1 Introduction
1.1 Definitions and results
Let
$(X,\mathcal {B}, \mu )$
be a standard Lebesgue space and let T be a free, ergodic,
$\mu $
-preserving action of a discrete amenable group G on X. It is natural to ask what properties of T are preserved by a generic extension
$( \overline {X}, \overline {\mu }, \overline {T} )$
(a precise definition of ‘generic extension’ is discussed in §3). For example, it was shown in [Reference Glasner, Thouvenot and WeissGTW21] that a generic
$\overline {T}$
has the same entropy as T and that if T is a non-trivial Bernoulli shift, then a generic
$\overline {T}$
is also Bernoulli. A system
$(X, \mu , T)$
is said to be dominant if it is isomorphic to a generic extension
$(\overline {X}, \overline {\mu }, \overline {T})$
. Thus, for example, the aforementioned results from [Reference Glasner, Thouvenot and WeissGTW21] together with Ornstein’s famous isomorphism theorem [Reference OrnsteinOrn70] imply that all non-trivial Bernoulli shifts are dominant. More generally, it has been shown in [Reference Austin, Glasner, Thouvenot and WeissAGTW21] that:
-
(1) if
$G = \mathbb {Z}$
, then
$(X,\mu ,T)$
is dominant if and only if it has positive Kolmogorov–Sinai entropy; and -
(2) for any G, if
$(X, \mu , T)$
has positive entropy, then it is dominant.
In this paper, we complete the picture by proving the following result.
Theorem 1.1. Let G be any discrete amenable group and let
$(X, \mu , T)$
be any free ergodic action with zero entropy. Then
$(X, \mu , T)$
is not dominant.
The proof of result (2) is based on the theory of ‘slow entropy’ developed by Katok and Thouvenot in [Reference Katok and ThouvenotKT97] (see also [Reference FerencziFer97]), and our proof of Theorem 1.1 uses the same ideas.
2 Slow entropy
Fix a Følner sequence
$(F_n)$
for G. For
$g \in G$
, write
$T^g x$
for the action of g on the point
$x \in X$
, and for a subset
$F \subseteq G$
, write
$T^F x = \{T^f x : f \in F \}$
. If
$Q = \{Q_1, \ldots , Q_k\}$
is a partition of X, then for
$x \in X$
denote by
$Q(x)$
the index of the cell of Q containing x. Sometimes we use the same notation to mean the cell itself; which meaning is intended will be clear from the context. Given a finite subset
$F \subseteq G$
, the
${(Q,F)}$
-name of x for the action T is the tuple
$Q_{T,F}(x) := (Q(T^f x))_{f \in F} \in \{1,2,\ldots ,k\}^F$
. Similarly, we also define the partition
$Q_{T,F} := \bigvee _{f \in F} T^{f^{-1}} Q$
, and in some contexts we use the same notation
$Q_{T,F}(x)$
to refer to the cell of
$Q_{T,F}$
containing x.
For a finite subset
$F \subseteq G$
and any finite alphabet
$\Lambda $
, the symbolic space
$\Lambda ^F$
is equipped with the normalized Hamming distance
$d_F(w,w') = ({1}/{|F|}) \sum _{f \in F} 1_{w(f) \neq w'(f)}$
.
Definition 2.1. Given a partition
$Q = \{Q_1, \ldots , Q_k\}$
, a finite set
$F \subseteq G$
, and
$\epsilon> 0$
, define
$$ \begin{align*} B_{\operatorname{Ham}}(Q,T,F,x,\epsilon) := \{y \in X : d_F(Q_{T,F}(y), Q_{T,F}(x)) < \epsilon \}. \end{align*} $$
We refer to this set as the ‘
$(Q,T,F)$
-Hamming ball of radius
$\epsilon $
centered at x’. Formally, it is the preimage under the map
$Q_{T,F}$
of the ball of radius
$\epsilon $
centered at
$Q_{T,F}(x)$
in the metric space
$([k]^F, d_F)$
.
