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Existence of \(C^{k}\)-Invariant Foliations for Lorenz-Type Maps

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Abstract

Under conditions similar to those in Shashkov and Shil’nikov (Differ Uravn 30(4):586–595, 732, 1994) we show that a \(C^{k+1}\) Lorenz-type map T has a \(C^{k}\) codimension one foliation which is invariant under the action of T. This allows us to associate T to a \(C^{k}\) one-dimensional transformation.

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References

  1. Afraĭmovič, V.S., Bykov, V.V., Sil’nikov, L.P.: The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR 234(2), 336–339 (1977)

    MathSciNet  Google Scholar 

  2. V.S. Afraĭmovich and Ya.B. Pesin: Dimension of Lorenz type attractors. In: Mathematical Physics Reviews. Soviet Sci. Rev. Sect. C Math. Phys. Rev., vol. 6, pp. 169–241. Harwood Academic Publishing, Chur (1987)

  3. Araújo, V., Melbourne, I.: Exponential decay of correlations for nonuniformly hyperbolic flows with a \(C^{1+\alpha }\) stable foliation, including the classical Lorenz attractor. Preprint (2015). arXiv:1504.04316

  4. Araújo, V., Pacifico, M.J.: Three-Dimensional Flows, vol. 53 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Heidelberg (2010)

  5. Araújo, V., Melbourne, I., Varandas, P.: Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps. Commun. Math. Phys. 340(3), 901–938 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Araújo, V., Varandas, P.: Robust exponential decay of correlations for singular-flows. Commun. Math. Phys. 311(1), 215–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Belickiĭ, G.R.: Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class. Funkcional. Anal. i Priložen. 7(4), 17–28 (1973)

    MathSciNet  Google Scholar 

  8. Brandão, P: On the structure of lorenz maps. Preprint (2014). arXiv:1402.2862

  9. Cohen, D.I.A.: Basic Techniques of Combinatorial Theory. Wiley, New York (1978)

    MATH  Google Scholar 

  10. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969). Enlarged and corrected printing: Pure Appl. Math. 10-I

  11. Guckenheimer, J.: A strange, strange attractor, in the Hopf bifurcation and its applications. Appl. Math. Ser. 19, 368–381 (1976)

    Google Scholar 

  12. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. In: Applied Mathematical Sciences, vol. 42. Springer, New York (1983)

  13. Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 59–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hirsch, M.W., Pugh, C.C.: Stable manifolds for hyperbolic sets. Bull. Am. Math. Soc. 75, 149–152 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jakobson, M.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  17. Martens, M., de Melo, W.: Universal models for Lorenz maps. Ergodic Theory Dyn. Syst. 21, 833–860 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martens, M., Winckler, B.: On the hyperbolicity of Lorenz renormalization. Commun. Math. Phys. 325(1), 185–257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Martens, M., Winckler, B: Physical measures for infinitely renormalizable Lorenz maps. Preprint (2014). arXiv:1412.8041

  20. Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)

    Book  Google Scholar 

  21. Morales, C.A., Pacifico, M.J., Pujals, E.R.: Strange attractors across the boundary of hyperbolic systems. Commun. Math. Phys. 211(3), 527–558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Palis, J., Brandão, P., Pinheiro, V.: On the finiteness of attractors for one-dimensional maps with discontinuities. Preprint (2016). arXiv:1401.0232

  23. Palis, J., Brandão, P., Pinheiro, V.: On the finiteness of attractors for piecewise \(C^2\) maps of the interval. Preprint (2016). arXiv:1506.00276

  24. Robinson, C: Differentiability of the stable foliation for the model Lorenz equations. In: Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), vol. 898 of Lecture Notes in Mathematics, pp. 302–315. Springer, Berlin (1981)

  25. Robinson, C.: Transitivity and invariant measures for the geometric model of the Lorenz equations. Ergodic Theory Dyn. Syst. 4(4), 605–611 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rovella, A.: The dynamics of perturbations of the contracting Lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24(2), 233–259 (1993)

  27. Rychlik, M.R.: Lorenz attractors through Šil’nikov-type bifurcation. I. Ergodic Theory Dyn. Syst. 10(4), 793–821 (1990)

  28. Shashkov, M.V., Shil’nikov, L.P.: On the existence of a smooth invariant foliation in Lorenz-type mappings. Differ. Uravn. 30(4):586–595, 732 (1994)

  29. Viana, M.: What’s new on Lorenz strange attractors? Math. Intell. 22(3), 6–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vidarte, J.: Smooth perturbation of Lorenz-Like flow. PhD Thesis, ICMC-USP (2014). http://www.teses.usp.br/teses/disponiveis/55/55135/tde-15072014-155326/en.php

  31. Williams, R.F.: The structure of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 73–99 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work is based on the Ph.D. Thesis of the second author. J. V. was partially supported by FAPESP 2009/17153-9. D.S. was partially supported by CNPq 305537/2012-1. The authors thank the referee for constructive and helpful comments and suggestions that improved this work. Furthermore, the authors thank Nancy Chachapoyas, Luis Mello, Leandro Gomes and Pouya Mehdipour for several useful comments.

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Authors and Affiliations

  1. Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos, SP, CEP 13560-970, Brazil

    Daniel Smania

  2. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS, 1303 Pinheirinho, Itajubá, MG, 37500-903, Brazil

    José Vidarte

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  1. Daniel Smania
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Correspondence to José Vidarte.

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Smania, D., Vidarte, J. Existence of \(C^{k}\)-Invariant Foliations for Lorenz-Type Maps. J Dyn Diff Equat 30, 227–255 (2018). https://doi.org/10.1007/s10884-016-9539-1

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