ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Golubencev A. F., Anikin V. M., Arkadaksky S. S. On some properties of the Frobenius - Perron operator for the Bernoulli shifts. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 2, pp. 67-73. DOI: 10.18500/0869-6632-2000-8-2-67-73

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2000no2p067.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
519.6:517.929
DOI: 
10.18500/0869-6632-2000-8-2-67-73

On some properties of the Frobenius - Perron operator for the Bernoulli shifts

Autors: 
Golubencev Aleksandr Fedorovich, Saratov State University
Anikin Valerij Mihajlovich, Saratov State University
Arkadaksky Sergej Sergeevich, Saratov State University
Abstract: 

The expansion of solutions of Frobenius — Perron equations of the Bernoulli shifts and conjugate maps in terms of the eigenfunctions of the same name operators are presented. The convergence of nonstationary solutions to the invariant density is discussed. it is marked that the Lyapunov exponent may be considered as a measure of the speed of convergence. The entire function as an element of Frobenius — Perron operator kernel is found.

Key words: 
-
Acknowledgments: 
The authors are grateful to S.V. Ershov for discussion of the paper and stimulating comments. The work was supported by the Federal Target Program "Integration" (project № А0057/1999).
Reference: 
  1. Ornstein DS. Ergodic Theory, Randomness and Dinamical Systems. New Haven: Yale Univ. Press; 1974. 141 p.
  2. Grenanger U, Freiberger W. A Short Course in Computational Probability and Statistics. Berlin: Springer; 1971. 168 p.
  3. Renyi А. Representations for the real numbers and their ergodic properties. Acta Mathematica Acad. Sc. Hung. 1957;8:477-493.
  4. Goloubentsev AF, Anikin VM. The explicit solutions of the Frobenius-Perron equation for the chaotic infinite maps. Int. J. Bifurcation and Chaos. 1998;8(5):1049-1051. DOI: 10.1142/S0218127498000863.
  5. Ulam SM. A Collection of Mathematical Problems. N.Y.: Interscience Publishers;1960. 150p.
  6. Golubentsev AF, Anikin VM, Bogomolov AV. Chaotic generators of biological rhythms. J. Biomed. Radioelectronics. 2000;2:38-41.
  7. Lasota А, Mackey MC. Probabilistic properties оf deterministic systems. Cambridge: Cambridge University Press, 1985. 358 p.
  8. Ershov SV, Malinetsky GG. On the solution of the inverse problem for the Frobenius-Perron equation.Computational Mathematics and Mathematical Physics. 1988;28(10):1491-1497.
  9. Antoniou I, Tasaki S. Spectral decomposition оf the Renyi map. J. Phys. A: Math. Gen. 1993;26(1):73-94. DOI: 10.1088/0305-4470/26/1/012.
  10. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. N.Y.: McGraw-Hill Book Company; 1968. 1152 p.
  11. Hardy GH. Divergent Series. Oxford: Oxford University Press; 1949. 394 p.
  12. Khurgin YaI, Yakovlev VP. Finite functions in physics and technology.М.: Nauka; 1971. 408 p. (in Russian).
  13. Jeffreys H. Theory of probability. Oxford: Oxford University Press, 1998. 470 p.
  14. Winter А. On the shape of the angular case оf Cauchy’s distribution curves. Ann. Math. Statist. 1947;18:589-593
Received: 
07.09.1999
Accepted: 
06.04.2000
Published: 
25.05.2000
  • 354 reads