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A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930202.png" /> is a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930203.png" /> is a topological group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930204.png" /> is a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930205.png" /> defining a left action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930207.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t0930209.png" /> is the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302011.png" />, then (using multiplicative notation for the operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302012.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302013.png" /> and
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{{TEX|done}}
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A triple $(W,G,F)$, where $W$ is a topological space, $G$ is a topological group and $F$ is a continuous function $G\times W\to W$ defining a left action of $G$ on $W$: If $w\in W$, $e$ is the identity element of $G$ and $g,h\in G$, then (using multiplicative notation for the operation in $G$) $F(e,w)=w$ and
  
<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/t/t093/t093020/t09302014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$F(gh,w)=F(g,F(h,w))\tag{1}$$
  
(in other words, if one denotes the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302015.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302017.png" />). Instead of a left action one often considers a right action. In this case the arguments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302018.png" /> are usually written in the other order (expressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302019.png" /> as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302020.png" />), and (1) is replaced by the condition
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(in other words, if one denotes the transformation $w\to F(g,w)$ by $T_g$, then $T_{gh}=T_gT_h$). Instead of a left action one often considers a right action. In this case the arguments of $F$ are usually written in the other order (expressing $F$ as a mapping $W\times G\to W$), and \ref{1} is replaced by the condition
  
<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/t/t093/t093020/t09302021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$F(w,gh)=F(F(w,g),h).\tag{2}$$
  
Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302023.png" /> one often writes simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302025.png" />. Then (1) and (2) are written in the form
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Instead of $F(g,w)$ or $F(w,g)$ one often writes simply $gw$ or $wg$. Then \ref{1} and \ref{2} are written in 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/t/t093/t093020/t09302026.png" /></td> </tr></table>
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$$(gh)w=g(hw),\quad w(gh)=(wg)h.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302027.png" /> is commutative, then there is no essential difference between a left and a right action. The most important cases are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302028.png" /> (the additive group of integers with the discrete topology; in this case one speaks of a (topological) cascade) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302029.png" /> (in this case one speaks of a (topological) flow). In the narrow sense, topological dynamical systems refer to these two cases. Sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302030.png" /> is not a group but a semi-group. Basically, however, one considers only the semi-group of non-negative integers (in other words, one considers iteration of some continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302031.png" />) or (more rarely) of non-negative real numbers.
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If $G$ is commutative, then there is no essential difference between a left and a right action. The most important cases are $G=\mathbf Z$ (the additive group of integers with the discrete topology; in this case one speaks of a (topological) cascade) and $G=\mathbf R$ (in this case one speaks of a (topological) flow). In the narrow sense, topological dynamical systems refer to these two cases. Sometimes $G$ is not a group but a semi-group. Basically, however, one considers only the semi-group of non-negative integers (in other words, one considers iteration of some continuous mapping $T\colon W\to W$) or (more rarely) of non-negative real numbers.
  
The term "topological dynamical system" (usually without the first adjective) belongs to [[Topological dynamics|topological dynamics]], while in topology the same object is called a continuous transformation group. The different terminologies are partly due to the fact that the two disciplines study different properties of the object, and impose different restrictions on it. Thus, a lot of results in topology concern a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302032.png" />, whereas in topological dynamics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302033.png" /> is usually taken to be locally compact, but never compact, and the interest is in the limiting behaviour of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302034.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302035.png" /> (that is, outside arbitrarily large compact parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302036.png" />), which even in the analytic case can be extremely complicated. In the theory of algebraic transformation groups (cf. [[Algebraic group of transformations|Algebraic group of transformations]]) one does not assume compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302037.png" />, but on the other hand there is a very strong condition of regularity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302038.png" /> as a mapping between algebraic varieties (where the ground field is usually assumed to be algebraically closed, so that in the classical case one is talking about regularity "in the entire complex domain" ). Combined with connectivity (and usually also reductivity) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302039.png" />, this enables one, as in the compact case, to obtain significant information about the possible types of mutually adjoining orbits (cf. [[Orbit|Orbit]]), and, in particular, to exclude various phenomena associated with complicated limiting behaviour of trajectories.
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The term "topological dynamical system" (usually without the first adjective) belongs to [[Topological dynamics|topological dynamics]], while in topology the same object is called a continuous transformation group. The different terminologies are partly due to the fact that the two disciplines study different properties of the object, and impose different restrictions on it. Thus, a lot of results in topology concern a compact group $G$, whereas in topological dynamics $G$ is usually taken to be locally compact, but never compact, and the interest is in the limiting behaviour of the trajectory $F(g,w)$ as $g\to\infty$ (that is, outside arbitrarily large compact parts of $G$), which even in the analytic case can be extremely complicated. In the theory of algebraic transformation groups (cf. [[Algebraic group of transformations|Algebraic group of transformations]]) one does not assume compactness of $G$, but on the other hand there is a very strong condition of regularity for $F$ as a mapping between algebraic varieties (where the ground field is usually assumed to be algebraically closed, so that in the classical case one is talking about regularity "in the entire complex domain"). Combined with connectivity (and usually also reductivity) of $G$, this enables one, as in the compact case, to obtain significant information about the possible types of mutually adjoining orbits (cf. [[Orbit|Orbit]]), and, in particular, to exclude various phenomena associated with complicated limiting behaviour of trajectories.
  
