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Sectional-hyperbolic systems

Published online by Cambridge University Press:  01 October 2008

R. METZGER  and
C. MORALES
R. METZGER
Affiliation:
Instituto de Matemática y Ciencias Afines IMCA, Jr. Ancash 536. Lima 1., Casa de las Trece Monedas, Perú (email: metzger@uni.edu.pe)
C. MORALES
Affiliation:
Instituto de Matematica, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970 Rio de Janeiro, Brazil (email: morales@impa.br)
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Abstract

We introduce a class of vector fields on n-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Li et al [Robust transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst.13(2) (2005), 239–269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Li et al are similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem 9.26 in Bonatti et al [Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1587 - 1597
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Araujo, V., Pacifico, M. J., Pujals, E. R. and Viana, M.. Singular-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc. in press.Google Scholar
[2]Afraimovich, V. S. and Hsu, S.-B.. Lectures on Chaotic Dynamical Systems (AMS/IP Studies in Advanced Mathematics, 28). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[3]Bautista, S. and Morales, C.. Existence of periodic orbits for singular-hyperbolic sets. Mosc. Math. J. 6(2) (2006), 265–297.CrossRefGoogle Scholar
[4]Beltran, J., Metzger, R. and Morales, C.. Ergodic properties of sectional-hyperbolic attractors, in preparation.Google Scholar
[5]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005.Google Scholar
[6]Bonatti, C., Gourmelon, N. and Vivier, T.. Perturbation of the derivative along periodic orbits. Ergod. Th. & Dynam. Sys. 26(5) (2006), 1307–1337.CrossRefGoogle Scholar
[7]Bonatti, Ch., Pumariño, A. and Viana, M.. Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris Sér. I Math. 325(8) (1997), 883–888.CrossRefGoogle Scholar
[8]Fowler, A. C. and Sparrow, C. T.. Bifocal homoclinic orbits in four dimensions. Nonlinearity 4(4) (1991), 1159–1182.CrossRefGoogle Scholar
[9]Hayashi, S.. Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows. Ann. of Math. (2) 145(1) (1997), 81–137.CrossRefGoogle Scholar
[10]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[11]Li, M., Gan, S. and Wen, L.. Robust transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst. 13(2) (2005), 239–269.CrossRefGoogle Scholar
[12]Milnor, J. W. and Stasheff, J. D.. Characteristic Classes (Annals of Mathematics Studies, 76). Princeton University Press, Princeton, NJ, 1974.CrossRefGoogle Scholar
[13]Morales, C. A., Pacifico, M. J. and Pujals, E. R.. On C 1 robust singular transitive sets for three-dimensional flows. C. R. Acad. Sci. Paris Ser. I Math. 326(1) (1998), 81–86.CrossRefGoogle Scholar
[14]Morales, C. A., 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
[15]Morales, C. A., Pacifico, M. J. and Pujals, E. R.. Singular hyperbolic systems. Proc. Amer. Math. Soc. 127(11) (1999), 3393–3401.CrossRefGoogle Scholar
[16]Shilnikov, L. P. and Turaev, D. V.. An example of a wild strange attractor. Mat. Sb. 189(2) (1998), 137–160 (in Russian with Russian summary). Engl. Transl. Sb. Math. 189(1–2) (1998), 291–314.Google Scholar
[17]Shilnikov, L. P.. Existence of a countable set of periodic motions in a four-dimensional space in an extended neighborhood of a saddle-focus. Dokl. Akad. Nauk SSSR 172 (1967), 54–57 (in Russian).Google Scholar
[18]Wiggins, S.. Global Bifurcations and Chaos. Analytical Methods (Applied Mathematical Sciences, 73). Springer, New York, 1988.CrossRefGoogle Scholar