# Shift dynamical system

A dynamical system $f ^ { t }$( or, in a different notation, $f ( t, \cdot )$) on a space of continuous functions $\phi : \mathbf R \rightarrow S$( $S$ is a metric space) equipped with the compact-open topology (that is, the topology of uniform convergence on segments), defined by

$$f ^ { t } \phi = T _ {t} \phi ,$$

where $T _ {t}$ is the shift operator by $t$, that is,

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

Thus, the trajectory of a point $\phi$ in a shift dynamical system is the set of all shifts of $\phi$, that is, of all functions of the form $\phi ( t + \tau )$ for $\tau \in \mathbf R$. The closure of the trajectory is the set of all functions of the form

$$\widetilde \phi ( \tau ) = \ \lim\limits _ {k \rightarrow \infty } \ \phi ( t _ {k} + \tau ),$$

where the limit is uniform on each segment. A shift dynamical system is equipped with normalized invariant measures (cf. Invariant measure); these exist by the Bogolyubov–Krylov theorem (Bogolyubov–Krylov invariant measures are concentrated on compact sets).

A shift dynamical system is used in the theory of dynamical systems mainly to construct examples (here $S$ is usually taken to be $\mathbf R$; Markov's example of a non-strictly ergodic system on a compact set each trajectory of which is everywhere dense, and others), and also in the theory of non-autonomous systems of ordinary differential equations, where $S$ is usually taken to be $\mathbf R ^ {n}$ or a space of mappings $\mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$( in the theory of linear homogeneous non-autonomous systems it is usual to take $S = \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} )$).

A shift dynamical system as defined above is often called a Bebutov system; cf. [a3]. The Bebutov–Kakutani theorem states that a dynamical system on a compact metric space is isomorphic to a subsystem of the Bebutov system with $S = \mathbf R$ if and only if the set of its invariant points is homeomorphic to a subset of $\mathbf R$( cf. [a5], and, for a generalization, [a4]).
Usually, by a shift dynamical system one understands a discrete-time system (a cascade) of the form $( \Omega _ {S} , \sigma )$; here $S$ denotes a finite non-empty set, $\Omega _ {S} = S ^ {\mathbf Z }$ is the space of all two-sided infinite sequences with elements in $S$, endowed with the usual product topology (this is just $C( \mathbf Z , S)$ with its compact-open topology when $S$ is considered with its discrete topology), and $\sigma$ is the shift operator by 1, that is, $( \sigma x ) _ {n} = x _ {n+} 1$ for $x = ( x _ {n} ) _ {n \in \mathbf Z } \in \Omega _ {S}$.
These (discrete) shift systems play an important role in ergodic theory and topological dynamics. For example, a Bernoulli system is a shift system endowed with the product measure on $S ^ {\mathbf Z }$ defined by a probability measure on $S$( cf. Bernoulli automorphism). The discrete shift systems and their subsystems (subshifts) are not only used for the construction of special examples (for an important method — substitution — cf. [a6]), they are also important for the study of the behaviour of a large class of cascades by "coding" their trajectories by means of elements of $\Omega _ {S}$ for a suitable set $S$( cf. Symbolic dynamics). More recently, it turned out that the methods used for the classification of so-called subshifts of finite type (see [a2]) are useful for information processing; see [a1].