Construction and description of the stationary measure of weakly dissipative dynamical systems

We consider the stationary measure of the dissipative dynamical system in a finite volume. A finite dissipation, however small, generally makes the measure singular, while at zero dissipation the measure is constant. Thus dissipative part of the dynamics is a singular perturbation producing an infinite change in the measure. This is a result of the infinite time of evolution that enhances the small effects of dissipation to form singularities. We show how to deal with the singularity of the perturbation and describe the statistics of the measure. We derive all the correlation functions and the statistics of "mass" contained in a small ball. The spectrum of dimensions of the attractor is obtained. The fractal dimension is equal to the space dimension, while the information dimension is equal to the Kaplan-Yorke dimension.