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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

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

( $  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====
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|dynamical system]] containing a [[Homoclinic point|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|chaos]] ([[#References|[a1]]], [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Guckenheimer,   P. Holmes,   "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.K. Nikol'skii,   "Treatise on the shift operator: spectral function theory" , Springer  (1986)  (Translated from Russian)</TD></TR></table>