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Original Articles

Recent progress on sectional-hyperbolic systems

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Pages 369-382 | Received 02 Dec 2014, Accepted 20 May 2015, Published online: 22 Jun 2015

References

  • Shil'nikov LP, Turaev D. An example of a wild strange attractor. Mat Sb. 1998;189(2):137–160; translation in Sb Math. 1998;189(1–2):291–314.
  • Morales CA, Pacífico MJ, Pujals ER. On C1 robust singular transitive sets for three-dimensional flows. C R Acad Sci Paris Sér I Math. 1998;326(1):81–86.
  • Morales CA, Pacífico MJ, Pujals ER. Global attractors from the explosion of singular cycles. C R Acad Sci Paris Sér I Math. 1997;325(12):1317–1322.
  • Morales CA. Sectional-Anosov flows. Monatsh Math. 2010;159(3):253–260.
  • Morales CA. Singular-hyperbolic attractors with handlebody basins. J Dyn Control Syst. 2007;13(1):15–24.
  • Morales CA. Examples of singular-hyperbolic attracting sets. Dyn Syst. 2007;22(3):339–349.
  • Morales CA, Pacifico MJ. A dichotomy for three-dimensional vector fields. Ergodic Theory Dyn Syst. 2003;23(5):1575–1600.
  • Morales CA, Pacifico MJ, Pujals ER. Singular hyperbolic systems. Proc Am Math Soc. 1999;127(11):3393–3401.
  • Morales CA, Pacifico MJ. Sufficient conditions for robustness of attractors. Pac J Math. 2004;216(2):327–342.
  • Morales CA. A note on periodic orbits for singular-hyperbolic flows. Discrete Contin Dyn Syst. 2004;11(2–3):615–619.
  • Bonatti C, Pumariño A, Viana M. Lorenz attractors with arbitrary expanding dimension. C R Acad Sci Paris Sér I Math. 1997;325(8):883–888.
  • Metzger R, Morales CA. Sectional-hyperbolic systems. Ergodic Theory Dyn Syst. 2008;28(5):1587–1597.
  • Li M, Gan S, Wen L. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin Dyn Syst. 2005;13(2):239–269.
  • Morales CA. Strong stable manifolds for sectional-hyperbolic sets. Discrete Contin Dyn Syst. 2007;17(3):553–560.
  • Hirsch MW, Pugh CC, Shub M. Invariant manifolds. Lecture notes in mathematics. Vol. 583. Berlin: Springer-Verlag; 1977.
  • Morales CA, Vilches M. On 2-Riemannian manifolds. SUT J Math. 2010;46(1):119–153.
  • Gähler S. Lineare 2-normierte Räume. Math Nachr. 1964;28:1–43.
  • Kawaguchi A. On areal spaces, I. Metric tensors in n-dimensional spaces based on the notion of two-dimensional area. Tensor NS. 1950;1:14–45.
  • Araujo A. Existência de atratores hiperbólicos para difeomorfismos de superficies [Existence of hyperbolic attractors for surface diffeomorphisms]. Prepublicações IMPA Série F, no. 23/88; 1988. Portuguese.
  • Crovisier S, Pujals ER. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Preprint arXiv:1011.3836v1 [math.DS]; 2010 Nov 16.
  • Katok A, Hasselblatt B. Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Vol. 54, Encyclopedia of mathematics and its applications. Cambridge: Cambridge University Press; 1995.
  • Palis J, Takens F. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors, Vol. 35, Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press; 1993.
  • Afraimovic VS, Bykov VV, Shilnikov LP. The origin and structure of the Lorenz attractor. Dokl Akad Nauk SSSR. 1977;234(2):336–339.
  • Guckenheimer J, Williams R. Structural stability of Lorenz attractors. Publ Math IHES. 1979;50:59–72.
  • Bautista S, Morales CA. Lectures on sectional-Anosov flows. Preprint IMPA Série D 84; 2011.
  • Bautista S, Morales C, Pacifico MJ. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete Contin Dyn Syst. 2007;19(4):761–775.
  • Carballo CM, Morales CA. Omega-limit sets close to singular-hyperbolic attractors. Illinois J Math. 2004;48(2):645–663.
  • Arbieto A, Morales CA, Senos L. On the sensitivity of sectional-Anosov flows. Math Z. 2012;270(1–2):545–557.
  • Bautista S, Morales CA. On the essential hyperbolicity of sectional-Anosov flows. Proc Am Math Soc. Forthcoming.
  • Bautista S, Morales CA. Existence of periodic orbits for singular-hyperbolic sets. Mosc Math J. 2006;6(2):265–297.
  • Labarca R, Pacífico MJ. Stability of singularity horseshoes. Topology. 1986;25(3):337–352.
  • Banks J, Brooks J, Cairns G, et al. On Devaney's definition of chaos. Am Math Mon. 1992;99(4):332–334.
