CHAPTER 6
Topological dynamics
In contrast to the theory of p.m.p. actions, topological dynamics is bedeviled by a
disconcerting variety of different properties. Even the basic equivalence between indecomposability and transitivity expressed in Proposition 1.5 fails in the topological framework, an observation which launches our discussion of minimality, topological transitivity, and recurrence in Section 6.1 (compare Section 1.1). Nevertheless, one can still
establish a structure theorem for minimal actions which exhibits a striking parallel with
the Furstenberg-Zimmer structure theorem for p.m.p. actions. The only difference is that,
in addition to the isometric and weakly mixing incontractible extensions which play the
role of the compact and weakly mixing extensions in the Furstenberg-Zimmer picture, the
tower decomposition must now also weave in proximal extensions, which have no counterpart in the p.m.p. theory. When these proximal extensions are trivial, the tower reduces
precisely to the Furstenberg-Zimmer form. This structure theory based on extensions is
explained in Section 6.3. The absolute versions of weak mixing and being isometric are
examined in Section 6.2 (compare Section 1.2), the latter under its purely topological
formulation as equicontinuity.
6.1. Minimality, topological transitivity, and Birkhoff recurrence
For an action G y X of a group on an ordinary set, the notions of irreducibility
(no nonempty proper G-invariant subsets) and transitivity (for any x, y ∈ X there is
an s ∈ G such that sx = y) are equivalent. When translating these concepts into the
measure-theoretic framework of p.m.p. actions G y (X, µ), the equivalence persists: by
Proposition 1.5, such an action is ergodic (no G-invariant sets of intermediate measure)
if and only if for all sets A, B ⊆ X of positive measure there is an s ∈ G such that
µ(sA ∩ B) > 0. In topological dynamics this equivalence breaks down and we end up
with two distinct properties, minimality and (topological) transitivity. The latter is more
akin to ergodicity from a Baire category viewpoint, where comeagerness plays the role
of full measure (see Proposition 6.9). Minimality on the other hand is a considerably
stronger property with much more substantial consequences, and it is generally harder to
exhibit than ergodicity.
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6. TOPOLOGICAL DYNAMICS
From now onward in this section G y X will always be a continuous action on a
compact Hausdorff space.
D EFINITION 6.1. The action G y X is minimal if X has no nonempty proper Ginvariant closed subsets. A closed G-invariant set A ⊆ X is minimal if the restriction
of the action to A is minimal. The action G y X is (topologically) transitive if for all
nonempty open sets U, V ⊆ X there exists an s ∈ G such that sU ∩ V 6= ∅.
As the brackets above indicate, we will drop the qualifier “topologically” when speaking of transitivity in topological dynamics. This will not cause any confusion since continuous actions of G on a compact Hausdorff space X are never transitive in the ordinary
sense when X is infinite, while the two notions of transitivity coincide when X is finite.
P ROPOSITION 6.2. The action G y X is minimal if and only if every orbit is dense.
P ROOF. For every x ∈ X the closed set Gx is G-invariant by continuity, and so
minimality implies that every orbit is dense. Conversely, if every orbit is dense then for
every nonempty closed G-invariant set A ⊆ X we take a point x ∈ A and observe that
X = Gx ⊆ A, so that A = X.
In general one cannot decompose a topological system into minimal pieces in the
manner of the ergodic decomposition for p.m.p. actions. A simple example is the action
of an infinite group G on itself by left translation compactified with a fixed point at infinity.
At least we know that minimal subsystems always exist:
P ROPOSITION 6.3. For every action G y X there is a nonempty closed G-invariant
subset of X on which the action is minimal.
P ROOF. The nonempty closed G-invariant subsets of X are partially ordered by inclusion, and every totally ordered collection of such sets is bounded below by its intersection.
Thus by Zorn’s lemma there is a minimal such set, and this set satisfies the requirements
of the proposition.
The word free will be used in in the same sense as for actions on ordinary sets:
D EFINITION 6.4. The action G y X is free if sx = x for some x ∈ X and s ∈ G
implies s = e.
As for ergodic p.m.p. actions, in the special case G = Z we have a simple dichotomy
for minimal actions: either (i) X is finite and the action is conjugate to the translation
action on Z/nZ where n = |X|, or (ii) X is infinite and the action is free.
E XAMPLE 6.5. Let θ ∈ [0, 1) and define the rotation homeomorphism Tθ : T →
T by Tθ z = e2πiθ z, as in Section 1.3. If θ is a nonzero rational number expressed in
6.1. MINIMALITY, TOPOLOGICAL TRANSITIVITY, AND BIRKHOFF RECURRENCE
161
lowest terms as p/q, then every point x is periodic (i.e., T n x = x for some n ≥ 1)
with smallest period q. On the other hand, if θ is irrational then Tθ is minimal. To see
this, let A be an open arc in T and z a point in T and let us argue that the orbit of
z intersects A. By the pigeonhole principle there are distinct n, m ∈ Z such that the
distance |T n z − T m z| is smaller than the diameter of A. Since θ is irrational, T n−m is a
nontrivial rotation. As the distance separating any two successive points in the forward
orbit z, T n−m z, T 2(n−m) z, T 3(n−m) z, . . . is equal to |T n z − T m z|, it follows that one of
the points in this list must be contained in A, as desired.
E XAMPLE 6.6. For θ ∈ R we define the homeomorphism T from T2 ∼
= R2 /Z2 to
itself by T (x, y) = (x + θ, x + y), with points in T2 expressed in terms of their representatives in R2 . The inverse of this map is (x, y) 7→ (x − θ, y − x + θ). These skew
transformations appeared in Section 1.3, where they were shown to be ergodic with respect to Haar measure in the case that θ is irrational.
We argue that the assumption of irrationality on θ also implies that T is minimal.
