Namespaces
Variants
Actions

Difference between revisions of "Shift operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849001.png" /> that depends on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849002.png" /> and acts in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849003.png" /> of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849004.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849005.png" /> is an Abelian [[Semi-group|semi-group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849006.png" /> is a set) in accordance with the formula
+
<!--
 +
s0849001.png
 +
$#A+1 = 18 n = 0
 +
$#C+1 = 18 : ~/encyclopedia/old_files/data/S084/S.0804900 Shift operator
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/s/s084/s084900/s0849007.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849008.png" /> is also called the operator of shift by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s0849009.png" />). The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490010.png" /> is often taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490012.png" /> (then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490013.png" /> is a shift in some space of functions of a real variable), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490015.png" /> (then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490016.png" /> is a shift in some space of sequences). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490017.png" /> and the corresponding set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084900/s08490018.png" /> are usually endowed with a certain structure (of a vector, topological vector, normed, metric, or probability space).
+
An operator $  T _ {t} $
 +
that depends on a parameter  $  t $
 +
and acts in a set  $  \Phi $
 +
of mappings  $  \phi : A \rightarrow E $(
 +
where  $  A $
 +
is an Abelian [[Semi-group|semi-group]] and  $  E $
 +
is a set) in accordance with the formula
  
A shift operator is used, in particular, in the theory of dynamical systems (see [[Shift dynamical system|Shift dynamical system]]; [[Bernoulli automorphism|Bernoulli automorphism]]). Also used is the terminology  "shift operator along the trajectories of differential equations" (see [[Cauchy operator|Cauchy operator]]).
+
$$
 +
T _ {t} \phi ( \cdot = \
 +
\phi ( \cdot + t)
 +
$$
  
 +
( $  T _ {t} $
 +
is also called the operator of shift by  $  t $).
 +
The semi-group  $  A $
 +
is often taken to be  $  \mathbf R $
 +
or  $  \mathbf R  ^ {+} $(
 +
then  $  T _ {t} $
 +
is a shift in some space of functions of a real variable),  $  \mathbf Z $
 +
or  $  \mathbf N $(
 +
then  $  T _ {t} $
 +
is a shift in some space of sequences). The set  $  E $
 +
and the corresponding set  $  \Phi $
 +
are usually endowed with a certain structure (of a vector, topological vector, normed, metric, or probability space).
  
 +
A shift operator is used, in particular, in the theory of dynamical systems (see [[Shift dynamical system|Shift dynamical system]]; [[Bernoulli automorphism|Bernoulli automorphism]]). Also used is the terminology  "shift operator along the trajectories of differential equations"  (see [[Cauchy operator|Cauchy operator]]).
  
 
====Comments====
 
====Comments====

Latest revision as of 08:13, 6 June 2020


An operator $ T _ {t} $ that depends on a parameter $ t $ and acts in a set $ \Phi $ of mappings $ \phi : A \rightarrow E $( where $ A $ is an Abelian semi-group and $ E $ is a set) in accordance with the formula

$$ T _ {t} \phi ( \cdot ) = \ \phi ( \cdot + t) $$

( $ T _ {t} $ is also called the operator of shift by $ t $). The semi-group $ A $ is often taken to be $ \mathbf R $ or $ \mathbf R ^ {+} $( then $ T _ {t} $ is a shift in some space of functions of a real variable), $ \mathbf Z $ or $ \mathbf N $( then $ T _ {t} $ is a shift in some space of sequences). The set $ E $ and the corresponding set $ \Phi $ are usually endowed with a certain structure (of a vector, topological vector, normed, metric, or probability space).

A shift operator is used, in particular, in the theory of dynamical systems (see Shift dynamical system; Bernoulli automorphism). Also used is the terminology "shift operator along the trajectories of differential equations" (see Cauchy operator).

Comments

The discrete dynamical systems generated by shift operators on sequence spaces are often easy to analyze. They are of great importance in dynamical systems theory, owing to the Smale–Birkhoff theorem: A discrete-time dynamical system containing a homoclinic point at which the stable and unstable manifolds interact transversely, must contain a compact invariant set on which the dynamics is isomorphic to a certain type of shift in which periodic orbits are dense. This is the best-known method for demonstrating deterministic chaos ([a1], [a2]).

References

[a1] J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)
[a2] S. Smale, "Diffeomorphisms with many periodic points" S.S. Cairns (ed.) , Differential and Combinatorial Topol. (Symp. in honor of M. Morse) , Princeton Univ. Press (1965) pp. 63–80
[a3] N.K. Nikol'skii, "Treatise on the shift operator: spectral function theory" , Springer (1986) (Translated from Russian)
How to Cite This Entry:
Shift operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shift_operator&oldid=48684
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article