intersection of homoclinic classes on sectional-hyperbolic sets. Discrete Contin. Dyn. Syst. 19(4) (2007), 761–775.CrossRefGoogle Scholar
[6]Bonatti, C., Diaz, L. and Viana, M.. Dynamics beyond uniform hyperbolicity. Mathematical Physics, III (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
[7]Carballo, C. and Morales, C.. Omega-limit sets close to sectional-hyperbolic attractors. Illinois J. Math. 48 (2004), 645–663.CrossRefGoogle Scholar
[8]de Melo, W. and Palis, J.. Geometric Theory of Dynamical Systems. An Introduction. Springer, New York, 1982. Translated from the Portuguese by A. K. Manning.Google Scholar
[9]Fried, D.. Transitive Anosov flows and pseudo-Anosov maps. Topology 22(3) (1983), 299–303.CrossRefGoogle Scholar
[10]Guckenheimer, J. and Williams, R.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Etudes Sci. 50 (1979), 59–72.CrossRefGoogle Scholar
[11]Hirsch, M., Palis, J., Pugh, C. and Shub, M.. Neighborhoods of hyperbolic sets. Invent. Math. 9 (1969/1970), 121–134.CrossRefGoogle Scholar
[12]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[13]Hunt, T. J. and MacKay, R. S.. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16(4) (2003), 1499–1510.CrossRefGoogle Scholar
[14]Lorenz, E.. Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[15]Marsden, J. E. and McCracken, M.. The Hopf Bifurcation and its Applications (Applied Mathematical Sciences, 19). Springer, New York, 1976, With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale.CrossRefGoogle Scholar
[16]Metzger, R. and Morales, C.. Sectional-hyperbolic systems. Ergod. Th. & Dynam. Sys. 28 (2008), 1587–1597.CrossRefGoogle Scholar
[17]Morales, C.. Sectional-Anosov flows. Monatsh. Math. to appear.Google Scholar
[18]Morales, C.. A singular-hyperbolic closing lemma. Michigan Math. J. 56(1) (2008), 29–53.CrossRefGoogle Scholar
[19]Morales, C.. The explosion of singular-hyperbolic attractors. Ergod. Th. & Dynam. Sys. 24 (2004), 577–591.CrossRefGoogle Scholar
[20]Morales, C. and Pacifico, M. J.. Mixing attractors for 3-flows. Nonlinearity 14 (2001), 359–378.CrossRefGoogle Scholar
[21]Morales, C. and Pacifico, M.. A spectral decomposition for sectional-hyperbolic sets. Pacific J. Math. 229(1) (2007), 223–232.CrossRefGoogle Scholar
[22]Morales, C. and Pacifico, M. J.. Sufficient conditions for robustness of attractors. Pacific J. Math. 216(2) (2004), 327–342.CrossRefGoogle Scholar
[23]Morales, C., Pacifico, M. J. and Pujals, E. R.. Singular-hyperbolic Systems. Proc. Amer. Math. Soc. 127 (1999), 3393–3401.CrossRefGoogle Scholar
[24]Morales, C., Pacifico, 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.CrossRefGoogle Scholar
[25]Morales, C., Pacifico, M. J. and Pujals, E. R.. On C1 robust singular transitive sets for three-dimensional flows. C. R. Acad. Sci. Paris Sér. I Math. 326(1) (1998), 81–86.CrossRefGoogle Scholar
[26]Pesin, Y. B.. Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic. Proceedings of the International Congress of Mathematicians, Vol. 1 and 2 (Berkeley, CA, 1986). American Mathematical Society, Providence, RI, 1987, pp. 1195–1209.Google Scholar
[27]Pesin, Y. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12(1) (1992), 123–151.CrossRefGoogle Scholar
[28]Noda, T.. Projectively Anosov flows with differentiable (un)stable foliations. Ann. Inst. Fourier (Grenoble) 50(5) (2000), 1617–1647 (in English with a French summary).CrossRefGoogle Scholar
[29]Shilnikov, L. P. and Turaev, D.. An example of a wild strange attractor. Mat. Sb. 189(2) (1998), 137–160 (in Russian). (Engl. transl. Sb. Math. 189(1–2) (1998), 291–314).Google Scholar
[30]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[31]Tucker, W.. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2(1) (2002), 53–117.CrossRefGoogle Scholar