# Shift dynamical system

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A dynamical system (or, in a different notation, ) on a space of continuous functions ( is a metric space) equipped with the compact-open topology (that is, the topology of uniform convergence on segments), defined by where is the shift operator by , that is, Thus, the trajectory of a point in a shift dynamical system is the set of all shifts of , that is, of all functions of the form for . The closure of the trajectory is the set of all functions of the form 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 is usually taken to be ; 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 is usually taken to be or a space of mappings (in the theory of linear homogeneous non-autonomous systems it is usual to take ).

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 if and only if the set of its invariant points is homeomorphic to a subset of (cf. [a5], and, for a generalization, [a4]).
Usually, by a shift dynamical system one understands a discrete-time system (a cascade) of the form ; here denotes a finite non-empty set, is the space of all two-sided infinite sequences with elements in , endowed with the usual product topology (this is just with its compact-open topology when is considered with its discrete topology), and is the shift operator by 1, that is, for .
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 defined by a probability measure on (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 for a suitable set (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].