CHAPTER 7
Tameness and independence
One of the main goals of this chapter is to introduce the combinatorial notion of independence (Definitions 7.1 and 7.7) and explain how it connects to the structure theory
in Chapters 2 and 6. This connection is starker in the case of measure-preserving dynamics but also requires more technicalities in the definitions and arguments, and so we will
concentrate our efforts on topological dynamics and only briefly discuss p.m.p. actions in
Section 7.4.
The structure theory for minimal actions is based around the concepts of weak mixing,
equicontinuity, proximality, and distality, and the way in which these relate to combinatorial independence can be understood through the intermediary functional-analytic notion
of tameness. Tameness for a continuous group action G y X involves the norm structure
of the linear span of partial orbits of functions in C(X), specifically in relation to the
Banach space ℓ1 , and is an expression of low dynamical complexity. It can be viewed
as a topological stand-in for measure-dynamical compactness, which is also defined using orbits of functions but in the space L2 , which is not naturally present in topological
dynamics.
In fact if one transposes the property of tameness into the measure-dynamical setting
then it is exactly the same as compactness (Proposition 7.20), while the complete lack of
tameness across orbits of functions characterizes weak mixing (Theorem 7.21). Thus if
one wishes to understand how the measure-dynamical dichotomy between weak mixing
and compactness might be interpreted in topological dynamics, tameness provides a direct
passage from one realm to the other. When G is Abelian and the canonical tower for a
minimal action most resembles the Furstenberg-Zimmer tower of a p.m.p. action (i.e.,
when proximal extensions are absent), we see a tight alignment between topological and
measurable dynamics, and in particular between tameness and the components of the
topological-dynamical structure theory: tameness and equicontinuity are equivalent (see
the paragraph following Theorem 7.19), as are topological weak mixing and the total lack
of tameness (Theorem 7.16). When proximal extensions are permitted in the canonical
tower, tameness is no longer equivalent to equicontinuity, but it still resides at the very
bottom end of the tower: minimal tame actions are always highly proximal extensions of
equicontinuous actions (Theorem 7.19).
177
178
7. TAMENESS AND INDEPENDENCE
The very idea of tameness, along with its role in illuminating the dichotomy between
weak mixing and compactness, is rooted in Rosenthal’s ℓ1 theorem in Banach space theory
(see Remark 7.13). This theorem asserts that a bounded sequence of vectors in a Banach
space is either weakly Cauchy (“compactness”) or contains a subsequence equivalent to
the standard basis of ℓ1 (“weak mixing”). Its proof requires a combinatorial Ramseytheoretic analysis of pairs of subsets of the dual unit ball which uniformly separate vectors
under evaluation, and reveals the close link between combinatorial independence and ℓ1
structure. We will explore this link in the present chapter at the local dynamical level via
the notion of IT-tuple, which, along with tameness, is introduced in Section 7.2. We note
that the connection between combinatorial independence and ℓ1 will also appear later in
the study of entropy in Chapter 10 but in a more quantitative form that is tied to the Elton
theorem in Banach space theory.
Section 7.1 is devoted to a Ramsey-theoretic dichotomy which forms the core of
Rosenthal’s ℓ1 theorem and is similarly fundamental to the study of IT-tuples and their
relation to tameness in Section 7.2 (in fact all of the ingredients necessary to prove Rosenthal’s theorem will appear in the course of the first two sections—see Remark 7.13).
Sections 7.3 and 7.4 show how tameness fits together with the structure theory of Sections 6.2 and 6.3. In Section 7.3 we relate topological weak mixing and independence
via IT-tuples, while in Section 7.4 we present the connections between equicontinuity and
tameness, along with a brief discussion of tameness in the measure-dynamical context.
7.1. Ramsey theory and a dichotomy of Rosenthal
We establish here the k-tuple version of a dichotomy of Rosenthal which asserts that
every sequence of pairs of subsets of a given set has a subsequence that either converges
or is independent in the following sense.
D EFINITION 7.1. Let X be a set. A collection {(Ai,1 , . . . , Ai,k ) : i ∈ I} of k-tuples
T
of subsets of X is said to be independent if i∈F Ai,ω(i) 6= ∅ for every finite set F ⊆ I
and ω ∈ {1, . . . , k}F .
