Topological Dynamics

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Curator: Joseph Auslander

Topological dynamics is the study of asymptotic or long term properties of families of maps of topological spaces.



Abstract topological dynamics is usually developed in the context of flows. A flow \((X,T)\) is a jointly continuous action of the topological group \(T\) on the topological space \(X\ .\) That is, there is a continuous map from \(T \times X \to X\ ,\) \((t,x) \mapsto tx\) with \(ex=x\) and \(t(sx)=(ts)x\) (where \(e\) is the identity of \(T\),\(t,s \in T\) and \(x\in X\)). Classical examples are the additive groups \(\Bbb Z\) equivalently the powers of a generating homomorphism, (a "cascade") and \(\Bbb R\ ,\) a one parameter group of homeomorphisms. The theory is most highly developed when the space \(X\) is compact Hausdorff, and we shall henceforth assume this. In fact, most examples of interest are on metric spaces, but there are reasons to develop the theory for Hausdorff spaces.

If \((X,T)\) is a flow, and \(x \in X\ ,\) the orbit of \(x\) is the set \(Tx=\{tx|t\in T\}\ .\) A subset \(K\) of \(X\) is said to be invariant if it is a union of orbits, equivalently \(tK=K\) for \(t\in T\ .\)

Minimal sets and minimal flows

The "irreducible" objects are minimal sets. A subset \(M\) of \(X\) is a minimal set if it is non-empty, closed, invariant, and minimal with respect to these properties. Equivalently, a non-empty subset \(M\) of \(X\) is minimal if it is the orbit closure of each of its points, \(\overline{Tx}=M\) for all \(x\in M\ .\) Obviously, distinct minimal sets are disjoint. It follows easily from Zorn's Lemma that minimal sets always exist for flows on compact Hausdorff spaces.

If \((X,T)\) is itself minimal (\(\overline{Tx}=X\) for every \(x\in X\)) we say that \((X,T)\) is a minimal flow.

A weaker notion than minimality is topological transitivity, every non-empty invariant open set is dense. The flow \((X,T)\) is point transitive if it has a dense orbit. A point \(x \in X\) whose orbit is dense is called a transitive point, so \((X,T)\) is minimal if and only if all points are transitive points. Clearly point transitivity implies topological transitivity, and if the space \(X\) is metrizable, the converse holds.

An intrinsic condition for minimality of an orbit closure is in terms of almost periodicity. A subset \(A\) of the group \(T\) is called syndetic if there is a compact subset \(K\) of \(T\) such that \(T=KA\ .\) If \(T=\Bbb Z\) syndetic coincides with relatively dense-there is an \(N>0\) such that every sequence of \(N\) consecutive integers contains at least one member of \(A\ .\) A similar characterization holds for \(\Bbb R\ .\)

The point \(x\) is almost periodic if for every neighborhood \(U\) of \(x\) the set \(A\) of "return times" to \(U\ ,\) \(A=\{t\in T|tx \in U\}\) is syndetic. The orbit closure \(\overline{Tx}\) is minimal if and only if \(x\) is an almost periodic point.

The morphisms in topological dynamics are homomorphisms or continuous equivariant maps. If \((X,T)\) and \((Y,T)\) are flows, a homomorphism is a continuous onto map \(\pi:X\to Y\) such that \(\pi(tx)=t\pi(x)\) for \(t\in T\) and \(x\in X\ .\) (If \((Y,T)\) is minimal, \(\pi\) is automatically onto.) In this case, we say that \(Y\) is a factor of \(X\) and that \(X\) is an extension of \(Y\ .\)

It is not the case that all questions in topological dynamics can be reduced to questions about minimal sets. (This is in contrast to the situation in ergodic theory, where one need consider only ergodic systems.) For one thing, it is not always the case that a flow decomposes into the union of minimal sets, and even when it does the minimal sets need not "fit together" nicely.

Nevertheless, the classification of minimal flows is an important issue for the subject, and this article will be largely devoted to this.


The equicontinuous minimal flows are completely classified. These are the flows for which the collection of maps defined by the elements of \(T\) form an equicontinuous family. If the space \(X\) is metric, this is equivalent to: if \(\varepsilon>0\) there is a \(\delta>0\) such that if \(d(x,y)<\delta\) then \(d(tx,ty)< \varepsilon\) for all \(t\in T\ .\) (In the general case, equicontinuity may be formulated in terms of the unique compatible uniformity.)

If the flow \((X,T)\) is equicontinuous, it follows easily that \((X,T)\) is pointwise almost periodic. That is, every orbit closure is minimal, so \(X\) is a union of minimal sets.

