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

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Anikin V. M., Remizov A. S., Arkadaksky S. S. Eigenfunctions and eigenvalues of the Perron–Frobenius operator of piece-wise linear chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, iss. 2, pp. 62-75. DOI: 10.18500/0869-6632-2007-15-2-62-75

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Eigenfunctions and eigenvalues of the Perron–Frobenius operator of piece-wise linear chaotic maps

Anikin Valerij Mihajlovich, Saratov State University
Remizov Aleksandr Sergeevich, Saratov State University
Arkadaksky Sergej Sergeevich, Saratov State University

A chaotic piece-wise linear map having arbitrary interchange of linear increasing and decreasing branches is introduced. Polynomial eigenfunctions for associated non-selfadjoint Perron–Frobenius operator are found. Odd eigenvalus of the operator depend on difference between numbers of increasing and decreasing map branches. This situation may determine transition of odd polynomials from set of eigenfunctions to null-space of the operator or lead to nonsimplicity of eigenvalues.

Key words: 
  1. 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. DOI: 10.18500/0869-6632-2006-14-2-16-34.
  2. Goloubentsev AF, Anikin VM, Arkadasky SS. On some properties of the Frobenius–Perron operator for the Bernoulli shifts. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(2):67–73. (in Russian)
  3. Golubencev AF, Anikin VM. Invariant subspaces for linear evolution operators of chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(1):3–17. DOI: 10.18500/0869-6632-2005-13-1-3-37.
  4. Ulam S. Unsolved mathematical problems. Moscow: Nauka; 1964. 168 p. (in Russian)
  5. Schuster G. Deterministic Chaos. An Introduction. Moscow: Mir; 1988.
  6. Neimark IuI, Landa PS. Stochastic and Chaotic Oscillations. Moscow: Nauka; 1987.
  7. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. (in Russian)
  8. Prigigine IP, Stengers I. Time, chaos, quantum. On the decision of the paradox of time. Moscow: Progress; 1994. (in Russian)
  9. Malinetskiy GG, Potapov AB. Modern problems of nonlinear dynamics. Moscow: Editorial URSS Publ.; 2000. 336 p.
  10. Blank ML.  Stability and Localization in Chaotic Dynamics. Moscow: MCCME; 2001. (in Russian)
  11. Lasota A, Mackey MC. Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press; 1985. Ch. 4.
  12. Iosifescu M, Kraaikamp C. Metrical theory of continued fractions. Kluwer Boston, Inc. 2002. Ch. 1, 2.
  13. Handbook of Mathematical Functions with Formulas, Graphs. Ed. by Abramowitz M, Stegun I. Moscow: Nauka; 1979.
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