Let C be a collection of finite subsets of N. Let F be a finite subset of N and A
an infinite subset of N. We write [F, A] for the collection of all infinite subsets of F ∪
(A\{1, . . . , max(F )}) which include F . An initial segment of A is a set of the form
{1, . . . , n} ∩ A for some n ∈ N. We say that A accepts F if every set in [F, A] has an
initial segment in C, and that A rejects F if no infinite subset of A accepts F . From these
definitions we observe the following basic facts:
(i) If A accepts (rejects) F then every infinite subset of A accepts (rejects) F .
(ii) Either we can find a B ∈ [F, A] which accepts F , or A rejects F .
7.1. RAMSEY THEORY AND A DICHOTOMY OF ROSENTHAL
179
(iii) If A rejects F , then A accepts only finitely many sets of the form F ∪{n} where
n ∈ A\{1, . . . , max(F )}.
To see (iii), note that if the set B of all n ∈ A\{1, . . . , max(F )} such that A accepts
F ∪ {n} were infinite, then given a C ∈ [F, B] we would have C ∈ [F ∪ {min(C\F )}, A]
so that C has an initial segment in C, which shows that B accepts F , contradicting the
rejection of F by A.
To establish the dichotomy of Lemma 7.3, we will argue that there exists an infinite
subset of N which either accepts ∅ or rejects each of its finite subsets. This will be done
in two steps, each of which involves a recursive construction. The first step we record as
a separate lemma:
L EMMA 7.2. Let C be a collection of finite subsets of N. Then there exists an infinite
set B ⊆ N each of whose finite subsets is either accepted or rejected by B.
P ROOF. We recursively construct infinite sets B1 ⊇ B2 ⊇ . . . and n1 < n2 < . . .
with nk ∈ Bk for each k as follows. To begin, by (ii) there is an infinite set B1 ⊆ N which
either accepts or rejects ∅. Choose n1 ∈ B1 . Assume now that we have defined B1 , . . . , Bk
and n1 , . . . , nk . Enumerating the subsets of {n1 , . . . , nk }, applying (ii) in succession 2k
times, and making use of (i), we can find an infinite set Bk+1 ⊆ Bk \{1, . . . , nk } such that
each finite subset of {n1 , . . . , nk } is either accepted or rejected by Bk+1 . To complete the
recursion choose nk+1 to be any element of Bk+1 . Now set B = {n1 , n2 , . . . }. Then by
(i) B either accepts or rejects ∅, and if F is a nonempty finite subset of B then Bk either
accepts or rejects F whenever nk > max(F ), so that by (i) again B either accepts or
rejects F .
For a set B we write Fin(B) for the collection of its finite subsets.
L EMMA 7.3. Let C be a collection of finite subsets of N. Then there is an infinite
set B ⊆ N such that either Fin(B) ∩ C = ∅ or every infinite subset of B has an initial
segment in C.
P ROOF. By Lemma 7.2 there is an infinite set B ⊆ N each of whose finite subsets is
either accepted or rejected by B. If B accepts ∅, then every infinite subset of B has an
initial segment in C and we are done. Suppose then that B does not accept ∅, in which
case it must reject ∅. We recursively construct a sequence n1 < n2 < n3 < . . . in B by
first using (iii) to obtain an n1 ∈ B so that B rejects {n1 } and then again using (iii) at
the kth stage to find an integer nk+1 > nk in B such that B rejects F ∪ {nk+1 } for every
F ⊆ {n1 , . . . , nk }. Set B ′ = {n1 , n2 , . . . }. Then B ′ rejects each of its finite subsets.
Thus Fin(B ′ ) ∩ C is empty, for any element of C included in B ′ would be accepted by
B′.
180
7. TAMENESS AND INDEPENDENCE
T HEOREM 7.4. Let {(An,1 , . . . , An,k )}∞
n=1 be a sequence of k-tuples of subsets of a
set X. Then there is an infinite set B ⊆ N such that either {(An,1 , . . . , An,k ) : n ∈ B}
is independent or for every x ∈ X there exists an i ∈ {1, . . . , k} for which the set
{n ∈ B : x ∈ An,i } is finite.