Suppose that this is not the case. Then by Proposition 6.3 there exists a nonempty proper
minimal T -invariant closed subset Z of T2 . Now the rotation transformation x 7→ x + θ
of T is a factor of T via the first coordinate projection map π : (x, y) 7→ x, and since this
rotation is minimal by Example 6.5 it follows that π(Z) = T. Thus for each x ∈ T the
fibre Zx := {y ∈ T : (x, y) ∈ Z} is nonempty.
Given an α ∈ R, define the homeomorphism Rα : T2 → T2 by Rα (x, y) = (x, y + α).
This commutes with T , so that the set Rα Z is invariant and minimal under T . Thus if the
T -invariant set Rα Z ∩ Z is nonempty then we have Rα Z = Z. It follows that the set
{α ∈ R : Rα Z ∩ Z 6= ∅} is a closed subgroup of R, and therefore must be equal modulo
Z to {0, 1/n, 2/n, . . . , (n−1)/n} for some integer n ≥ 1 since Z is proper in T2 . We thus
deduce that each Zx is a coset of T modulo the subgroup {0, 1/n, 2/n, . . . , (n − 1)/n}.
Consider next the maps Sn , Tn : T2 → T2 defined by Sn (x, y) = (x, ny) and
Tn (x, y) = (x + θ, y + nx). Observe that Tn Sn = Sn T . For each x ∈ T the fibre
{y ∈ T : (x, y) ∈ Sn Z} is equal to nZx and hence is a singleton, which implies that Sn Z
is the graph of a continuous function g : T → T. Since Tn (Sn Z) = Sn (T Z) = Sn Z, for
every x ∈ T we have g(x + θ) = g(x) + nx. But this is impossible because x 7→ g(x) and
x 7→ g(x + θ) lie in the same homotopy class of continuous maps T → T while x 7→ nx
is homotopically nontrivial. We therefore conclude that T is minimal.
E XAMPLE 6.7. Let Y be a compact Hausdorff space and consider Y G with the product
topology. We define the (left) shift action G y Y G by (sy)t = ys−1 t for all (yt ) ∈ Y G . As
the constant functions in Y G are fixed points, this action is neither minimal (unless Y is a
singleton) nor free (unless G = {e}). It is however transitive when G is infinite, as is easy
to see using the standard basis for the product topology. Assigning a constant function in
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6. TOPOLOGICAL DYNAMICS
Y G its unique value in Y gives a homeomorphism from the set of fixed points to Y , and
so the space Y is, up to homeomorphism, a complete invariant for the shift actions of a
fixed G.
E XAMPLE 6.8. Consider the free group Fr on a finite generating set S of cardinality
r ≥ 2. Write ∂Fr for its Gromov boundary, which consists of all infinite reduced words
s1 s2 s3 · · · in S ∪ S −1 . For each t ∈ Fr \{e} write Wt for the set of all elements in ∂Fr
which begin with t expressed as a reduced word. These sets form a clopen basis for a
topology under which ∂Fr is a Cantor set. We then define a continuous action Fr y ∂Fr
by concatenation and reduction. It is easy to see that every orbit is dense, so that the action
is minimal by Proposition 6.2. One can moreover verify that a point s1 s2 s3 · · · in ∂Fr is
fixed by some nontrivial group element if and only if it is eventually periodic, by which
we mean that there are n, k ∈ N such that sm+k = sm for all m ≥ n. Thus the action
is not free. Note however that the set of all points which are not eventually periodic is a
dense Gδ (in fact for any continuous action G y X on a compact Hausdorff space the
G-invariant set of points in X with trivial stabilizer is a Gδ ), and so using Baire category
language one might describe the action as being generically free.
In view of Proposition 6.2, the following result suggests that, when X is metrizable,
transitivity can be thought of as a generic version of minimality in the sense of Baire
category. We thus do get an analogue of the equivalence of ergodicity and measuretheoretic transitivity for p.m.p. actions ((i)⇔(iii) of Proposition 1.5) granted that we use
comeagerness as a substitute for the requirement that something hold almost everywhere
in measure.
P ROPOSITION 6.9. Suppose that X is metrizable. Then the following are equivalent:
(i) the action G y X is transitive,
(ii) there is a dense orbit,
(iii) the set of points in X with dense orbit is a dense Gδ .
P ROOF. (i)⇒(iii). Fix a countable basis {Ui : i ∈ I} for the topology on X. For i ∈ I
write Ωi for the set of all x ∈ X such that sx ∈ Ui for some s ∈ G. Then Ωi is open,
T
and it is dense by transitivity. Therefore the Gδ set i∈I Ωi is dense by the Baire category
theorem, and it consists precisely of those points with dense orbit.
(iii)⇒(ii). Trivial.
(ii)⇒(i). Take an x ∈ X with dense orbit. Then for all nonempty open sets U, V ⊆ X
we can find s, t ∈ G such that sx ∈ U and tx ∈ V , in which case ts−1 U ∩ V 6= ∅. We
thus obtain (i).
Example 6.7 and the Bernoulli actions in Section 1.3 together illustrate that topological transitivity is more akin to ergodicity than is minimality, and occurs with a similar
6.1. MINIMALITY, TOPOLOGICAL TRANSITIVITY, AND BIRKHOFF RECURRENCE
163
natural prevalence. In fact, looking beyond Examples 6.5, 6.6, and 6.8 it becomes difficult to find simple occurrences of minimality. Every countable group G admits a free
minimal action, as one can take a minimal closed subset of the canonical action G y βG
on the Stone-Cech compactification, which is free (as can be seen by taking for a given
t ∈ G\{e} a maximal set H ⊆ G satisfying H ∩ tH = ∅ and then defining E1 = H,
E2 = tH, and E3 = G\(E1 ∪E2 ), and then checking that Ei ∩tEi = ∅ for each i = 1, 2, 3,
which means that the closures of these sets in βG, which partition βG, have the same
property). It is harder to prove that every countable group G admits a free minimal action
with an invariant regular Borel probability measure (such an action of G always exists
on the Cantor set [72]). It is also largely a mystery which connected compact manifolds
admit minimal actions of a given G, even when G = Z.