Equicontinuous minimal flows are homogeneous spaces of compact topological groups. The proof of this fact depends on a construction of more general interest. If \((X,T)\) is a flow (not necessarily minimal or equicontinuous) the group \(T\) may be identified with the collection of self maps of \(X\) it defines. With this identification \(T\) is a subset of \(X^X\) the collection of all maps from \(X\) to itself. \(X^X\) provided with the product topology is compact, by Tychonoff's theorem., so \(E\) the closure of \(T\) in \(X^X\) is compact. Moreover, it can be shown that \(E\) is a semigroup under composition of maps (the enveloping semigroup of \((X,T)\)). Dynamical properties of flows can be correlated with algebraic and topological properties of the enveloping semigroup.

In general the maps in \(E\) need not be continuous, one to one, nor onto. It can be shown that the flow \((X,T)\) is equicontinuous if and only if \(E\) is a group of homeomorphisms, in which case \(E\) is a compact topological group. If in addition \((X,T)\) is minimal, and if we fix an \(x\in X\) then \(F=\{p \in E|px=x\}\) is a closed subgroup of \(E\ ,\) and \(X\) is homeomorphic to the homogeneous space \(E/F\ .\) (When the acting group \(T\) is abelian, \(E\) is abelian, and \(X\) itself has the structure of a topological group.)

Proximality, distality, and the Furstenberg theorem

An important generalization of equicontinuity is distality. We first define the proximal relation. If \((X,T)\) is a flow, the points \(x\) and \(y\) are said to be proximal if there is a net \(\{t_n\}\) in \(T\) and a \(z\in X\) such that \(t_nx \to z\) and \(t_ny \to z\ .\) If \(X\) is a metric space \(x\) and \(y\) are proximal if and only if for every \(\varepsilon >0\) there is a \(t\in T\) such that \(d(tx,ty)<\varepsilon\ .\) Let \(P\) denote the proximal relation. \(P\) is obviously reflexive, symmetric, and \(T\) invariant (if \((x,y)\in P\) and \(t\in T\) then \((tx,ty)\in P\)). In general \(P\) is not an equivalence relation (this is the case if and only if the enveloping semigroup has a unique minimal left ideal) nor is it closed. (When \(P\) is closed it is in fact an equivalence relation.)

An elementary but useful observation is that proximality and almost periodicity in the product flow are incompatible. That is, if \((x,y) \in P\) and \((x,y)\) is an almost periodic point for \((X \times X,T)\) then \(x=y\ .\) It follows that for actions of abelian groups, there are no non-trivial "proximal" minimal flows (\(P=X\times X\)) since \((x,tx)\) is almost periodic for \(t\in T\ .\)

The flow \((X,T)\) is said to be distal if there are no non-trivial proximal pairs, \(P=\Delta\ .\) It is immediate that an equicontinuous flow is distal, but the converse fails, even under the assumption of minimality.

The flow \((X,T)\) is distal if and only if its enveloping semigroup is a group. It follows that a distal flow is pointwise almost periodic (so a topologically transitive distal flow is necessarily minimal) and that a flow is distal if and only if the product flow \((X \times X,T)\) is pointwise almost periodic. This in turn implies that a factor of a distal flow is distal.

More generally, if \((X,T)\) is a flow, \(x\in X\) and \(K\) a minimal set with \(K \subset \overline{Tx}\ ,\) there is a point \(y\in K\) with \(x\) and \(y\) proximal. This result has combinatorial applications.

The structure of distal minimal flows is given by a deep theorem due to Hillel Furstenberg. The formulation of Furstenberg's structure theorem is based on the relativization of the various dynamical notions. Let \(\pi:X\to Y\) be a homomorphism, and let \(R(\pi)\) be the equivalence relation defined by \(\pi\ :\) \(R(\pi)=\{(x,x')|\pi(x)=\pi(x')\}\ .\) One can speak of the extension \(\pi\) being equicontinuous, distal, or proximal, by restricting to consideration of \(R(\pi)\ .\) Thus \(\pi\) is equicontinuous if (in the metric case) for every \(\varepsilon>0\) there is a \(\delta>0\) such that whenever \((x,x')\in R(\pi)\) and \(d(x,x')<\delta\) then \(d(tx,tx')<\varepsilon\) for all \(t\in T\ .\) Similarly, \(\pi\) is distal if \(R(\pi) \cap P=\Delta\ ,\) and \(\pi\) is proximal if \(R(\pi)\subset P\ .\) Of course an equicontinuous extension is distal. (If \(Y=1\) the one point flow, then clearly \(\pi\) is equicontinuous, distal, or proximal if and only if \((X,T)\) has the corresponding property.)