P ROOF. For each nonempty finite set F ⊆ N we define σF ∈ {1, . . . , k}F so that
σF (ni ) ≡ i mod k where n1 < n2 < · · · < nm are the elements of F . Define C to be the
T
collection of all nonempty finite sets F ⊆ N such that n∈F An,σF (n) = ∅.
By Lemma 7.3 we can find an infinite set B ⊆ N such that either (i) Fin(B) ∩ C = ∅
or (ii) every infinite subset of B has an initial segment in C. We will show that these two
cases produce the dichotomy in the theorem statement. Suppose first that (i) holds. Let
n1 < n2 < . . . be an enumeration of the elements of B and write B0 for the set of all ni
such that i ≡ 0 mod k. Suppose that we are given a nonempty finite set F ⊆ B0 and a
σ ∈ {1, . . . , k}F . As B0 consists of every kth element of B starting at nk , it is possible
to construct a finite set F ′ ⊆ B such that F ′ ⊇ F and σF ′ restricts to σ on F . Then
T
T
n∈F An,σ(n) ⊇
n∈F ′ An,σF ′ (n) 6= ∅, showing that the collection {(An,1 , . . . , An,k ) : n ∈
B0 } is independent.
Suppose now that (ii) holds, and suppose that there exists an x ∈ X for which the
set Di := {n ∈ B : x ∈ An,i } is infinite for each i = 1, . . . , k. Then we can construct
a sequence m1 < m2 < . . . such that for each j ∈ N we have mj ∈ Di where i ≡
j mod k. By assumption this sequence has an initial segment F in C. But then we have
T
x ∈ n∈F An,σF (n) = ∅, a contradiction.
7.2. Tameness and IT-tuples
Throughout G y X is a continuous action on a compact Hausdorff space.
D EFINITION 7.5. Let f ∈ C(X). We say that a set M ⊆ G is an ℓ1 -isomorphism
set for f if the set {sf }s∈M is equivalent to the standard basis of ℓ1 (M ). This means that
there are λ1 , λ2 > 0 such that for every finite set F ⊆ M and scalars cs for s ∈ F one has
X
X
X
≤ λ2
|cs | ≤ λ1
c
(sf
)
|cs |,
s
s∈F
s∈F
s∈F
or equivalently that the map that sends the standard basis vector es ∈ ℓ1 (M ) to sf extends
to a Banach space isomorphism from ℓ1 (M ) to the closed linear span of {sf }s∈M in
C(X).
D EFINITION 7.6. A function f ∈ C(X) is said to be tame if it does not have an
infinite ℓ1 -isomorphism set. The action G y X is said to be tame if every function in
C(X) is tame.
7.2. TAMENESS AND IT-TUPLES
181
D EFINITION 7.7. Let G y X be an action and A = (A1 , . . . , Ak ) a tuple of subsets of X. We say that a set M ⊆ G is an independence set for A if the collection
T
{(s−1 A1 , . . . , s−1 Ak ) : s ∈ M } is independent (Definition 7.1), i.e., s∈F s−1 Aω(s) 6= ∅
for every finite set F ⊆ M and ω ∈ {1, . . . , k}F .
D EFINITION 7.8. A tuple x = (x1 , . . . , xk ) ∈ X k is an IT-tuple (or IT-pair in the case
k = 2) if for every product neighbourhood U1 × · · · × Uk of x the tuple (U1 , . . . , Uk ) has
an infinite independence set. We denote the set of IT-tuples of length k by ITk (X, G).
Our next goal is to collect some properties of IT-tuples in Proposition 7.14. To this
end we first establish a few lemmas.
L EMMA 7.9. A tuple A = (A1 , . . . , Ak ) of closed subsets of X has an infinite independence set if and only if there is an infinite set M ⊆ G such that for every infinite set
M ′ ⊆ M there exists an x ∈ X for which the sets {s ∈ M ′ : sx ∈ Aj } for j = 1, . . . , k
are all infinite.
P ROOF. For the forward direction take M to be an infinite independence set for A
and use the compactness of A1 × · · · × Ak in the form of the finite intersection property.
The reverse direction is immediate from Theorem 7.4.
L EMMA 7.10. Let A = (A1 , . . . , Ak ) be a tuple of closed subsets of X with an infinite
independence set. Let A1,1 and A1,2 be closed subsets of X such that A1 = A1,1 ∪ A1,2 .