If X contains no isolated points, then every transitive action G y X is recurrent in
the sense that for all nonempty open sets U ⊆ X the set {s ∈ G : sU ∩ U 6= ∅} is infinite,
which in the case that X is metrizable can be verified by picking an x ∈ U with dense
orbit and recursively constructing a sequence {sn } in G such that for every n we have
sn x ∈ U \{s1 x, . . . , sn−1 x}. It is also useful to localize the idea of recurrence to points:
D EFINITION 6.10. A point x ∈ X is recurrent if for every neighbourhood U of x the
set {s ∈ G : sx ∈ U } is infinite.
P ROPOSITION 6.11. Suppose that the action G y X is transitive and that X is
metrizable and has no isolated points. Then the set of recurrent points in X is a dense
Gδ .
P ROOF. Fix a compatible metric d on X. Take an enumeration s1 , s2 , . . . of the elements of G, and for n, m ∈ N define the open set
Am,n = {x ∈ X : d(si x, x) < 1/m for some i ≥ n}.
Then Am,n contains every point x with dense orbit, since for every s ∈ G the set of all
i ∈ N for which d(si x, sx) < 1/m is infinite in view of the fact that sx is not an isolated
T
point. It follows by Proposition 6.9 that the intersection n,m∈N Am,n is a dense Gδ , and
it consists precisely of the recurrent points.
As a nontrivial shift action G y Y G demonstrates, not every point need be recurrent
for a transitive action on a space without isolated points. On the other hand, we show next
that if the action is minimal then every point is almost periodic, which for infinite G is a
refinement of recurrence.
D EFINITION 6.12. A point x ∈ X is almost periodic if for every neighbourhood U of
x the set {s ∈ G : sx ∈ U } is syndetic (Definition C.15).
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6. TOPOLOGICAL DYNAMICS
P ROPOSITION 6.13. For an action G y X, a point x ∈ X is almost periodic if and
only if the restriction of the action to Gx is minimal. In particular, X decomposes into a
(disjoint) union of minimal sets if and only if every point is almost periodic.
P ROOF. We may assume that Gx = X. Suppose first that the restriction of the action
to Gx is minimal. Let U be an open neighbourhood of x. Then {sU : s ∈ G} is an
open cover of X, for otherwise the complement of its union is a nonempty proper closed
G-invariant subset of X, contradicting minimality. Thus by compactness there is a finite
set F ⊆ G such that {sU : s ∈ F } covers X. Now for every t ∈ G there exists an s ∈ F
such that tx ∈ sU and hence s−1 tx ∈ U , so that t = s(s−1 t) ∈ F · {s ∈ G : sx ∈ U }.
Thus {s ∈ G : sx ∈ U } is syndetic.
In the converse direction, suppose now that x is almost periodic. To establish minimality, we suppose to the contrary that there exists a nonempty proper closed G-invariant
subset A of X. Then x ∈
/ A, for otherwise Gx ⊆ A and hence Gx 6= X. Since A is
closed, we can find a closed neighbourhood V of x such that V ∩ A = ∅. Since x is
almost periodic there is a finite set F ⊆ G such that for every s ∈ G there is a t ∈ F for
S
which tsx ∈ V . Since the open set X\ t∈F t−1 V contains A and hence is nonempty, it
must contain ux for some u ∈ G by the density of Gx in X. But then tux ∈
/ V for all
t ∈ F , contradicting our choice of F . We thus conclude that the action is minimal.
Combining Propositions 6.3 and 6.13 we obtain the following.
T HEOREM 6.14 (Birkhoff recurrence). For every action G y X the set of almost
periodic points is nonempty. In particular, if G is infinite then there exists a recurrent
point.
Note that the set of almost periodic points can be a singleton even if X is infinite.
Consider for instance the case that G is infinite and acts on itself by left translation and
fixes ∞ in the one-point compactification G ∪ {∞}.
The compactness principle which permits us to localize to points in Theorem 6.14
will come into play once again in our study of higher-order recurrence properties based
on combinatorial independence.
6.2. Weak mixing and equicontinuity
The following are topological analogues of the ergodic-theoretic properties of mixing,
weak mixing, and compactness. Recall that a family Ω of maps from a compact Hausdorff
space X to itself is said to be equicontinuous if for every neighbourhood U of the diagonal
in X × X there is another neighbourhood V of this diagonal such that for all f ∈ Ω and
all x, y ∈ X satisfying (x, y) ∈ V one has (f (x), f (y)) ∈ U . If X has a compatible
6.2. WEAK MIXING AND EQUICONTINUITY
165
metric d one can express this by saying that for every ε > 0 there is a δ > 0 such that for
all f ∈ Ω and all x, y ∈ X satisfying d(x, y) < δ one has d(f (x), f (y)) < ε.
D EFINITION 6.15. The action G y X is
(i) mixing if G is infinite and for all nonempty open sets U, V ⊆ X there is a finite
set F ⊆ G such that sU ∩ V 6= ∅ for all s ∈ G\F ,
(ii) weakly mixing if the product action G y X × X is transitive, i.e., for all
nonempty open sets U1 , U2 , V1 , V2 ⊆ X there is an s ∈ G such that sU1 ∩V1 6= ∅
and sU2 ∩ V2 6= ∅,
(iii) equicontinuous if the homeomorphisms x 7→ sx for s ∈ G form an equicontinuous family of maps from X to itself.
Clearly mixing implies weak mixing. Note also that if X is metrizable and G y X
is equicontinuous, then there is a compatible metric d on X under which the action is
isometric, i.e., d(sx, sy) = d(x, y) for all x, y ∈ X and s ∈ G. To construct such a metric
d, start with any compatible metric d0 on X and set d(x, y) = sups∈G d0 (sx, sy) for all
x, y ∈ X. Conversely, if the action is isometric for some compatible metric then it is
clearly equicontinuous.
