#### For citation:

Anikin V. M., Arkadaksky S. S., Kuptsov S. N., Remizov A. S. Polynomial eigenfuctions of the perron–frobenius operator. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2016, vol. 24, iss. 4, pp. 6-16. DOI: 10.18500/0869-6632-2016-24-4-6-16

# Polynomial eigenfuctions of the perron–frobenius operator

In the paper, we reveal the structure of polynomial functions of the eigenfunctions and the kernel of the Perron–Frobenius operator for one-dimensional chaotic maps that iterative functions have the following properties: they are piecewise-linear ones; they have full branches transforming the domain of its definition to the full range of the mapping; the have arbitrary slope of branches; they have not some gaps between the branches. Knowledge of solution of the spectral problem allows us to find analytically the rate of establishment of the invariant distribution in the, the rate of decay of correlations in a dynamic system, which has chaotic properties, to construct the function decomposition similar to the Euler–Maclaurin decomposition. For solving the spectral problem, we introduce a combined approach based on the method of generating function for the operator eigenfunctions and the method of undetermined coefficients. The new results of the paper is a general solution of the spectral problem for piecewise linear maps having arbitrary skew of linear branches of the mapping. We present the solution for polynomial eigenfunctions and eigenvalues of Perron–Frobenius operator associated to arbitrary piece-wise linear chaotic maps with full branches without «gaps» (finite intervals where iterative function is equal to zero). A general form of the functions of the operator kernel is written. The factoring generating function for the eigenfunctions allows us to find an universal set of coefficients that are calculated recursively and form polynomial eigenfunctions. These solutions include partial spectral solutions for Bernoulli shifts and other sawtooth maps.

- Anikin V.M., Goloubentsev A.F. Analytical models of deterministic chaos / Preface by D. I. Trubetskov. Moscow: FIZMATLIT, 2007. 328 p. (in Russian).
- Anikin V.M., Arkadaksky S.S., Remisov A.S. Non-selfadjoint operators in chaotic dynamics. Saratov: Izd-vo Sarat. un-ta, 2015. 96 p. (in Russian).
- Anikin V.M. Spectral problems for the Perron–Frobenius operator // Izvestiya VUZ. AND. 2009. Vol. 17, Iss. 4. P. 35–48 (in Russian).
- Babenko K.I. Fundamental of numerical analysis. Moscow: Nauka, 1986. 744 p. (in Russian).
- Knuth D.E. The art of computer programming. 3d ed. Vol. 2. Seminumerical Algorithms. Addison-Wesley Longman, Inc., 1998.
- Anikin V.M. The Gauss map: evolution and probability properties. Saratov: Saratov University Press, 2007. 80 p. (in Russian).
- Anikin V.M., Arkadasky S.S., Kuptsov S.N., Remisov A.S., Vasilenko L.P. Relaxation properties of chaotic dynamical systems: operator approach // Bulletin of the Russian Academy of science: Physics. 2009. Vol. 73, Iss. 12. P. 1632–1637.
- Anikin V.M., Arkadaksky S.S., Kuptsov S.N., Remisov A.S., Vasilenko L.P. Classification of one-dimensional chaotic models // Bulletin of the Russian Academy of science: Physics. 2009. Vol. 73, Iss. 12. P. 1681–1683.
- Prigogine I. Stengers I. Time, Chaos, Quantum: Towards the Resolution of the Paradox of Time. Moscow: URSS, 2000. 240 p. (in Russian).
- Anikin V.M., Trubetskov D.I. Problems of deterministic chaos theory in A. F. Goloubentsev’s works // Izvestiya VUZ. AND. 2013. Vol. 21, Iss. 5. P. 120–123 (in Russian).
- Goloubentsev A.F., Anikin V.M., Arkadasky S.S. On some properties of the Frobenius–Perron operator for the Bernoulli shifts // Izvestiya VUZ. AND. 2000. Vol. 8, Iss. 2. P. 67–73 (in Russian).
- Goloubentsev A.F., Anikin V.M. Invariant subspaces for linear evolution operators of chaotic maps // Izvestiya VUZ. AND. 2005. Vol. 13, Iss. 1–2. P. 3–17 (in Russian).
- Anikin V.M., Arkadaksky S.S., Remisov A.S. Analytical solution of spectral problem for the Perron–Frobenius operator of piece-wise linear chaotic maps // Izvestiya VUZ. AND. 2006. Vol. 14, Iss. 2. P. 16–34 (in Russian).
- Anikin V.M., Arkadaksky S.S., Remisov A.S. Features of solving spectral problem for the Perron–Frobenius operator caused by critical combination of chaotic map parameters // Theoretical Physics. 2007. Vol. 8. P. 176–183 (in Russian).
- Anikin V.M., Remisov A.S., Arkadaksky S.S. Eigenfunctions and eigenvalues of the Perron–Frobenius operator of piece-wise linear chaotic maps // Izvestiya VUZ. AND. 2007. Vol. 15, Iss. 2. P. 62–75 (in Russian).
- Anikin V.M., Arkadasky S.S., Remisov A.S., Kuptsov S.N., Vasilenko L.P. Investigation of structure of invariant density for Renyi map by Gauss method // Izvestiya VUZ. AND. 2008. Vol. 16, Iss. 6. P. 46–56 (in Russian).
- Anikin V.M., Arkadaksky S.S., Kuptsov S.N., Remisov A.S., Vasilenko L.P. Lyapunov exponent for chaotic 1D maps with uniform invariant distribution // Bulletin of the Russian Academy of science: Physics. 2008. Vol. 72, Iss. 12. P. 1684–1688.
- Abramovits M., Steegan I. Handbook of Special Functions. Moscow: Nauka, 1979. P. 611 (in Russian).

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