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

A [[Dynamical system|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)|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|diffeomorphism]], a [[Homeomorphism|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).