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''in the theory of dynamical systems, discrete-time dynamical system''
 
''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206701.png" /> (or the additive semi-group of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206702.png" />) on some phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206703.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206704.png" /> is associated, such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
S _ {n+m} (w)  = \
 +
S _ {n} ( S _ {m} (w) )
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206706.png" />. Therefore, every transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206707.png" /> can be obtained from the single transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206708.png" /> by means of iteration and (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c0206709.png" />) inversion:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067010.png" /></td> </tr></table>
+
$$
 +
S _ {n}  = (S)  ^ {n} \ \
 +
\textrm{ for }  n > 0 ,\ \
 +
S _ {n}  = ( S  ^ {-1} )  ^ {-n} \ \
 +
\textrm{ for }  n < 0 .
 +
$$
  
Thus, the study of a cascade reduces essentially to the study of the properties of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067011.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067012.png" /> is usually endowed with some structure which is preserved by the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067013.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067015.png" /> be defined almost everywhere and that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067016.png" />, equation (*) holds for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067017.png" />). In these cases the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020670/c02067018.png" /> 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).
+
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).

Latest revision as of 05:47, 18 May 2022


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).

How to Cite This Entry:
Cascade. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cascade&oldid=17447
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article