The asymptotic number of attractors in the random map model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as its image. We derive here explicit formulae for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulae increases exponentially with n; therefore, they are not directly applicable to study the behaviour of systems where n is large. However, our formulae can be used to derive useful asymptotic expressions, as we show