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Shift dynamical system

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A dynamical system ( 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} ) ).

See also Singular exponents; Central exponents.

Comments

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]).

Markov's example, mentioned above, can be found in [a7], Chapt. VI, 9.35. For the use of Bebutov systems for non-autonomous systems of ordinary differential equations, see [a8].

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].

References

[a1] R.L. Adler, D. Coppersmith, M. Hassner, "Algorithms for sliding block codes" IEEE Trans. Inform. Theory , 29 (1983) pp. 5–22
[a2] R.L. Adler, B. Marcus, "Topological entropy and equivalence of dynamical systems" , Amer. Math. Soc. (1979)
[a3] H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press (1981)
[a4] O. Hajek, "Representations of dynamical systems" Funkcial. Ekvac. , 114 (1971) pp. 25–34
[a5] S. Kakutani, "A proof of Bebutov's theorem" J. Differential Equations , 4 (1968) pp. 194–201
[a6] J.C. Martin, "Substitution minimal flows" Amer. J. Math. , 93 (1971) pp. 503–526
[a7] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[a8] G.R. Sell, "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold (1971)
How to Cite This Entry:
Shift dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shift_dynamical_system&oldid=48683
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article