E XAMPLE 6.16. The rotations of Example 6.5 are isometric for the Euclidean metric
when T is viewed as the unit circle in R2 , and hence are equicontinuous. The shift action
of Example 6.7 on the other hand is easily seen to be mixing when G is infinite. We will
see in Example 6.25, in the context of our discussion on distality, that the minimal skew
tranformations from Example 6.6 are neither weakly mixing nor equicontinuous.
P ROPOSITION 6.17. If G y Y is an equicontinuous factor of a weakly mixing action
G y X, then Y is a singleton. In particular, if an action G y X is both weakly mixing
and equicontinuous then X is a singleton.
P ROOF. Suppose to the contrary that |Y | > 1. Then we can find a neighbourhood
ε of the diagonal in Y × Y and nonempty open sets U, V ⊆ Y such that U × V and
ε are disjoint, and since the action is equicontinuous we may assume by shrinking U if
necessary that (sx, sy) ∈ ε for all s ∈ G and x, y ∈ U . Writing ϕ : X → Y for the
factor map and setting Ũ = ϕ−1 (U ) and Ṽ = ϕ−1 (V ), it follows that there is no s ∈ G
for which we have both sŨ ∩ Ṽ 6= ∅ and sŨ ∩ Ũ 6= ∅, contradicting weak mixing.
The simple dichotomy that governs the relation between weak mixing and compactness in measurable dynamics breaks down in the topological framework once we move
beyond the above observation. The next results highlight the role of amenability and
invariant probability measures in determining when a non-weakly-mixing action has a
nontrivial equicontinuous factor, and also demonstrate how this relates to the problem of
whether weak mixing implies weak mixing of all orders in the following sense.
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6. TOPOLOGICAL DYNAMICS
D EFINITION 6.18. The action G y X is weakly mixing of all orders if for every
n ∈ N the product action G y X n is transitive.
For p.m.p. actions, Theorem 1.24 shows that weak mixing is equivalent to weak mixing of all orders as defined by substituting ergodicity for transitivity above. As we will
see in Example 6.22, this is no longer true in the topological setting. It is however true
for minimal actions admitting an invariant regular Borel probability measure, as we show
in Proposition 6.20, as well as for all actions of Abelian groups, as we show in Proposition 6.21. Later in Theorem 7.16 we will characterize weak mixing of all orders in terms
of a local independence property.
In the following two results, by an action G y X being nontrivial we mean that X is
not a singleton.
L EMMA 6.19. Let G y X be a minimal action and let G y Y be a transitive action
with an invariant regular Borel probability measure µ of full support. Suppose that the
product action G y X ×Y fails to be transitive. Then the action G y X has a nontrivial
equicontinuous factor.
P ROOF. Suppose that the product action G y X × Y is not transitive. Then we can
find a proper closed G-invariant subset A of X × Y with nonempty interior. For every
x ∈ X set Ax = {y ∈ Y : (x, y) ∈ A}. Let π : X → L1 (Y, µ) be the map x 7→ 1Ax .
With G acting isometrically on L1 (Y, µ) via the composition sf (y) = f (s−1 y), we have
s · 1Ax = 1Asx for all x ∈ X and s ∈ G, so that π is equivariant.
Next we show that π is continuous, so that the action of G on its image is an equicontinuous factor of G y X. Let w ∈ X and ε > 0. By regularity there is an open neighbourhood U of Aw in Y such that µ(U \Aw ) < ε/2. Since A is closed, the map x 7→ Ax
is upper semicontinuous for the Vietoris topology, and so there exists a neighbourhood V
of w in X such that for all x ∈ V we have Ax ⊆ U , in which case µ(Ax ) ≤ µ(Aw ) + ε/2.
Thus the function h on X given by x 7→ µ(Ax ) is upper semicontinuous. Since h is constant on orbits, each of which is dense by minimality, it follows that h is globally constant.
Hence for every x ∈ V we have
k1Ax − 1Aw k1 ≤ µ(U \Ax ) + µ(U \Aw ) < ε,
showing that π is continuous.
It remains to prove that the image of π is not a singleton, so that the factor is nontrivial.
The images of (X ×Y )\A and the interior of A under the projection map X ×Y → Y are
open, nonempty, and G-invariant, and hence are also dense because G y Y is transitive.
Thus these two images contain a common point y. Taking x, w ∈ X such that (x, y) ∈
(X × Y )\A and (w, y) lies in the interior of A we then have 1Ax 6= 1Aw in L1 (Y, µ) since
µ has full support. Hence |π(X)| > 1.
6.2. WEAK MIXING AND EQUICONTINUITY
167
P ROPOSITION 6.20. Let G y X be a minimal action possessing an invariant regular
Borel probability measure µ. Then
(i) the action is either weakly mixing or has a nontrivial equicontinuous factor,
(ii) the action is weakly mixing if and only if it is weakly mixing of all orders.
P ROOF. (i) By Proposition 6.17 we see that both properties cannot occur simultaneously. Suppose that the product action G y X × X is not transitive. Since minimality
implies that µ has full support, we conclude by Lemma 6.19 that G y X has a nontrivial
equicontinuous factor.
(ii) For the nontrivial direction, suppose that the action is weakly mixing but not
weakly mixing of all orders. Take the smallest n > 2 such that the product action
G y X n fails to be transitive. Using the product action G y X n−1 , which is transitive and has the invariant probability measure µn−1 of full support, we apply Lemma 6.19
to deduce that the action G y X has a nontrivial equicontinuous factor, contradicting
(i).