Then one of the tuples (A1,1 , . . . , Ak ) and (A1,2 , . . . , Ak ) has an infinite independence set.
P ROOF. Suppose to the contrary that neither (A1,1 , . . . , Ak ) nor (A1,2 , . . . , Ak ) has an
infinite independence set. Take an M ⊆ G as given by Lemma 7.9 for A. Applying
Lemma 7.9 twice in succession we can find infinite subsets M1 ⊇ M2 of M such that
for every x ∈ X and i = 1, 2 at least one of the sets {s ∈ Mi : sx ∈ A1,i } and
{s ∈ Mi : sx ∈ Aj } for j = 2, . . . , k is finite. By our choice of M there exists an x ∈ X
such that for each j = 1, . . . , k the set {s ∈ M2 : sx ∈ Aj } is infinite, which means that
{s ∈ M1 : sx ∈ A1,1 } and {s ∈ M2 : sx ∈ A1,2 } are both finite. But the union of these
latter two sets contains the infinite set {s ∈ M2 : sx ∈ A1 }, a contradiction.
For λ > 0, a collection {vi }i∈I of vectors in a Banach space is said to be λ-equivalent
to the standard basis of ℓ1 if there are λ1 , λ2 > 0 with λ2 /λ1 ≤ λ such that for all finite
sets F ⊆ I and scalars ci for i ∈ F one has
X
X
X
λ1
|ci | ≤ ci vi ≤
λ
|ci |.
2
i∈F
i∈F
i∈F
L EMMA 7.11. Let D0 , D1 be closed disks in the complex plane with respective centres
z0 , z1 and common radius r > 0 such that r ≤ |z0 − z1 |/8. Let V be a set, and let {fi }∞
i=1
182
7. TAMENESS AND INDEPENDENCE
be a bounded sequence of functions in ℓ∞ (V ) such that the collection {(Ai,0 , Ai,1 )}∞
i=1 is
∞
independent, where Ai,j = {v ∈ V : fi (v) ∈ Dj }. Then {fi }i=1 is (C/r)-equivalent to
the standard basis of ℓ1 , where C = supi∈N kfi k.
P ROOF. We may assume by multiplying the functions fi by |z1 − z0 |/(z1 − z0 ) that
z1 − z0 is real and positive. It suffices to show that for any complex scalars c1 , . . . , cn we
P
P
P
have k ni=1 ci fi k ≥ r ni=1 |ci |. Writing ci = ai + ibi we may assume that ni=1 |ai | ≥
Pn
i=1 |bi |.
Consider σ ∈ {0, 1}n such that σ(i) is 0 or 1 depending on whether ai < 0 or ai ≥ 0.
T
T
By independence there exist v ∈ ni=1 Ai,σ(i) and w ∈ ni=1 Ai,1−σ(i) . Since re(fi (v) −
fi (w)) ≥ dist(D0 , D1 ) ≥ 6r when σ(i) = 1 and re(fi (w) − fi (v)) ≥ dist(D0 , D1 ) ≥ 6r
when σ(i) = 0, we have
X
n
n
X
re
ai (fi (v) − fi (w))/2 ≥ 3r
|ai |.
i=1
i=1
Note also that
X
X
n
n
n
n
X
X
im
bi (fi (v) − fi (w))/2 ≤
|bi ||im((fi (v) − fi (w))/2| ≤ r
|bi | ≤ r
|ai |.
i=1
i=1
i=1
Therefore
n
X
n
X
ci fi ≥ re
ci (fi (v) − fi (w))/2
i=1
i=1
i=1
= re
≥ 2r
X
n
i=1
n
X
ai (fi (v) − fi (w))/2 − im
|ai | ≥ r
i=1
n
X
|ci |.
X
n
i=1
bi (fi (v) − fi (w))/2
i=1
L EMMA 7.12. Let V be a set and let ε > 0. Let {fn }∞
n=1 be a bounded sequence of
∞
functions in ℓ (V ) with no pointwise convergent subsequence. Then there exist closed
disks D0 , D1 ⊆ C with respective centres z0 , z1 and common radius r > 0 such that
r ≤ ε|z0 − z1 | and an infinite set J ⊆ N such that the collection {(fn−1 (D0 ), fn−1 (D1 )) :
n ∈ J} is independent.
