Noise-Induced Phases of Iterated Functions

We consider iterating functions that consist of the identity map $\mathrm{}\left[f\right(x)=x]$ plus a random function. The random function in every iteration is different and has a mean value of zero. We investigate the behavior of the entire iterated function. It is demonstrated that there are three distinct classes of random functions that generate three "phases" of the iterated function. These phases show universal properties independent of the precise form of the added random function. The physical interpretation of the model in terms of aggregation is discussed and an application of the above ideas is made to the problem of particles in a random potential that is varying in space and time.