We remark that if G is amenable then (i) and (ii) in the above proposition hold for
any minimal action, since the existence of an invariant regular Borel probability measure
is automatic in this case by Theorem 3.4. Also, the existence of weakly mixing invariant
regular Borel probability measure of full support implies topological weak mixing, but
the converse is false, even if we upgrade the hypothesis of weak mixing to mixing (see
4.27 in [57]).
When G is Abelian we can establish (ii) of Proposition 6.20 for general actions using
a more direct argument:
P ROPOSITION 6.21. Suppose that G is Abelian. Then the action G y X is weakly
mixing if and only if it is weakly mixing of all orders.
P ROOF. To prove the nontrivial direction, suppose that the action is weak mixing. For
nonempty open sets U, V ⊆ X we define
N (U, V ) = {s ∈ G : sU ∩ V 6= ∅}.
Let U, V, Y, Z be nonempty open subsets of X, and let us show that there are nonempty
open sets A, B ⊆ X such that N (A, B) ⊆ N (U, V )∩N (Y, Z). Since the action is weakly
mixing we can find an s ∈ G such that the sets A := U ∩ s−1 Y and B := V ∩ s−1 Z are
both nonempty. Now if t ∈ N (A, B) then
t(U ∩ s−1 Y ) ∩ (V ∩ s−1 Z) 6= ∅,
which means on the one hand that tU ∩ V 6= ∅ and on the other, using Abelianness, that
s−1 (tY ∩ Z) = ts−1 Y ∩ s−1 Z 6= ∅ and hence tY ∩ Z 6= ∅. Thus t ∈ N (U, V ) ∩ N (Y, Z)
and so N (A, B) ⊆ N (U, V ) ∩ N (Y, Z), as desired.
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6. TOPOLOGICAL DYNAMICS
It now follows that for any nonempty open sets U1 , . . . , Un , V1 , . . . , Vn ⊆ X we have
i=1 N (Ui , Vi ) 6= ∅ , for if n = 1 or 2 we can apply weak mixing directly while if
n > 2 we can reduce to the case n − 1 by appealing to the first paragraph and then
repeat this process until we reach the case n = 2. Since N (U1 × · · · × Un , V1 × · · · ×
T
Vn ) = ni=1 N (Ui , Vi ) we conclude that the product action G y X n is transitive for every
n ∈ N.
Tn
The following example shows that the hypothesis of an invariant regular Borel probability measure in Proposition 6.20 cannot be removed.
E XAMPLE 6.22. Consider the free group Fr on a finite generating set S of cardinality
r ≥ 2. We will argue that the boundary action Fr y ∂Fr from Example 6.8 is weakly
mixing but the product action Fr y ∂Fr ×∂Fr ×∂Fr is not transitive. Note that the action
Fr y ∂Fr does not admit an invariant Borel probability measure, as can be easily deduced
from the fact that for any a ∈ S and nonempty compact set K ⊆ ∂Fr \{a−1 a−1 a−1 · · · }
we have an K → {aaa · · · } in the Vietoris topology as n → ∞. As before, for each
t ∈ Fr \{e} write Wt for the set of infinite words in ∂Fr which begin with t expressed as
a reduced word in S ∪ S −1 .
Let u, v, x, y ∈ Fr \{e}. Since |S| ≥ 2 we can construct a z ∈ Fr such that, expressing
u, v, x, y, z as reduced words,
(i) the word s := yzu−1 does not reduce,
(ii) the word zu−1 (which is reduced by (i)) is longer than the word x, so as to
ensure that the reduction of sx begins with y, and
(iii) the word yz (which is reduced by (i)) is longer than the word v, so as to ensure
that the reduction of uz −1 y −1 v begins with u.
Then sWx ∩ Wy 6= ∅ by (ii) and sWu ∩ Wv 6= ∅ by (iii), showing that the action Fr y ∂Fr
is weakly mixing.
Now pick distinct a, b ∈ S and suppose that there is an s ∈ Fr such that sWa ∩ Wa ,
sWb ∩ Wb−1 , and sWb−1 ∩ Wb are all nonempty. The nonemptiness of sWa ∩ Wa implies
that s, as a reduced word, must begin with a or end with a−1 , while the nonemptiness of
both sWb ∩ Wb−1 and sWb−1 ∩ Wb means that s cannot end with a−1 . Thus s must begin
with a. But then the nonemptiness of sWb ∩ Wb−1 forces s to end with b−1 while the
nonemptiness of sWb−1 ∩ Wb forces s to end with b, a contradiction. Therefore the action
Fr y ∂Fr × ∂Fr × ∂Fr fails to be transitive.
What is responsible for the behaviour of the boundary action Fr y ∂Fr in Example 6.22 is the high compressibility of clopen sets that places us far outside the realm of
6.3. PROXIMALITY, DISTALITY, AND STRUCTURE THEOREMS
169
analogy with p.m.p. actions in terms of mixing properties. As we will see in the next section, even actions admitting an invariant regular Borel probability measure with full support can exhibit compressible behaviour between points along their orbits, a phenomenon
known as proximality, which already complicates the parallel with measurable dynamics.
Nevertheless, by properly taking proximality into account, one can still establish a general structure theorem for minimal actions, although the practical utility of this theorem
is largely restricted to the case where proximality is absent.
6.3. Proximality, distality, and structure theorems
Example 6.22 and Proposition 6.20 together suggest that the assumption of compactness on the space in our topological-dynamical set-up, while furnishing a sufficient degree
of “finiteness” to guarantee Birkhoff recurrence, will not by itself be a strict enough analogue of the condition of preserving a probability measure in order to imply a structure
theorem that shares the formal simplicity of the Furstenberg-Zimmer tower for p.m.p.
actions. In fact, contrary to what Proposition 6.20 might suggest, even if we restrict to
actions admitting an invariant regular Borel probability measure the picture is still much
more complicated than in the measurable setting, as we are forced to take into account
the phenomenon of compressibility between points along their orbits. This compressibility and the corresponding incompressibility are captured in the following definitions of
proximality and distality. What is remarkable is that minimal actions on compact metrizable spaces admit a tower decomposition that exactly parallels the Furstenberg-Zimmer
decomposition modulo proximal extensions, as we will see in Theorem 6.34.