P ROOF. Let {(Dk,0 , Dk,1 )}∞
k=1 be an enumeration of the pairs of closed disks in C
which have a common rational radius r > 0 and complex rational centres z0 , z1 satisfying
r ≤ ε|z0 − z1 |. Suppose that the conclusion of the lemma does not hold. Then, by
Theorem 7.4, for every k ∈ N and infinite set J ⊆ N there exists an infinite set J ′ ⊆ J
such that for each v ∈ V at least one of the sets {n ∈ J ′ : fn (v) ∈ Dk,0 } and {n ∈ J ′ :
fn (v) ∈ Dk,1 } is finite. We can thus recursively construct a sequence J1 ⊇ J2 ⊇ . . .
7.2. TAMENESS AND IT-TUPLES
183
of infinite subsets of N such that for all k ∈ N and v ∈ V at least one of the sets {n ∈
Jk : fn (v) ∈ Dk,0 } and {n ∈ Jk : fn (v) ∈ Dk,1 } is finite. Choose n1 < n2 < . . . with
nk ∈ Jk for each k. Then for all k ∈ N and v ∈ V there is at least one i ∈ {0, 1} such
that the set of all n ∈ {n1 , n2 , . . . } with fn (v) ∈ Dk,i is finite, which means that {fnk }∞
k=1
converges pointwise, contradicting our assumption.
R EMARK 7.13. Lemmas 7.12 and 7.11 combine to yield one form of Rosenthal’s ℓ1
theorem: for a set V , every bounded sequence in ℓ∞ (V ) has a subsequence that either
converges pointwise or is equivalent to the standard basis of ℓ1 .
P ROPOSITION 7.14.
(i) Let (A1 , . . . , Ak ) be a tuple of closed subsets of X which has an infinite independence set. Then there exists an IT-tuple (x1 , . . . , xk ) such that xj ∈ Aj for
all j = 1, . . . , k.
(ii) A function f ∈ C(X) is untame if and only if there is an IT-pair (x, y) with
f (x) 6= f (y).
(iii) The action G y X is tame if and only if there are no nondiagonal pairs in
IT2 (X, T ).
(iv) ITk (X, G) is a closed G-invariant subset of X k .
(v) If Z is a closed G-invariant subset of X then ITk (Z, G) ⊆ ITk (X, G).
(vi) If π : X → Y is a G-factor map then (π × · · · × π)(ITk (X, G)) = ITk (Y, G).
P ROOF. (i). Suppose that this assertion is not true. Then, since X k is compact and
Hausdorff, every tuple (x1 , . . . , xk ) such that xj ∈ Aj for all j = 1, . . . , k has a closed
product neighbourhood in X k without an infinite independence set. It follows by the
compactness of A1 × · · · × Ak that we can then find, for each j = 1, . . . , k, a finite
S
collection Ωj of closed subsets of Aj with Ωj = Aj so that whenever Bj is a member
of Ωj for j = 1 . . . , k the tuple (B1 , . . . , Bk ) does not have an infinite independence set.
But then a recursive application of Lemma 7.10 shows that at least one of these tuples
must have an infinite independence set, a contradiction.
(ii). For the forward direction, it suffices by part (i) to show the existence of a pair
(A, B) of disjoint closed subsets of X which has an infinite independence set and satisfies
f (A) ∩ f (B) = ∅. Let M = {sj : j ∈ N} ⊆ G be an infinite ℓ1 -isomorphism set for f .
Then for every ω ∈ {0, 1}M the assignment sj f 7→ w(j) extends to a bounded linear functional on the linear span of {sj f }∞
j=1 , and hence to a bounded linear functional on C(X)
by the Hahn-Banach theorem. These functionals show that the sequence {sj f }∞
j=1 has no
weakly convergent subsequence. Since bounded linear functionals on C(X) correspond
to Radon measures on X, it follows by the dominated convergence theorem that {sj f }j∈N
has no pointwise convergent subsequence. Consequently by Lemma 7.12, for which we
view C(X) as a subspace of ℓ∞ (X), we can find disjoint closed disks D1 , D2 ⊆ C such
184
7. TAMENESS AND INDEPENDENCE
that the pair (f −1 (D1 ), f −1 (D2 )) has an infinite independence set, thus fulfilling our requirements.