Now it is immediate that if \((Y,T)\) is distal and \(\pi\) is distal, then \((X,T)\) is distal. However, it is not the case that an equicontinuous extension of an equicontinuous flow is equicontinuous (an example will be given below). In fact, this observation is the key to the structure theorem. Start with the one point flow and extend it equicontinuously, to obtain an equicontinuous flow. Extend this flow equicontinuously, to obtain a distal flow. Continue to extend equicontinuously (possibly transfinitely often, as defined below) always remaining in the class of distal flows.

This is reasonably straightforward. What is remarkable and deep is that every distal minimal flow is obtained in this manner, starting with the one point flow, and successively extending equicontinuously. The precise statement of the Furstenberg structure theorem follows.

Let \((X,T)\) be a distal minimal flow. Then there is an ordinal number \(\eta\) and a family of minimal flows \((X_{\alpha},T)\) for \(\alpha \leq \eta\) such that \((X_{\eta},T)=(X,T)\ ,\) \((X_0,T)\) is the trivial (one point) flow, \((X_{\alpha +1},T)\) is an equicontinuous extension of \((X_{\alpha},T)\ ,\) and if \(\alpha\) is a limit ordinal, then \((X_{\alpha},T)\) is the inverse limit of the flows \((X_{\beta},T)\) for \(\beta <\alpha\ .\)

A non obvious consequence is that a distal minimal flow always has a non-trivial equicontinuous factor (namely \((X_1,T)\)).

At the opposite extreme from distal and equicontinuous are the weakly mixing flows. These are defined as the flows for which the product flow \((X\times X,T)\) is topologically transitive. It follows that weakly mixing flows have no non-trivial equicontinuous factor (so in light of the previous paragraph no distal factor). If \(T\) is abelian this latter property characterizes weakly mixing minimal flows.

The Galois theory of minimal flows

A partial classification is provided by the Galois theory of minimal flows, initiated by Robert Ellis. To this end, we need to introduce the universal minimal flow. For every group there is a unique universal minimal flow \((M,T)\ .\) Its defining property is that every minimal flow \((X,T)\) is a factor of \((M,T)\ .\) Let \(G\) denote the group of automorphisms of \((M,T)\ .\) The group \(G\) is "as large as possible" in the sense that if \((m,n)\) is an almost periodic point of the product flow \((M\times M,T)\) then there is an \(\alpha \in G\) such that \(\alpha(m)=n\ .\)

Let \((X,T)\) be a minimal flow, with \(\pi:M \to X\) a homomorphism. The (Ellis) group of \((X,T)\) is the subgroup of \(G\) defined by \(\mathcal G (X,\pi)=\{\alpha \in G|\pi\alpha=\pi\}\ .\) (A different homomorphism gives rise to a conjugate subgroup.) Clearly if \((Y,T)\) is a factor of \((X,T)\ ,\) with \(\theta:X \to Y\) a homomorphism, then \(\mathcal G(X,\pi)\) is a subgroup of \(\mathcal G(Y,\theta \pi)\ .\) The converse holds modulo a proximal extension. That is, if \((X,T)\) and \((Y,T)\) are minimal flows with \(\mathcal G(X) \subset \mathcal G(Y)\) then there is a proximal extension \((X',T)\) of \((X,T)\) with \(Y\) a factor of \(X'\ .\) It follows that if \(\mathcal G (X)=\mathcal G (Y)\) then \(X\) and \(Y\) have a common proximal extension.

The group \(G\) can be endowed with a compact \(T_1\) (but not Hausdorff) topology, and a subgroup \(A\) of \(G\) is the Ellis group of some minimal flow if and only if \(A\) is closed. A number of dynamical properties are "Ellis group invariants". That is, they depend only on the Ellis group of the minimal flow. One such is the property of proximal being an equivalence relation, and proximal closed is another such. If the acting group \(T\) is abelian the minimal flows \((X,T)\) and \((Y,T)\) are disjoint (meaning the product flow \((X \times Y,T)\) is minimal) if and only if the product of their Ellis groups is \(G\ .\)

If \(T\) is abelian, weak mixing is an Ellis group property. Let \(G'\) be the intersection of the closed neighborhoods of the identity in \(G\ .\) \(G'\) is a closed normal subgroup of \(G\) and the quotient group \(G/G'\) is Hausdorff. If \(\mathcal G (X)G'=G\ ,\) then \((X,T)\) is weak mixing, and if \(T\) is abelian, the converse holds as well.

A general structure theorem for minimal flows combines equicontinuous, proximal, and weakly mixing extensions. We omit the precise statement. (A homomorphism \(\pi:X \to Y\) is weak mixing if the relation \(R(\pi)\) is topologically transitive, so if \(Y=1\ ,\) the trivial flow, then \((X,T)\) is weak mixing.) If the extensions are equicontinuous and proximal this leads to the class of PI (proximal isometric) flows. A subclass of the latter are the point distal flows (those minimal flows which have a distal point–one which has no other point proximal to it). These theorems were inspired by, and generalize the Furstenberg structure theorem. Their proofs make essential use of the Galois theory.


