Shift operator

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)