D EFINITION 6.23. A pair (x, y) ∈ X × X is proximal if the closure of its orbit under
the product action G y X ×X intersects the diagonal, and the action G y X is proximal
if every pair in X × X is proximal. A pair in X × X is distal if it is not proximal, and
the action G y X is distal if every nondiagonal pair in X × X is distal. Note that if
X is metrizable with compatible metric d then a pair (x, y) ∈ X × X is proximal if
inf s∈G d(sx, sy) = 0 and distal if inf s∈G d(sx, sy) > 0.
E XAMPLE 6.24. The boundary action Fr y ∂Fr in Example 6.8 is proximal, for if
(x, y) ∈ ∂Fr × ∂Fr and S denotes the standard generating set for Fr then we can take
an a ∈ S ∪ S −1 which differs from the first letters of both x and y and observe that
(an x, an y) → (aaa · · · , aaa · · · ) as n → ∞.
E XAMPLE 6.25. The minimal skew transformation T (x, y) = (x + θ, y + x) of
T ∼
= R2 /Z2 in Example 6.6 (with θ irrational) is distal but not equicontinuous. Let
d denote the metric on either R2 /Z2 or R/Z which takes the smallest Euclidean distance between representatives. For distality, observe that for two points (x, y) and (x, y ′ )
2
170
6. TOPOLOGICAL DYNAMICS
with the same first coordinate we have d(T n (x, y), T n (x, y ′ )) = d((x, y), (x, y ′ )) for all
n ∈ Z, while if (x, y) and (x′ , y ′ ) are two points with distinct first coordinates then
d(T n (x, y), T n (x′ , y ′ )) ≥ d(x, x′ ) for all n ∈ Z. To see that T fails to be equicontinuous,
by minimality we can find a sequence ni → ∞ such that T ni (0, 0) = (ni θ, 21 ni (ni −
1)θ) → (0, 0), in which case
T ni ( 2n1 i , 0) = ( 2n1 i + ni θ, 12 + 12 ni (ni − 1)θ) → (0, 21 ).
Thus for large enough i the points ( 2n1 i , 0) and (0, 0), the distance between which tends
to 0 as i → ∞, are mapped under T ni to points at distance at least 41 from each other,
showing that T is not equicontinuous.
P ROPOSITION 6.26. Let G y X be an action. Let x ∈ X. Then there exists an
almost periodic point x′ ∈ X such that the pair (x, x′ ) is proximal.
P ROOF. Consider the collection C of subsets A ⊆ X such that the inclusion map from
A into X is an almost periodic point for the product action G y X A . This condition is
equivalent to saying that, given a finite set E ⊆ A and a neighbourhood Ua of a for each
a ∈ E, the set {s ∈ G : sa ∈ Ua for all a ∈ E} is syndetic. Ordering C by inclusion, we
observe that the union of any totally ordered subcollection of C is again in C, and so by
Zorn’s lemma C has a maximal element B.
Let w ∈ X B be the inclusion map from B into X. By Theorem 6.14 there is an
almost periodic point (w0 , x0 ) ∈ G(w, x) for the product action G y X B × X. Since
w is almost periodic, by Proposition 6.13 we have w ∈ Gw0 . By compactness we can
then find an x′ ∈ X such that (w, x′ ) ∈ G(w0 , x0 ). Since (w0 , x0 ) is almost periodic so is
(w, x′ ) by Proposition 6.13. Then x′ ∈ B, for otherwise we would have an identification
′
X B × X = X B∪{x } which would produce a contradiction with the maximality of B in
C. Therefore w has x′ as one of its coordinates, and since (w, x′ ) ∈ G(w0 , x0 ) ⊆ G(w, x)
we deduce that (x′ , x′ ) ∈ G(x′ , x) for the product action G y X × X, so that (x, x′ ) is a
proximal pair.
The following is immediate from Propositions 6.26 and 6.13.
P ROPOSITION 6.27. If G y X is distal then every point is almost periodic, in which
case X decomposes into a (disjoint) union of minimal sets.
P ROPOSITION 6.28. The action G y X is distal if and only if every point in X × X
is almost periodic for the product action G y X × X.
P ROOF. If G y X is distal then evidently so is G y X × X, and thus every point
in X × X is almost periodic by Proposition 6.27. Conversely, suppose that every point in
X × X is almost periodic. Let (x, y) be a nondiagonal pair in X × X, and suppose that
6.3. PROXIMALITY, DISTALITY, AND STRUCTURE THEOREMS
171
it is proximal. Then there is a w ∈ X such that (w, w) ∈ G(x, y). Since (x, y) is almost
periodic it follows by Proposition 6.13 that (x, y) ∈ G(w, w), which is impossible. Thus
(x, y) is a distal pair, and so G y X is distal.
Minimal distal actions on metrizable spaces are subject to the following structure
theorem that parallels the decomposition of distal p.m.p. actions into a tower of compact
extensions (see Theorem 2.15 and Remark 2.17). First we need to define the analogue of
compactness.
D EFINITION 6.29. An extension π : X → Y of minimal G-actions is isometric if
S
there is a continuous real-valued function d on the closed subset y∈Y (π −1 (y) × π −1 (y))
of X × X such that, for each y ∈ Y ,
(i) the restriction of d to π −1 (y) × π −1 (y) is a metric, and
(ii) d(sx, sx′ ) = d(x, x′ ) for all s ∈ G and x, x′ ∈ π −1 (y).
One can also formulate the above condition in an abstract topological way using
neighbourhoods of the diagonal in X × X so that it formally reduces to equicontinuity in the absolute case and does not require metrizability on the fibres, and indeed the
term equicontinuous is also used to describe such extensions.