For the acting group \(\Bbb Z\) a "trivial" minimal flow is a permutation of a finite set. The simplest non-trivial example of a minimal \(\Bbb Z\) action is the irrational rotation of the circle \(\Bbb K\ ,\) \(\varphi(z)=\alpha z\ ,\) where \(\alpha\) and \(z\) are complex numbers of absolute value \(1\ ,\) and \(\alpha\) not a root of unity. On the two torus \(\Bbb K^2\ ,\) \(\varphi(z,w)=(\alpha z,\beta w)\) defines a minimal action if and only if \(\alpha ^m\beta ^n \neq 1\) except when \(m=n=0\ .\) (Both of these cascades are equicontinuous.) An important generalization of the latter are the skew products on \(\Bbb K^2\ ,\) \(\varphi(z,w)=(\alpha z, \theta(z)w)\ ,\) where as above \(\alpha\) is not a root of unity and \(\theta:\Bbb K \to \Bbb K\) is continuous. The cascade defined by \(\varphi\) is minimal if and only if the functional equation \(f(\alpha z)=\theta(z)^n f(z)\) has no continuous solution for \(n \neq 0\ .\) In particular, \(\varphi (z,w)=(\alpha z,zw)\) defines a minimal cascade, which is distal and not equicontinuous. Moreover, it is an equicontinuous extension of the (equicontinuous) irrational rotation of the circle, illustrating the Furstenberg structure theorem.

The subject of symbolic dynamics, which originated from the study of geodesics on surfaces of negative curvature, provides a rich supply of examples of cascades. Let \(\Omega=\{0,1\}^{\Bbb Z}\) the space of two sided infinite sequences of \(0\)s and \(1\)s, provided with the product topology. Let \(\sigma\) be the shift homeomorphism of \(\Omega\ ,\) \(\sigma(\omega)(n)=\omega(n+1)\ .\) The cascade \((\Omega, \sigma)\) is topologically transitive, but clearly not minimal. To obtain a minimal system it is sufficient to construct an almost periodic point \(\omega\ ,\) in which every finite block occurs "syndetically often", equivalently with bounded gaps. In this case the orbit closure of \(\omega\) is a minimal cascade. A historically important example is the Morse sequence, which has a number of definitions. One such, which has inspired more general constructions, is by "substitution". If \(b\) is a finite word consisting of \(0\)s and \(1\)s let \(b^*\) be the word obtained by substituting \(01\) for \(0\) and \(10\) for \(1\ .\) (For example if \(b=011\) then \(b^*=011010\ .\)) Now let \(b_1=0\) and inductively define \(b_{n+1}= b_n^*\ .\) The Morse sequence is the infinite sequence obtained by this process (and then reflecting to obtain a two sided sequence). Its orbit closure, the Morse minimal set, is an example of a PI flow-its analysis requires both equicontinuous and proximal extensions.

A minimal and equicontinuous real action on the two torus, which is closely related to the irrational cascades discussed above is defined by \(\varphi_t(z,w)=(e^{2\pi i t}z, e^{2 \pi i\mu t}w)\) for \(t\in \Bbb R\) and \(\mu\) irrational. Important examples of distal but not equicontinuous real actions are the flows on nilmanifolds, which are homogeneous spaces of nilpotent Lie groups. At the other extreme are the horocycle flows which are minimal weakly mixing real actions on the unit tangent bundle of a surface.

As was mentioned above, abelian groups do not admit minimal proximal actions. There are minimal proximal flows for free groups as well as \(SL(2, \Bbb R)\) and other matrix groups. The existence of such examples has been applied to study the exponential growth of groups.

A brief bibliography follows. References [Gottschalk and Hedlund, 1955], [Ellis,1969], and [Auslander.1988] develop the theory as outlined in this article. [Furstenberg,1981] and [Glasner, 1976] present connections of topological dynamics to number theory and group theory, respectively. [Glasner, 2003] is an important recent book, which exhibits connections as well as analogies of topological dynamics and ergodic theory.


  • J.Auslander, Minimal flows and their extensions, North Holland (Notas de Mathematica, 153), Amsterdam, 1988
  • R.Ellis, Lectures in topological dynamics, W.A.Benjamin, New York, 1969
  • H.Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981
  • E.Glasner, Proximal flows, Lecture Notes in Mathematics, 517, Springer-Verlag, 1976
  • E.Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, Vol.101, Amer.Math.Soc., Providence, R.I., 2003
  • W.H.Gottschalk and G.A.Hedlund, Topological dynamics, A.M.S. Colloquium Publications, Vol. XXXVI, Amer. Math. Soc., Providence, R.I., 1955

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