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} ) $).

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)