The term "topological dynamical system" should be preferred to the commonly used "continuous dynamical system (a flow, a cascade)" , because "continuity" can mean also: a) metric continuity, cf. [[Continuous flow|Continuous flow]] 1); and b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093020/t09302040.png" />. (When a topological dynamical system is taken in the narrow sense, one says that a flow is the case of continuous time, and a cascade is the case of discrete time. One sometimes speaks of a continuous and a discrete dynamical system, respectively.)
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The term "topological dynamical system" should be preferred to the commonly used "continuous dynamical system (a flow, a cascade)", because "continuity" can mean also: a) metric continuity, cf. [[Continuous flow|Continuous flow]] 1); and b) $G=\mathbf R$. (When a topological dynamical system is taken in the narrow sense, one says that a flow is the case of continuous time, and a cascade is the case of discrete time. One sometimes speaks of a continuous and a discrete dynamical system, respectively.)
  
  

Revision as of 16:18, 19 August 2014

A triple $(W,G,F)$, where $W$ is a topological space, $G$ is a topological group and $F$ is a continuous function $G\times W\to W$ defining a left action of $G$ on $W$: If $w\in W$, $e$ is the identity element of $G$ and $g,h\in G$, then (using multiplicative notation for the operation in $G$) $F(e,w)=w$ and

$$F(gh,w)=F(g,F(h,w))\tag{1}$$

(in other words, if one denotes the transformation $w\to F(g,w)$ by $T_g$, then $T_{gh}=T_gT_h$). Instead of a left action one often considers a right action. In this case the arguments of $F$ are usually written in the other order (expressing $F$ as a mapping $W\times G\to W$), and \ref{1} is replaced by the condition

$$F(w,gh)=F(F(w,g),h).\tag{2}$$

Instead of $F(g,w)$ or $F(w,g)$ one often writes simply $gw$ or $wg$. Then \ref{1} and \ref{2} are written in the form

$$(gh)w=g(hw),\quad w(gh)=(wg)h.$$

If $G$ is commutative, then there is no essential difference between a left and a right action. The most important cases are $G=\mathbf Z$ (the additive group of integers with the discrete topology; in this case one speaks of a (topological) cascade) and $G=\mathbf R$ (in this case one speaks of a (topological) flow). In the narrow sense, topological dynamical systems refer to these two cases. Sometimes $G$ is not a group but a semi-group. Basically, however, one considers only the semi-group of non-negative integers (in other words, one considers iteration of some continuous mapping $T\colon W\to W$) or (more rarely) of non-negative real numbers.

The term "topological dynamical system" (usually without the first adjective) belongs to topological dynamics, while in topology the same object is called a continuous transformation group. The different terminologies are partly due to the fact that the two disciplines study different properties of the object, and impose different restrictions on it. Thus, a lot of results in topology concern a compact group $G$, whereas in topological dynamics $G$ is usually taken to be locally compact, but never compact, and the interest is in the limiting behaviour of the trajectory $F(g,w)$ as $g\to\infty$ (that is, outside arbitrarily large compact parts of $G$), which even in the analytic case can be extremely complicated. In the theory of algebraic transformation groups (cf. Algebraic group of transformations) one does not assume compactness of $G$, but on the other hand there is a very strong condition of regularity for $F$ as a mapping between algebraic varieties (where the ground field is usually assumed to be algebraically closed, so that in the classical case one is talking about regularity "in the entire complex domain"). Combined with connectivity (and usually also reductivity) of $G$, this enables one, as in the compact case, to obtain significant information about the possible types of mutually adjoining orbits (cf. Orbit), and, in particular, to exclude various phenomena associated with complicated limiting behaviour of trajectories.

The term "topological dynamical system" should be preferred to the commonly used "continuous dynamical system (a flow, a cascade)", because "continuity" can mean also: a) metric continuity, cf. Continuous flow 1); and b) $G=\mathbf R$. (When a topological dynamical system is taken in the narrow sense, one says that a flow is the case of continuous time, and a cascade is the case of discrete time. One sometimes speaks of a continuous and a discrete dynamical system, respectively.)


Comments

For references cf. also Topological dynamics.

References

[a1] J. de Vries, "Topological transformation groups" , I , Math. Centre (1975) Zbl 0315.54002 Zbl 0299.54030
[a2] R. Ellis, "Lectures on topological dynamics" , Benjamin (1969) MR0267561 Zbl 0193.51502
[a3] I.U. Bronshtein, "Extensions of minimal transformation groups" , Sijthoff & Noordhoff (1979) (Translated from Russian) MR0550605
How to Cite This Entry:
Topological dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_dynamical_system&oldid=21952
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article