  • Guckenheimer J. Sensitive dependence to initial conditions for one-dimensional maps. Comm Math Phys. 1979;70(2):133–160.
  • Lorenz EN. Predictability: does the flap of a butterfly's wings in Brazil set off a tornado in Texas? American Association for the Advancement of Science. Cambridge, MA: MIT; 1972.
  • Polo F. Sensitive dependence on initial conditions and chaotic group actions. Proc Am Math Soc. 2010;138(8):2815–2826.
  • Ruelle D. Microscopic fluctuations and turbulence. Phys Lett A. 1979;72(2):81–82.
  • Sinai YG. Chaos theory yesterday, today and tomorrow. J Stat Phys. 2010;138(1–3):2–7.
  • Bautista S, Morales CA. Characterizing omega-limit sets which are closed orbits. J Differ Equ. 2008;245(3):637–652.
  • Morales CA. An improved sectional-Anosov closing lemma. Math Z. 2011;268(1–2):317–327.
  • Apaza E, Mejia B, Morales CA. Topological properties of sectional-Anosov flows. Discrete Contin Dyn Syst. 2015;35(10):4735–4741.
  • Sodero T. Sectional-Anosov flows on certain compact 3-manifolds. Bull Braz Math Soc (NS). 2011;42(3):439–454.
  • Morales CA. A note on periodic orbits for singular-hyperbolic flows. Discrete Contin Dyn Syst. 2004;11(2–3):615–619.
  • Bautista S. The geometric Lorenz attractor is a homoclinic class. Bol Mat (NS). 2004;11(1):69–78.
  • Arroyo A, Pujals ER. Dynamical properties of singular-hyperbolic attractors. Discrete Contin Dyn Syst. 2007;19(1):67–87.
  • Araujo V, Pacifico MJ. Three-dimensional flows. With a foreword by Marcelo Viana. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in mathematics and related areas]. 3rd Series, Vol. 53, A series of modern surveys in mathematics. Heidelberg: Springer; 2010.
  • Yang J. Gibbs cu-states and applications. Paper presented at: The International Conference on Dynamical Systems during the Semester on Dynamics Beyond Hyperbolicity. IMPA; 2013 August–November; Rio de Janeiro, Brazil.
  • Arbieto A, Morales CA, A.M. Lopez B. Homoclinic classes for sectional-hyperbolic sets. Preprint arxiv.org/abs/1408.3947v1.
  • Nakai N. Existence of periodic orbits for singular-hyperbolic Lyapunov stable sets. Preprint 2009 (unpublished).
  • Nakai N. Existence of periodic orbits for singular-hyperbolic Lyapunov stable sets. Preprint arXiv:1501.04339.
  • Reis J. Existence of infinitely many periodic orbits for sectional-Anosov flows [dissertation]. Rio de Janeiro: UFRJ; 2011.
  • Lopes, A.M. Existence of periodic orbits for higher dimensional sectional-Anosov flows [dissertation]. Rio de Janeiro: Departamento de Matemática, Instituto de Matemática, Universidade Federal do Rio de Janeiro; 2015.
  • Colmenarez W. Medidas físicas para atractores hiperbólicos singulares [Existence of physical measures for singular-hyperbolic attractors]. Trabajo de Ascenso presentado para optar a la categoría de Profesor ASOCIADO en el escalafón del Personal Docente y de Investigación, Universidad Centroccidental Lisandro Alvarado, Decanato de Ciencias y Tecnología, Barquisimeto, Venezuela; 2008. Spanish.
  • Araujo V, Pacifico MJ, Pujals ER, et al. Singular-hyperbolic attractors are chaotic. Trans Am Math Soc. 2009;361(5):2431–2485.
  • Metzger R, Morales CA. Stochastic stability of sectional-Anosov flows. Preprint arXiv:1505.01761.
  • Crovisier S, Yang D. On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis. Preprint arXiv:1404.5130v1 [math.DS]; 2014 Apr 21.
  • Palis J, Pugh CC. Fifty problems in dynamical systems. In: Palis J, Pugh CC, editors. Dynamical systems — Warwick 1974 (Proceedings of symposium on applied topology and dynamical systems, University of Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday). Lecture notes in mathematics, Vol. 468. Berlin: Springer; 1975. p. 345–353.
  • Arbieto A, Morales CA, Santiago B. On Araujo's theorem for flows. J Dyn Control Syst. Forthcoming.
  • Morales CA. Existence of attractors for three-dimensional flows. 2014. Preprint arXiv:1405.5069.
  • Arbieto A, Morales CA, Santiago B. Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows. Math Ann. 2015;361(1–2):67–75.

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