For the reverse direction, suppose that (x, y) is an IT-pair with f (x) 6= f (y). Take
closed disks D1 , D2 ⊆ C having centres f (x) and f (y), respectively, and satisfying the
condition in Lemma 7.11. Since (x, y) is an IT-pair, we can find an infinite independence
set M ⊆ G for the pair (f −1 (D1 ), f −1 (D2 )). By Lemma 7.11, the set M −1 is an ℓ1 isomorphism set for f .
(iii). This follows from (ii).
(iv) and (v). Clear from the definitions.
(vi). Write πk for the k-fold product π × · · · × π. For a tuple (A1 , . . . , Ak ) of subsets
of Y , an independence set for (π −1 (A1 ), . . . , π −1 (Ak )) is also an independence set for
(A1 , . . . , Ak ), and so πk (ITk (X, G)) ⊆ ITk (Y, G). Suppose now that y is a tuple in
ITk (Y, G). Let Ω be the collection of closed product neighbourhoods of y in Y k . Then
for every A1 × · · · × Ak ∈ Ω the tuple (A1 , . . . , Ak ) has an infinite independence set,
which is then also an independence set for (π −1 (A1 ), . . . , π −1 (Ak )), so that π −1 (A1 ) ×
T
· · ·×π −1 (Ak ) contains an IT-tuple by part (i). Since (A1 ,...,Ak )∈Ω π −1 (A1 )×· · ·×π −1 (Ak )
is equal to the closed set πk−1 (y), and ITk (X, G) is closed by part (iv), we deduce that
πk−1 (y) contains an IT-tuple. Therefore πk (ITk (X, G)) = ITk (Y, G).
7.3. Weak mixing and independence
Recall from Definition 6.15 that the action G y X is weakly mixing if for all nonempty
open sets U1 , U2 , V1 , V2 ⊆ X there is an s ∈ G such that sU1 ∩ V1 6= ∅ and sU2 ∩ V2 6= ∅,
The following lemma says that when G is Abelian we can cut down the number of sets in
this condition to two, which enables us to relate weak mixing to IT-pairs in Theorem 7.16.
L EMMA 7.15. Suppose that G is Abelian. Then the action G y X is weakly mixing if
and only if for all nonempty open sets U, V ⊆ X there is an s ∈ G such that sU ∩ U 6= ∅
and sU ∩ V 6= ∅.
P ROOF. The forward direction is trivial, and so let us assume the stated condition and
show that it implies weak mixing. Thus let U1 , U2 , V1 , V2 ⊆ X be nonempty open sets and
let us find an s ∈ G such that sU1 ∩ V1 6= ∅ and sU2 ∩ V2 6= ∅. By hypothesis we can find
an s1 ∈ G such that A := s1 U1 ∩ U2 6= ∅, then an s2 ∈ G such that B := s2 A ∩ s1 V1 6= ∅,
and then an s3 ∈ G such that s3 B ∩ B 6= ∅ and s3 B ∩ V2 6= ∅. Setting s = s2 s3 we
observe that
s1 (sU1 ∩ V1 ) = ss1 U1 ∩ s1 V1 ⊇ s(s1 U1 ∩ U2 ) ∩ s1 V1
= s3 s2 A ∩ s1 V1 ⊇ s3 B ∩ s1 V1 ⊇ s3 B ∩ B 6= ∅
7.3. WEAK MIXING AND INDEPENDENCE
185
and
s1 sU1 ∩ sU2 ∩ V2 = s(s1 U1 ∩ U2 ) ∩ V2 = s3 s2 A ∩ V2 ⊇ s3 B ∩ V2 6= ∅
so that sU1 ∩ V1 6= ∅ and sU2 ∩ V2 6= ∅.
Recall from Definition 6.18 that the action G y X is said to be weakly mixing of all
orders if the product action G y X k is transitive for every k ∈ N.
T HEOREM 7.16. Consider the following conditions:
(i)
(ii)
(iii)
(iv)
the action G y X is weakly mixing,
IT2 (G, X) = X 2 ,
the action G y X is weakly mixing of all orders,
ITk (G, X) = X k for every k ∈ N.
