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Chaos for successive maxima map implies chaos for the original map

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Abstract

\(\tau \) is a continuous map on a metric compact space \(X\). For a continuous function \(\phi :X\rightarrow \mathbb R\), we consider a one-dimensional map \(T\) (possibly multi-valued) which sends a local \(\phi \)-maximum on \(\tau \) trajectory to the next one: consecutive maxima map. The idea originated with famous Lorenz’s paper on strange attractor. We prove that if \(T\) has a horseshoe disjoint from fixed points, then \(\tau \) is in some sense chaotic, i.e., it has a turbulent trajectory and thus a continuous invariant measure.

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References

  1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, Textbooks in Mathematical Sciences. Springer, New York (1997)

    Google Scholar 

  2. Banks, J., Dragan, V., Jones, A.: Chaos, A Mathematical Introduction. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  3. Boyarsky, A., Góra, P.: Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension, Probability and Its Applications. Birkhäuser Boston Inc, Boston, MA (1997)

    Google Scholar 

  4. Boyarsky, A., Góra, P., Lioubimov, V.: On the existence of ergodic continuous invariant measures for folding transformations. Ergod. Theory Dyn. Syst. 20(1), 47–53 (2000)

    Article  MATH  Google Scholar 

  5. Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  6. Brown, J.R.: Ergod. Theory Topol. Dyn. Academic Press, NY (1976)

    Google Scholar 

  7. Cheng, W.-C.R.C.-C.H.N.G., Newhouse, S.E.: Pre-image entropy. Ergod. Theory Dyn. Syst. 25(4), 1091–1113 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, vol. 527. Springer, Berlin, Heidelberg, New York (1976). doi:10.1007/BFb0082364

    Google Scholar 

  9. Devaney, R.L.: A First Course in Chaotic Dynamical Systems. Theory and Experiment. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA (1992)

  10. Langevin, R., Walczak, P.: Entropie dune dynamique [Entropy of a dynamic]. C. R. Acad. Sci. Paris Sér. I Math. 312(1), 141–144 (1991)

    MATH  MathSciNet  Google Scholar 

  11. Langevin, R., Przytycki, F.: Entropie de l’image inverse d’une application [Entropy of the inverse image of a mapping]. Bull. Soc. Math. France 120(2), 237–250 (1992)

    MATH  MathSciNet  Google Scholar 

  12. Lasota, A., Yorke, J.A.: On the existence of invariant measures for transformations with strictly turbulent trajectories. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25(3), 233–238 (1977)

    MATH  MathSciNet  Google Scholar 

  13. Lasota, A., Pianigiani, G.: Invariant measures on topological spaces. Boll. Un. Mat. Ital. B (5) 14(2), 592–603 (1977)

    MATH  MathSciNet  Google Scholar 

  14. Lasota, A., Yorke, J.A.: On the existence of invariant measures for transformations with strictly turbulent trajectories. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25(3), 233–238 (1977)

    MATH  MathSciNet  Google Scholar 

  15. Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Applied Mathematical Sciences, vol. 97, 2nd edn. Springer, New York (1994)

    Google Scholar 

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

    Article  Google Scholar 

  17. Misiurewicz, M.: Horseshoes for mappings of the interval. Bull. Acad. Polon. Sci. Sér. Sci. Math. 27(2), 167–169 (1979)

    MATH  MathSciNet  Google Scholar 

  18. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Inst. Hautes Études Sci. Publ. Math. 53, 17–51 (1981)

  19. Nitecki, Z., Przytycki, F.: Preimage entropy for mappings. Int. J. Bifurcat. Chaos 9(9), 1815–1843 (1999)

  20. Robinson, C.: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)

    Google Scholar 

  21. Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Applied Mathematical Sciences, 41. Springer, New York, Berlin (1982)

    Book  Google Scholar 

  22. Sherwell, D., Visaya, V.: Complexity of multivalued maps. World Acad. Sci. Eng. Technol. 53, 717–720 (2011)

    Google Scholar 

  23. Tucker, W.: The Lorenz attractor exists. Comptes Rendus de l’Académie des Scienes Série I Mathématique 328, 1197–1202 (1999)

    MATH  Google Scholar 

  24. Visaya, M.V.V.: A lower estimate of the topological entropy from a one-dimensional reconstruction of time series. J. Math. Kyoto Univ. 46(3), 637–655 (2006)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors are very grateful to anonymous reviewers for comments which helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, QC, H3G 1M8, Canada

    Abraham Boyarsky & Paweł Góra

  2. Dipartimento di Matematica, Universita’ di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Roma, Italy

    Peyman Eslami

  3. Department of Mathematics, Honghe University, Mengzi, 661100, Yunnan, China

    Zhenyang Li

  4. Department of Mathematics, Baylor University, Waco, TX, 76798–7328, USA

    Jonathan Meddaugh & Brian E. Raines

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  1. Abraham Boyarsky
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  2. Peyman Eslami
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  3. Paweł Góra
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  4. Zhenyang Li
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  5. Jonathan Meddaugh
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  6. Brian E. Raines
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Correspondence to Paweł Góra.

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The research of the authors was supported by NSERC Grants.

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Boyarsky, A., Eslami, P., Góra, P. et al. Chaos for successive maxima map implies chaos for the original map. Nonlinear Dyn 79, 2165–2175 (2015). https://doi.org/10.1007/s11071-014-1802-6

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