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Anikin V. M., Arkadaksky S. S., Remizov A. S., Kuptsov S. N., Vasilenko L. P. Investigation of structure of invariant density for Renyi map by Gauss method. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 6, pp. 46-56. DOI: 10.18500/0869-6632-2008-16-6-46-56

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Investigation of structure of invariant density for Renyi map by Gauss method

Anikin Valerij Mihajlovich, Saratov State University
Arkadaksky Sergej Sergeevich, Saratov State University
Remizov Aleksandr Sergeevich, Saratov State University
Kuptsov Sergej Nikolaevich, Saratov State University
Vasilenko Leonid Petrovich, Saratov State University

It is shown that the structure of the invariant density for Renyi map xn+1 = = βxn mod 1, (1 < β < 2) may be clarified by action of the Perron–Frobenius operator on the uniform distribution. The invariant density is presented by finite linear combination of characteristic functions defined on the unit interval according to special rule. Some algebraic equations with entire coefficients are formulated for parameter β corresponding values definition.

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  1. Lasota A, Mackey MC. Probabilistic Properties of Deterministic Systems. Cambridge: Cambridge University Press; 1985. 358 p. DOI: 10.1017/CBO9780511897474.
  2. Blank ML. Stability and Localization in Chaotic Dynamics. Moscow: MCCME; 2001. 351 p. (in Russian).
  3. Anikin VM, Golubentsev AF. Analytical Models of Deterministic Chaos. Moscow: FIZMATLIT; 2007. 328 p. (in Russian).
  4. Anishchenko VS, Vadivasova TE, Okrokvertskhov GA, Strelkova GI. Correlation analysis of deterministic and noisy chaos. J. Commun. Technol. Electron. 2003;48(7):750–760.
  5. Anikin VM, Arkadakskij SS, Remizov AS. Analytical solution of spectral problem for the Perron – Frobenius operator of piece-wise linear chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(2):16–34 (in Russian). DOI: 10.18500/0869-6632-2006-14-2-16-34.
  6. Anikin VM, Remizov AS, Arkadakskij SS. Eigenfunctions and eigenvalues of the Perron–Frobenius operator of piece-wise linear chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(2):62–75 (in Russian). DOI: 10.18500/0869-6632-2007-15-2-62-75.
  7. Renyi A. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 1957;8(3–4):477–493. DOI: 10.1007/BF02020331.
  8. Rokhlin VA. Exact endomorphisms of the Lebesgue space. Izvestiya: Mathematics. 1961;25(4):499–530 (in Russian).
  9. Rokhlin VA. Selected Works. Moscow: MCCME; 1999. 318 p. (in Russian).
  10. Kosyakin AA, Sandler EA. Ergodic properties of a class of piecewise smooth transformations of an interval. Russian Mathematics (Izvestiya VUZ. Matematika). 1972;(3):32–40 (in Russian).
  11. Lasota A, Yorke JA. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 1973;186:481–488. DOI: 10.2307/1996575.
  12. Li TJ, Yorke JA. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 1978;235:183–192. DOI: 10.2307/1998213.
  13. Li TJ, Yorke JA. Ergodic maps on [0,1] and nonlinear pseudo-random numbers generators. Nonlinear Analysis. Theory, Methods and Applications. 1978;2(4):473–481. DOI: 10.1016/0362-546X(78)90054-8.
  14. Hofbauer F, Keller G. Equilibrium states for piecewise monotonic transformations. Ergodic Theory and Dynamical Systems. 1982;2(1):23–43. DOI: 10.1017/S014338570000955X.
  15. Gelfond AO. On one general property of number systems. Izvestiya: Mathematics. 1959;23(6):809–814 (in Russian).
  16. Parry W. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 1960;11(3–4):401–416. DOI: 10.1007/BF02020954.
  17. Anikin VM. Gaussian Mapping: Evolutionary and Probabilistic Properties. Saratov: Saratov University Publishing; 2007. 80 p. (in Russian).
  18. Mori H, So BC, Ose T. Time-correlation functions of one-dimensional transformations. Progress Theor. Phys. 1981;66(4):1266–1283. DOI: 10.1143/PTP.66.1266.
  19. Golubencev AF, Anikin VM. Invariant subspaces for linear evolution operators of chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(1):3–17 (in Russian). DOI: 10.18500/0869-6632-2005-13-1-3-37.
  20. Anikin VM, Arkadaksky SS. Piecewise linear mappings with non-uniform invariant distribution. Radio Engineering. 2005;(4):78–85 (in Russian).
  21. Parry W. Representations for real numbers. Acta Math. Acad. Sci. Hungar. 1964;15(1–2):95–105. DOI: 10.1007/BF01897025.
  22. Gora P. Invariant densities for generalized β-maps. Ergodic Theory and Dynamical Systems. 2007;27(5):1583–1598. DOI: 10.1017/S0143385707000053.
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