E XAMPLE 6.30. As shown in Example 6.25, the minimal skew transformation T (x, y) =
(x+θ, y+x) of T2 ∼
= R2 /Z2 from Example 6.6 (with θ irrational) is distal but not equicontinuous. It is, however, an isometric extension of an equicontinuous action, for one can
consider the first coordinate projection map (x, y) 7→ x from T2 onto T, which factors
T onto rotation by θ and is isometric in the sense of Definition 6.29, as witnessed by
the metric on the fibres T ∼
= R/Z which takes the smallest Euclidean distance between
representatives. We can express this as a tower
T2 → T → {pt}
of isometric extensions, which motivates the following distal structure theorem of Furstenberg.
T HEOREM 6.31. Let G y X be a minimal distal action with X metrizable. Then
there exist a countable ordinal λ and a tower
(23)
X = Yλ → · · · → Y2 → Y1 → Y0 = {pt}
of isometric extensions. The action at a limit ordinal is the inverse limit of the actions
below it.
From the above structure theorem one can show the existence of an invariant Borel
probability measure, and by Proposition 6.27 we know that distal actions decompose into
172
6. TOPOLOGICAL DYNAMICS
a union of minimal components. Thus if we exclude proximality from the picture we
obtain a close alignment between topological and measurable dynamics.
As shift actions G y Y G suggest, one might expect that any kind of chaotic action
will exhibit some degree of proximality. Indeed for actions of sofic groups, distality
implies that the topological entropy is either zero or −∞. Thus even though proximality
has no counterpart for p.m.p. actions, its ghost is present in any p.m.p. action of a sofic
group with positive entropy, since any topological model for such an action has positive
entropy by the variational principle.
An Abelian group admits no nontrivial minimal proximal actions. Indeed suppose
that G y X is a minimal proximal action of an Abelian group. Then given x ∈ X and
t ∈ G there is a net {sj } in G such that (sj x, sj tx) converges to a diagonal pair in X × X,
which we may assume to be (x, x) by minimality. Then {tsj x} converges to both x and
tx, so that tx = x. Thus x is a fixed point and so X = {x} by minimality. On the other
hand, every nonamenable group admits a nontrivial minimal proximal action, and some
amenable groups do as well (see [55]). Those groups which do not admit a nontrivial
minimal proximal action are called strongly amenable in [55], and every minimal action
of such a group with positive entropy has both proximal and distal pairs off the diagonal.
We have thus seen that proximal behaviour can occur on the one hand as a symptom
of positive entropy like in the shift action G y {0, 1}G , which is saturated by independence, and on the other as a reflection of setwise compressibility like in the boundary
action Fr y ∂Fr of a free group, which exhibits an almost total lack of independence.
Interestingly, the following result applies to many minimal actions within each of these
otherwise very different classes. Generalizing the concept of proximal pair, for an action
G y X we say that a tuple (x1 , . . . xn ) ∈ X n is proximal if the closure of its orbit under
the product action G y X n intersects the diagonal.
P ROPOSITION 6.32. Let G y X be a minimal action such that for every integer
n ≥ 2 the set of proximal tuples in X n is dense. Then for every minimal action G y Y
the product action G y X × Y is transitive. In particular, the action G y X is weakly
mixing.
P ROOF. Let G y Y be a minimal action. To show that G y X × Y is transitive,
it suffices to prove, given a closed G-invariant set W ⊆ X × Y with nonempty interior,
that W = X × Y . Take nonempty open sets U ⊆ X and V ⊆ Y such that U × V ⊆ W .
By minimality the open sets sV for s ∈ G cover Y , and so by compactness we can find
S
s1 , . . . , sn ∈ G such that Y = ni=1 si V . Set Q = s1 U × · · · × sn U . By the density
hypothesis, Q contains a proximal tuple (x1 , . . . , xn ).
Let x ∈ X and y ∈ Y and let us show that (x, y) ∈ W . By proximality there is a net
{tj } in G such that tj (x1 , . . . , xn ) → (x, . . . , x), and applying the pigeonhole principle
6.3. PROXIMALITY, DISTALITY, AND STRUCTURE THEOREMS
173
we may assume by passing to a subnet that there is an i0 ∈ {1, . . . , n} such that for every
j we have t−1
j y ∈ si0 V , in which case
(tj xi0 , y) = tj (xi0 , t−1
j y) ∈ tj si0 W = W.
Thus (x, y) ∈ W and hence W = X × Y , as desired.
Note that if a minimal action G y X is proximal (e.g., the boundary action Fr y
∂Fr in Examples 6.8 and 6.24) then it automatically satisfies the hypothesis of the above
proposition. Indeed if G y X is any proximal action then for all integers n ≥ 2 every
tuple in X n is proximal. We verify this by induction. Suppose that every tuple in X n is
proximal for some n ≥ 2. Let (x1 , . . . , xn+1 ) be a point in X n+1 . Then its orbit closure
contains a point of the form (y, y, . . . , y, z), and the orbit closure of the pair (y, z) in
X × X contains a diagonal pair (w, w). Therefore (w, w, . . . , w) is contained in the orbit
closure of (y, y, . . . , y, z) and hence also of (x1 , . . . , xn+1 ). Thus every tuple in X n+1 is
proximal.
Despite the complications revealed by all of the above discussion, one can still establish a general structure theorem for minimal actions on metrizable spaces (Theorem 6.34)
that reproduces the distal structure theorem as a special case. In fact this is in direct
parallel with the Furstenberg-Zimmer structure theorem for p.m.p. actions, with a single
weakly mixing extension at the top and a tower of isometric extensions and inverse limits
underneath, except that now everything must be done modulo proximal extensions, which
don’t exist in the p.m.p. realm. Below the topmost weak mixing extension, the tower alternates between isometric and proximal extensions, and also trusses together with a second
tower of proximal extensions over the action. We are thus technically decomposing the
action at the top of this second tower.
