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

A dynamical system defined by the action of the additive group of integers $\mathbf Z$ (or the additive semi-group of natural numbers $\mathbf N$) on some phase space $W$. 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 $S _ {n} : W \rightarrow W$ is associated, such that

$$\tag{* } S _ {n+m} (w) = \ S _ {n} ( S _ {m} (w) )$$

for all $w \in W$. Therefore, every transformation $S _ {n}$ can be obtained from the single transformation $S _ {1}$ by means of iteration and (if $n < 0$) inversion:

$$S _ {n} = (S) ^ {n} \ \ \textrm{ for } n > 0 ,\ \ S _ {n} = ( S ^ {-1} ) ^ {-n} \ \ \textrm{ for } n < 0 .$$

Thus, the study of a cascade reduces essentially to the study of the properties of the transformation $S$ 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 $W$ is usually endowed with some structure which is preserved by the transformations $S _ {n}$. For example, $W$ 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 $S _ {n}$ be defined almost everywhere and that for every $n , m$, equation (*) holds for almost-all $w$). In these cases the transformation $S$ 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).

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