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A rational transformation conjugated to the Arnold-Thom Cat map

An application of the Schröder method to hyperbolic toral maps

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Abstract

We compute a vector-valued rational map conjugated to the Arnold-Thom Cat map by means of Schröder’s method. The corresponding invariant density and the spectrum of Rényi information are analytically determined. We show that for the analogous n-dimensional case the invariant density is the product of n Cauchy densities. Hence, the spectrum of Rényi information is n times the corresponding Rényi information of a Cauchy density. We also work out two three-dimensional linear toral maps as an example of the general case.

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References

  1. L. Kocarev, S. Lian, Chaos-based Cryptography Theory, Algorithms and Applications. Studies in Computational Intelligence, Vol. 354 (Springer-Verlag, 2011)

  2. K. Young, J.D. Scargle, The Astrophys. J. 468, 617 (1996)

    Article  ADS  Google Scholar 

  3. J.H. Hannay, M.V. Berry, Physica D 1, 267291 (1980)

    MathSciNet  Google Scholar 

  4. J.P. Keating, F. Mezzadri, New Directions in Quantum Chaos Proceedings of the International School of Physics “Enrico Fermi”, Course CXLIII, Società Italiana di Fisica (2000), p. 449

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

  6. H. Broer, F. Takens, Dynamical Systems and Chaos (Springer Science+Business Media, LLC 2011)

  7. P. Collet, J.P. Eckmann, Concepts and Results in Chaotic Dynamics: A Short Course (Springer-Verlag, Berlin, Heidelberg, 2006), p. 230

  8. B. Hasselblatt, A. Katok, A First Course in Dynamics with a Panorama of Recent Developments (Cambridge University Press, NY, USA, 2003)

  9. B. Hasselblatt, A. Katok, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995)

  10. E. Schröder, Math. Ann. 3, 296 (1871). Available through http://gdz.sub.uni-goettingen.de/en (in German, search by title)

    Article  MATH  Google Scholar 

  11. Y.B. Pesin, Dimension Theory in Dynamical Systems Contemporary views and applications (The University of Chicago Press, Chicago, USA, 1997)

  12. C. Beck, F. Schlogl, Thermodynamic of Chaotic Systems (Cambridge University Press, Cambridge, 1993)

  13. J.R. Luévano, Rev. Mex. Fís. S 58(1), 21 (2012)

    Google Scholar 

  14. J.R. Luévano, E. Piña, J. Phys. A: Math. Theor. 41, 265101 (2008)

    Article  ADS  Google Scholar 

  15. K. Sogo, A. Masumizu, Phys. Lett. A 375, 3511 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  16. J. Honerkamp, Statistical Physics. An Advanced Approach with Applications, second edition (Springer-Verlag, Berlin Heidelberg, 2002)

  17. V. Arnold, Seminar at ICTP (2008), Personal WEB site: http://www.pdmi.ras.ru/arnsem/

  18. N. Leonenko, L. Pronzato, V. Savani, The Ann. Stat. 36, 2153 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Unidad Azcapotzalco, CP 02200, México, DF, México

    J.-R. Luévano

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  1. J.-R. Luévano
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Correspondence to J.-R. Luévano.

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Luévano, JR. A rational transformation conjugated to the Arnold-Thom Cat map. Eur. Phys. J. Spec. Top. 223, 2959–2968 (2014). https://doi.org/10.1140/epjst/e2014-02309-5

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  • DOI: https://doi.org/10.1140/epjst/e2014-02309-5

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