Topological dynamical system

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A triple , where is a topological space, is a topological group and is a continuous function defining a left action of on : If , is the identity element of and , then (using multiplicative notation for the operation in ) and


(in other words, if one denotes the transformation by , then ). Instead of a left action one often considers a right action. In this case the arguments of are usually written in the other order (expressing as a mapping ), and (1) is replaced by the condition


Instead of or one often writes simply or . Then (1) and (2) are written in the form

If is commutative, then there is no essential difference between a left and a right action. The most important cases are (the additive group of integers with the discrete topology; in this case one speaks of a (topological) cascade) and (in this case one speaks of a (topological) flow). In the narrow sense, topological dynamical systems refer to these two cases. Sometimes 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 ) 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 , whereas in topological dynamics is usually taken to be locally compact, but never compact, and the interest is in the limiting behaviour of the trajectory as (that is, outside arbitrarily large compact parts of ), 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 , but on the other hand there is a very strong condition of regularity for 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 , 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) . (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.)


For references cf. also Topological dynamics.


[a1] J. de Vries, "Topological transformation groups" , I , Math. Centre (1975)
[a2] R. Ellis, "Lectures on topological dynamics" , Benjamin (1969)
[a3] I.U. Bronshtein, "Extensions of minimal transformation groups" , Sijthoff & Noordhoff (1979) (Translated from Russian)
How to Cite This Entry:
Topological dynamical system. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article