Definition 2.2. Given
$\epsilon> 0$
, the Hamming
$\epsilon $
-covering number of
$\mu $
is defined to be the minimum number of
$(Q,T,F)$
-Hamming balls of radius
$\epsilon $
required to cover a subset of X of
$\mu $
-measure at least
$1-\epsilon $
, and is denoted by
$\operatorname {\mathrm {cov}}(Q,T,F,\mu ,\epsilon )$
.
Lemma 2.3. Let
$\varphi : (X,T,\mu ) \to (Y, S, \nu )$
be an isomorphism. Also let Q be a finite partition of X, let F be a finite subset of G, and let
$\epsilon> 0$
. Then
$\operatorname {\mathrm {cov}}(Q,T,F,\mu ,\epsilon ) = \operatorname {\mathrm {cov}}(\varphi Q, S, F, \nu , \epsilon )$
.
Proof. It is immediate from the definition of isomorphism that for
$\mu $
-almost every
${x,x' \in X}$
,
$$ \begin{align*} d_F( Q_{T,F}(x), Q_{T,F}(x') ) = d_F( (\varphi Q)_{S,F}(\varphi x), (\varphi Q)_{S,F}(\varphi x') ). \end{align*} $$
Therefore, it follows that
$\varphi ( B_{\operatorname {Ham}}(Q,T,F,x,\epsilon ) ) = B_{\operatorname {Ham}}(\varphi Q, S, F, \varphi x, \epsilon )$
for
$\mu $
-almost every x. Thus, any collection of
$(Q,T,F)$
-Hamming balls in X covering a set of
$\mu $
-measure
$1-\epsilon $
is directly mapped by
$\varphi $
to a collection of
$(\varphi Q, S, F)$
-Hamming balls in Y covering a set of
$\nu $
-measure
$1-\epsilon $
. Therefore,
$\operatorname {\mathrm {cov}}(Q,T,F,\mu ,\epsilon ) \geq \operatorname {\mathrm {cov}}(\varphi Q, S, F, \nu , \epsilon )$
. The reverse inequality holds by doing the same argument with
$\varphi ^{-1}$
in place of
$\varphi $
.
The goal of the rest of this section is to show that for a given action
$(X,T,\mu )$
, the sequence of covering numbers
$\operatorname {\mathrm {cov}}(Q, T, F_n, \mu , \epsilon )$
grows at a rate that is bounded uniformly for any choice of partition Q. A key ingredient is an analog of the classical Shannon–McMillan theorem for actions of amenable groups [Reference OllagnierMO85, Theorem 4.4.2].
Theorem 2.4. Let G be a countable amenable group and let
$(F_n)$
be any Følner sequence for G. Let
$(X,T,\mu )$
be an ergodic action of G and let Q be any finite partition of X. Then
$$ \begin{align*} \frac{-1}{|F_n|} \log \mu ( Q_{T, F_n}(x) ) \xrightarrow{L^1(\mu)} h(\mu, T, Q) \quad \text{as } n \to \infty, \end{align*} $$
where h denotes the entropy. In particular, for any fixed
$\gamma> 0$
,
$$ \begin{align*} \mu \{ x : \exp((-h-\gamma) |F_n|) < \mu(Q_{T,F_n}(x)) < \exp((-h+\gamma)|F_n|) \} \to 1 \quad \text{as } n \to \infty. \end{align*} $$
Lemma 2.5. For any partition P, any Følner sequence
$(F_n)$
and any
$\epsilon> 0$
, let
$\ell (T, P, \mu , n, \epsilon )$
be the minimum number of
$P_{T, F_n}$
-cells required to cover a subset of X of measure more than
$ 1-\epsilon $
. Then
$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{|F_n|} \log \ell(T, P, \mu, n, \epsilon) \leq h(\mu, T, P). \end{align*} $$
Proof. Let
$h = h(\mu , T, P)$
. Let
$\gamma> 0$
. By Theorem 2.4, for n sufficiently large depending on
$\gamma $
, we have
$$ \begin{align*} \mu \{ x \in X : \mu(P_{T, F_n}(x)) \geq \exp((-h-\gamma)|F_n|) \}> 1-\epsilon. \end{align*} $$
Let
$X'$
denote the set in the above equation. Let
$\mathcal {G}$
be the family of cells of the partition
$P_{T, F_n}$
that meet
$X'$
. Then clearly
$\mu ( \bigcup \mathcal {G} )> 1-\epsilon $
and
$|\mathcal {G}| < \exp ((h+\gamma ) |F_n|)$
. Therefore,
$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{|F_n|} \log \ell(T, P, \mu, n, \epsilon) \leq h + \gamma, \end{align*} $$
and this holds for arbitrary
$\gamma $
, so we are done.
