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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
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<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>
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{{TEX|auto}}
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{{TEX|done}}
  
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,
+
A [[Dynamical system|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
  
<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>
+
$$
 +
f ^ { t } \phi  = T _ {t} \phi ,
 +
$$
  
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
+
where  $  T _ {t} $
 +
is the [[Shift operator|shift operator]] by  $  t $,  
 +
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/s08489013.png" /></td> </tr></table>
+
$$
 +
T _ {t} \phi ( \cdot )  = \phi ( \cdot + t).
 +
$$
 +
 
 +
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|Invariant measure]]); these exist by the Bogolyubov–Krylov theorem (Bogolyubov–Krylov invariant measures are concentrated on compact sets).
 
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" />).
+
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|Singular exponents]]; [[Central exponents|Central exponents]].
 
See also [[Singular exponents|Singular exponents]]; [[Central exponents|Central exponents]].
 
 
  
 
====Comments====
 
====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]]]).
+
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 $  S = \mathbf R $
 +
if and only if the set of its invariant points is homeomorphic to a subset of $  \mathbf R $(
 +
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]]].
 
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" />.
+
Usually, by a shift dynamical system one understands a discrete-time system (a [[Cascade|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 <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]]].
+
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|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 $  \Omega _ {S} $
 +
for a suitable set $  S $(
 +
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====
 
====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>
 
<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>

Latest revision as of 08:13, 6 June 2020


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)
How to Cite This Entry:
Shift dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shift_dynamical_system&oldid=17867
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article