Cascade

in the theory of dynamical systems, discrete-time dynamical system

A dynamical system defined by the action of the additive group of integers (or the additive semi-group of natural numbers ) on some phase space . According to the general definition of the action of a group (or semi-group), this means that with each integer (or natural number) a transformation is associated, such that

(*)

for all . Therefore, every transformation can be obtained from the single transformation by means of iteration and (if ) inversion:

Thus, the study of a cascade reduces essentially to the study of the properties of the transformation generating it, and in this sense cascades are the simplest dynamical systems. For this reason, cascades have been very thoroughly investigated, although in applications, mostly continuous-time dynamical systems (cf. Flow (continuous-time dynamical system)) are encountered. Usually, the main features of cascades are the same for flows, but cascades are somewhat simpler to deal with technically; at the same time, the results obtained for them can often be carried over to flows without any particular difficulty, sometimes by means of a formal reduction of the properties of flows to those of cascades, but more often by a modification of the proofs.

As for arbitrary dynamical systems, the phase space is usually endowed with some structure which is preserved by the transformations . For example, can be a smooth manifold, a topological space or a measure space; the cascade is then said to be smooth, continuous or measurable, respectively (although in the latter case, one often modifies the definition, demanding that each be defined almost everywhere and that for every , equation (*) holds for almost-all ). In these cases the transformation generating the cascade is a diffeomorphism, a homeomorphism or an automorphism of the measure space (if one has a group of transformations), or else a smooth mapping, a continuous mapping or an endomorphism of the measure space (if the cascade is a semi-group).