D EFINITION 6.33. An extension π : X → Y of G-actions is said to be (topologically) weakly mixing if the product action of G on {(x, y) ∈ X × X : π(x) = π(y)} is
topologically transitive. It is incontractible if for every n ∈ N and y ∈ Y there is a dense
set of points in π −1 (y) × · · · × π −1 (y) which are almost periodic for the product action
G y X n . It is proximal if every pair (x, y) ∈ X × X such that π(x) = π(y) is proximal.
Incontractibility is a kind of rigidity condition that must be imposed on the weakly
mixing extension in Theorem 6.34 in order to ensure that the tower is canonically defined (it is also satisfied automatically by the isometric extensions). Indeed even an action
can be both weakly mixing and proximal (e.g., the boundary action Fr y ∂Fr in Examples 6.8, 6.22, and 6.24), but an extension that is both proximal and incontractible is
trivial. Thus incontractibility serves to isolate the weakly mixing behaviour that is not
purely proximal, which then appears in a canonical way at the top of the tower in the form
of a weakly mixing incontractible extension.
174
6. TOPOLOGICAL DYNAMICS
T HEOREM 6.34 (canonical tower). Let G y X be a minimal action with X metrizable. Then there are a countable ordinal λ and a canonically defined commutative diagram
Xλ
···
/
X3
/
πλ
π3
Yλ
X2
/
/
···
π2
~
~
/
θ4
Y3
X1
/
/
Z3
π1
~
/
θ3
Y2
X0
/
/
Z2
π0
~
/
θ2
Y1
/
/
Z1
/
θ1
Y0
X
/
{pt}
of extensions where π0 , π1 , . . . , πλ are incontractible, πλ is weakly mixing, θ1 , θ2 , . . . , θλ
are isometric, and the remaining horizontal extensions are proximal. At a limit ordinal η
the extension Xη → Yη is the inverse limit of the extensions Xν → Yν for ν < η.
If the proximal extensions are all trivial in the above picture, so that only the incontractible extensions remain, then we have the G-equivariant identification Xλ = X and
the tower reduces to
X → Yλ → · · · → Y2 → Y1 → Y0 = {pt}
where the leftmost extension is weakly mixing and the rest are isometric. Thus the incontractibility condition on extensions is precisely what substitutes for measure-preservingness
if we are looking for a canonical tower decomposition of exactly the same form as the
Furstenberg-Zimmer tower for p.m.p. actions. If the weakly mixing extension is also
trivial, we are back in the distal case of Theorem 6.31. One can also have the intermediate situation in which the proximal extensions are all trivial in the top tower but not
necessarily in the bottom one. Such actions are called PI (“proximal-isometric”).
6.4. Notes and references
Topological dynamics traces its roots back to Poincaré and his development of the
qualitative theory of differential equations. It evolved into a formal discipline through the
work of Birkhoff, Morse, Hedlund, and others starting in the 1920s and 1930s. General
references include [61, 39, 27, 144]. An encyclopedic treatise is [29].
In [117] Poincaré undertook an analysis of the orientation-preserving transformations
of the circle. To each such transformation he associated a “rotation number” by lifting
the transformation to the universal cover R and taking a limit of the average displacement under iteration. If the transformation is transitive, he showed that it is conjugate
to an irrational rotation as in Example 6.5. Nontransitive transformations with irrational
rotation number were later described by Markley in [98]. When the rotation number is
rational, every periodic orbit has the same period and the possibilities for the orbits can
be explicitly described. See Chapter 11 of [78] for details.
6.4. NOTES AND REFERENCES
175
The topology of the circle also severely restricts the kinds of actions of other groups
that can exist on it. An important result of Tits asserts that a finitely generated linear group
is either amenable (in fact virtually solvable) or contains a free group on two generators.
In [97] Margulis established an analogous dichotomy for general groups of transformations of T. It says that if G is a subgroup of the group of homeomorphisms from T to
itself, then either (i) there exists a G-invariant probability measure on T, or (ii) G contains
a free group on two generators. If the action of G on T is minimal, then the first of these
two alternatives occurs precisely when the action is equicontinuous, in which case G has
a commutative subgroup of index at most 2 and is in particular amenable.
Moving beyond the circle, two natural classes of study are the higher-dimensional
compact manifolds and the zero-dimensional spaces. The main difference now is that
mixing-type phenomena enter the picture, for instance through invariants like entropy
(Chapter 8), and any attempt at a classification becomes hopeless, whether because the
possibilities are too abundant (as for Z-actions on the Cantor set) or too mysterious (as
in the question of which groups admit actions on a given compact manifold). Spaces
of infinite covering dimension also naturally arise in the theory of algebraic actions, as
explored in Chapter 13.
An important device in the qualitative anaylsis of differential equations, and also in the
study of more general kinds of dynamical systems, is the coding of dynamics in symbolic
form. The basic objects of symbolic dynamics are the actions on zero-dimensional spaces
which arise as shift actions G y {1, . . . , k}G and their restrictions to closed invariant
sets. For an introduction to this extensive subject see [93].
The argument in Example 6.6 is taken from Section 1.7 of [49], and the proof of
Proposition 6.26 from [16]. Lemma 6.19 is Proposition A.1 in [13] but stated here for
general G. Proposition 6.21 first appeared in measure-dynamical form as Proposition II.3
of Furstenberg’s seminal article [47] on disjointness, and the same argument applies to
the topological setting. The structure theorem for minimal distal actions, Theorem 6.31,
was established by Furstenberg in [46]. Proposition 6.32 is taken from Section II.2 of
Glasner’s book [55] on proximal actions. The structure theorem for general minimal
actions, Theorem 6.34, was attained in stages through the work of Veech [138, 139],
Glasner, Ellis and Shapiro [40], and McMahon [101]. For additional accounts see [5, 56].