Ulam method and fractal Weyl law for Perron-Frobenius operators - Nonlinear Sciences > Chaotic DynamicsReport as inadecuate




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Abstract: We use the Ulam method to study spectral properties of the Perron-Frobeniusoperators of dynamical maps in a chaotic regime. For maps with absorption weshow that the spectrum is characterized by the fractal Weyl law recentlyestablished for nonunitary operators describing poles of quantum chaoticscattering with the Weyl exponent $ u=d-1$, where $d$ is the fractal dimensionof corresponding strange set of trajectories nonescaping in future times. Incontrast, for dissipative maps we find the Weyl exponent $ u=d-2$ where $d$ isthe fractal dimension of strange attractor. The Weyl exponent can be alsoexpressed via the relation $ u=d 0-2$ where $d 0$ is the fractal dimension ofthe invariant sets. We also discuss the properties of eigenvalues andeigenvectors of such operators characterized by the fractal Weyl law.



Author: Leonardo Ermann, Dima L. Shepelyansky

Source: https://arxiv.org/



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