View source for Shift dynamical system

A [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848901.png" /> (or, in a different notation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848902.png" />) on a space of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848903.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848904.png" /> is a metric space) equipped with the compact-open topology (that is, the topology of uniform convergence on segments), defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848905.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848906.png" /> is the [[Shift operator|shift operator]] by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848907.png" />, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848908.png" /></td> </tr></table>

Thus, the trajectory of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s0848909.png" /> in a shift dynamical system is the set of all shifts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489010.png" />, that is, of all functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489012.png" />. The closure of the trajectory is the set of all functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489013.png" /></td> </tr></table>

where the limit is uniform on each segment. A shift dynamical system is equipped with normalized invariant measures (cf. [[Invariant measure|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489014.png" /> is usually taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489015.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489016.png" /> is usually taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489017.png" /> or a space of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489018.png" /> (in the theory of linear homogeneous non-autonomous systems it is usual to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489019.png" />).

See also [[Singular exponents|Singular exponents]]; [[Central exponents|Central exponents]].



====Comments====
A shift dynamical system as defined above is often called a Bebutov system; cf. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489020.png" /> if and only if the set of its invariant points is homeomorphic to a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489021.png" /> (cf. [[#References|[a5]]], and, for a generalization, [[#References|[a4]]]).

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

Usually, by a shift dynamical system one understands a discrete-time system (a [[Cascade|cascade]]) of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489022.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489023.png" /> denotes a finite non-empty set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489024.png" /> is the space of all two-sided infinite sequences with elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489025.png" />, endowed with the usual product topology (this is just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489026.png" /> with its compact-open topology when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489027.png" /> is considered with its discrete topology), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489028.png" /> is the shift operator by 1, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489030.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489031.png" /> defined by a probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489032.png" /> (cf. [[Bernoulli automorphism|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. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489033.png" /> for a suitable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084890/s08489034.png" /> (cf. [[Symbolic dynamics|Symbolic dynamics]]). More recently, it turned out that the methods used for the classification of so-called subshifts of finite type (see [[#References|[a2]]]) are useful for information processing; see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.L. Adler,   D. Coppersmith,   M. Hassner,   "Algorithms for sliding block codes"  ''IEEE Trans. Inform. Theory'' , '''29'''  (1983)  pp. 5–22</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.L. Adler,   B. Marcus,   "Topological entropy and equivalence of dynamical systems" , Amer. Math. Soc.  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Furstenberg,   "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Hajek,   "Representations of dynamical systems"  ''Funkcial. Ekvac.'' , '''114'''  (1971)  pp. 25–34</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Kakutani,   "A proof of Bebutov's theorem"  ''J. Differential Equations'' , '''4'''  (1968)  pp. 194–201</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.C. Martin,   "Substitution minimal flows"  ''Amer. J. Math.'' , '''93'''  (1971)  pp. 503–526</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.R. Sell,   "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold  (1971)</TD></TR></table>