Then (iv)⇔(iii)⇒(ii), and when G is Abelian all of the conditions are equivalent.
P ROOF. (iii)⇒(iv). We may assume that k ≥ 2, for the case k = 2 implies the
case k = 1 by considering pairs of the form (U, U ) where U is an open subset of X.
Let U = (U1 , . . . , Uk ) be a tuple of nonempty open subsets of X. We will show by a
recursive procedure that it has an infinite independence set. Observe first that if x is an
isolated point in x ∈ X then the orbit of (x, x) for the product action G y X × X is both
open and G-invariant, which means that X = {x} since this product action is transitive
by (iii). We may thus assume that X does not contain any isolated points, and so we may
assume that U1 and U2 are disjoint by shrinking them if necessary.
Obviously every singleton in G is an independence set for U . We can thus recursively
construct an infinite independence set for U once we know, given any nonempty finite
independence set F ⊆ G for U , that we can find an independence set F ′ for U that
enlargens F by one element. Write Λ for {1, . . . , k}F ×{1, . . . , k}, and define the product
subsets
Y \
Y
V0 =
s−1 Uω(s) , V1 =
Uj
(ω,j)∈Λ s∈F
(ω,j)∈Λ
of X Λ . By (iii) there exists a t ∈ G such that V0 ∩t−1 V1 6= ∅, so that if we set F ′ = F ∪{t}
T
′
then we have s∈F ′ s−1 Uω(s) 6= ∅ for all ω ∈ {1, . . . , k}F . Since U1 and U2 are disjoint,
we must have t ∈
/ F , and so F ′ is indeed a larger independence set.
(iv)⇒(iii). Clear from the definitions.
(iv)⇒(ii). Trivial.
Finally, if G is Abelian then the equivalence (i)⇔(iii) is Proposition 6.21, and (ii)⇒(i)
follows from Lemma 7.15 in view of the definition of IT-pair.
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In the above theorem, conditions (iii) and (iv) are not equivalent in general to (i),
as Example 6.22 illustrates. That example also show that the following fails if G is not
Abelian.
P ROPOSITION 7.17. Suppose that G is Abelian. Let G y X be an action which is
both tame and weakly mixing. Then X is a singleton.
P ROOF. Combine Proposition 7.14(ii) and Theorem 7.16.
7.4. When tameness and equicontinuity are equivalent
In the last section we saw how weak mixing is related to the lack of tameness via the
local notion of IT-tuple. Here we describe how tameness fits into the general structure
theory discussed in Section 6.3 of Chapter 6. Most of the results we will merely state, as
their proofs rely heavily on structure theorems and lie beyond the scope of the book. We
assume throughout that our compact space X is metrizable.
For minimal tame actions G y X, it turns out that distality is equivalent to equicontinuity:
T HEOREM 7.18. Every minimal distal tame action G y X is equicontinuous.
When G is Abelian we can say more: for minimal actions the properties of tameness
and equicontinuity coincide up to proximal extensions. That is, in the canonical tower of
extensions (see Theorem 6.34) for a tame minimal G-action, there is no weakly mixing
extension at the top and at most one isometric extension. In fact the following theorem
tells us that even more is true. A G-extension π : X → Y is said to be highly proximal
if for every nonempty open set U ⊆ X and every y ∈ Y there exists an s ∈ G such that
π −1 (z) ⊆ sU .
T HEOREM 7.19. Suppose that G is Abelian. Then every tame minimal action G y X
is a highly proximal extension of an equicontinuous action and is uniquely ergodic.
In particular, if the proximal extensions are all trivial in the canonical tower, i.e., the
tower contains only incontractible extensions and reduces to the form
(24)
X → Yλ → · · · → Y2 → Y1 → Y0 = {pt}
where the leftmost extension is weak mixing the others are isometric, the properties of
tameness and nullness (see Section ??) are both equivalent to equicontinuity. Recall that
equicontinuity for nontrivial actions of this type means that the tower (24) collapses to a
single isometric extension.
