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Harmonic limits of dynamical systems

Pages: 1432 - 1439, Issue Special, September 2011

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Tobias Wichtrey - Institute of Mathematics, University of Augsburg, 86135 Augsburg, Germany (email)

Abstract: In this paper, we analyze the rotational behaviour of dynamical systems, particulary of solutions of ODEs. With rotational behaviour we mean the existence of rotational factor maps, i. e., semi-conjugations to rotations in the complex plane. In order to analyze this kind of rotational behaviour, we introduce harmonic limits lim$_(T\to\infty)1/T\int^T_0 e^(itw)f(\Phi_tx)$dt. We discuss the connection between harmonic limits and rotational factor maps, and some properties of the limits, e. g., existence under the presence of an invariant measure by the Wiener-Wintner Ergodic Theorem. Finally, we look at linear differential equations (autonomous and periodic), and show the connection between the frequencies of the rotational factor maps and the imaginary parts of the eigenvalues of the system matrix (or of the Floquet exponents in the periodic case).

Keywords:  Dynamical systems, ergodic theory, rotational behaviour, linear differential equations
Mathematics Subject Classification:  Primary: 37A30; Secondary: 34A30, 34D08

Received: June 2010;      Revised: June 2011;      Published: October 2011.