Polynomial eigenfuctions of the Perron--Frobenius operator

V. M. Anikin ; S. S. Arkadaksky ; S. N. Kuptsov ; A. S. Remisov

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 24 (2016) no. 4, pp. 6-16.

Résumé

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 ystem, 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.

Détail

Keywords: Piece-wise linear chaotic maps, the Perron–Frobenius operator, polynomial eigenfunctions, the kernel.