Ulam method and fractal Weyl law for PerronFrobenius operators
Abstract
We use the Ulam method to study spectral properties of the PerronFrobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d_{0}/2 where d_{0} is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.
 Publication:

European Physical Journal B
 Pub Date:
 June 2010
 DOI:
 10.1140/epjb/e2010001440
 arXiv:
 arXiv:0912.5083
 Bibcode:
 2010EPJB...75..299E
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Dynamical Systems
 EPrint:
 4 pages, 5 figures. Research done at Quantware http://www.quantware.upstlse.fr/