Eur. Phys. J. B 75, 299-304 (2010)
https://doi.org/10.1140/epjb/e2010-00144-0

Ulam method and fractal Weyl law for Perron-Frobenius operators

Laboratoire de Physique Théorique du CNRS (IRSAMC), Université de Toulouse, UPS, 118 route de Narbonne, 31062 Toulouse Cedex 4, France

Corresponding author: ^a dima@irsamc.ups-tlse.fr

Received: 4 February 2010
Revised: 21 April 2010
Published online: 12 May 2010

Abstract

We use the Ulam method to study spectral properties of the Perron-Frobenius 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 = d-1, 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 = d₀/2 where d₀ 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.