Ulam method and fractal Weyl law for Perron--Frobenius operators

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 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 $\nu=d-1$, where $d$ is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent $\nu=d/2$ where $d$ is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation $\nu=d_0/2$ 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.