This exact same picture holds for p.m.p. actions G y (X, µ), for which the concept
of proximality doesn’t exist. One can moreover dispense with the hypothesis that G be
7.4. WHEN TAMENESS AND EQUICONTINUITY ARE EQUIVALENT
187
Abelian. In accord with minimal actions lacking proximal extensions in their canonical
tower, the Furstenberg-Zimmer tower for a p.m.p. action G y (X, µ) has the form
(25)
X → Yλ → · · · → Y2 → Y1 → Y0 = {pt}
where the leftmost extension is weak mixing and the other extensions are compact (see
Section 2.2 of Chapter 2). Tameness for p.m.p. actions, which we define next, turns out
to be equivalent to compactness (Theorem 7.21), meaning that the tower (25) collapses to
a single compact extension if X is not already trivial.
In analogy with Definition 7.6 but allowing for L2 -perturbations in order to ensure the
appropriate measure-theoretic robustness, we say that a function f ∈ L∞ (X) is tame if
it can be approximated arbitrarily well in L2 -norm by functions in L∞ (X) which lack an
infinite ℓ1 -isomorphism set. The action G y (X, µ) is said to be tame if every function
in L∞ (X) is tame. The following proposition and theorem show that tameness is directly
tied to compactness and weak mixing.
P ROPOSITION 7.20. A function in L∞ (X) is tame if and only if it is compact in L2 (X)
under the Koopman representation.
In order to guarantee the above equivalence it is necessary to use perturbations in the
definition of tameness for a function f ∈ L∞ (X) and not merely ask that f itself have an
infinite ℓ1 -isomorphism set. Indeed every free ergodic Z-action has a weakly mixing (and
even mixing) topological model G y X [92], in which case every nonconstant function
f ∈ C(X) has an infinite ℓ1 -isomorphism set by Theorem 7.16 and Proposition 7.14.
T HEOREM 7.21. The action G y (X, µ) is weakly mixing if and only if every nonscalar element of L∞ (X) has an infinite ℓ1 -isomorphism set, and is compact if and only
if it is tame.
One can also define combinatorial independence in a way that is compatible with the
measure-theoretic context by requiring the independence to be observable modulo sets of
small measure. However, if one wants to develop a local analysis of tameness as in the
topological setting then one should not ask for infinite independence sets as in the definition of IT-tuple (Definition 7.8), but rather for independence sets of a fixed nonzero density in any given finite subset of G. As a consequence the analysis acquires a quantitative
character that shifts it away from Ramsey techniques and more towards the study of independence in entropy theory (see [85] for details). This in fact has a topological-dynamical
counterpart in the concepts of nullness and IN-tuples, which involve independence along
arbitrarily large finite subsets of G and which, although closely related to tameness and ITtuples, do not coincide with these in general. In p.m.p. dymamics one does not have this
distinction between null and tame behaviour. What is curious however is that the relationship of tameness to weak mixing and compactness in p.m.p. dynamics gets expressed at
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the function level in a way that is reminiscient of topological-dynamical tameness (Theorem 7.21) but at the spatial level in a way that is more suggestive of topological-dynamical
nullness (see Theorems 5.5 and 5.7 of [86]). We will examine nullness and IN-tuples in
Section ?? of Chapter 10.
7.5. Notes and references
Rosenthal’s ℓ1 -theorem, from which all of the material in this chapter springs, is from
[120]. The complex version is due to Dor [30]. Theorem 7.4 is the k-tuple version of
Theorem 2 in [120]. Following a standard procedure as in [135], we have deduced this
from the Ramsey-theoretic Lemma 7.3, which is a general version of a result of NashWilliams [108] that appears in the article [54] of Galvin and Prikry, whose proof we have
followed. See [62] for more on Ramsey methods in Banach space theory.
The notion of tameness (Definition 7.6), which established the link between Rosenthal’s ℓ1 -theorem and dynamics, was introduced by Köhler in [88]. IT-tuples were defined
in [85], from where Theorem 7.16 is extracted. Theorem 7.18 is due to Glasner [58], who
first used tameness in conjunction with the Ellis semigroup to show that the induced action
on the space of Borel probability measures on X with the weak∗ topology is again distal,
and then appealed to his earlier result that distal affine actions are automatically equicontinuous, which relies on some structure theory. Theorem 7.19 was established both by
Huang in [73] and by Kerr and Li in [85], and another proof was given by Glasner in [59].
Proposition 7.20 and Theorem 7.21 are from [86].