At this point, fix for all time
$\epsilon = 1/100$
. We can also now omit
$\epsilon $
from all of the notation defined previously, because it will never change. In addition, assume from now on that the system
$(X, T, \mu )$
has zero entropy.
Lemma 2.6. If
$(F_n)$
is a Følner sequence for G and A is any finite subset of G, then
$(AF_n)$
is also a Følner sequence for G.
Proof. First, because A is finite and
$(F_n)$
is Følner we have
$$ \begin{align*} \lim_{n \to \infty} \frac{|AF_n|}{|F_n|} = 1. \end{align*} $$
Now fix any
$g \in G$
and observe that
$$ \begin{align*} \frac{|gAF_n \,\triangle\, AF_n|}{|AF_n|} \leq \frac{|gAF_n \,\triangle\, F_n| + |F_n \,\triangle\, AF_n|}{|F_n|} \cdot \frac{|F_n|}{|AF_n|} \to 0 \quad \text{as } n \to \infty, \end{align*} $$
which shows that
$(AF_n)$
is a Følner sequence.
Lemma 2.7. Let
$b(m,n) \geq 0$
be real numbers satisfying:
-
•
$\lim _{n \to \infty } b(m,n) = 0$
for each fixed m; and -
•
$b(m+1, n) \geq b(m,n)$
for all
$m,n$
.
Then there exists a sequence
$(a_n)$
such that
$a_n \to 0$
and for each fixed m,
$b(m,n) \leq a_n$
for n sufficiently large (depending on m).
Proof. For each m, let
$N_m$
be such that
$b(m,n) < 1/m$
for all
$n> N_m$
. Without loss of generality, we may assume that
$N_m < N_{m+1}$
. Then we define the sequence
$(a_n)$
by
${a_n = b(1,n)}$
for
$n \leq N_2$
and
$a_n = b(m,n)$
for
$N_m < n \leq N_{m+1}$
. We have
$a_n \to 0$
because
${a_n < 1/m}$
for all
$n> N_m$
. Finally, the fact that
$b(m+1, n) \geq b(m,n)$
implies that for every fixed m,
$a_n \geq b(m,n)$
as soon as
$n> N_m$
.
Proposition 2.8. There is a sequence
$(a_n)$
such that:
-
(1)
$\limsup _{n \to \infty } ({1}/{|F_n|}) \log a_n = 0$
; and -
(2) for any finite partition Q, there exists an N such that
$\operatorname {\mathrm {cov}}(Q,T,F_n, \mu ) \leq a_n$
for all
$n> N$
.
Proof. Because T has zero entropy, there exists a finite generating partition for T (see, for example, [Reference SewardSew19, Corollary 1.2] or [Reference RosenthalRos88, Theorem 2
${}^{\prime }$
]). Fix such a partition P and let
$Q = \{Q_1, \ldots , Q_r\}$
be any given partition. Because P is generating, there is an integer m and another partition
$Q' = \{Q^{\prime }_1, \ldots , Q^{\prime }_r \}$
such that
$Q'$
is refined by
$P_{T, F_m}$
and
$$ \begin{align*} \mu \{ x : Q(x) \neq Q'(x) \} < \frac{\epsilon}{4}. \end{align*} $$
By the mean ergodic theorem, we can write
$$ \begin{align*} &d_{F_n}(Q_{T, F_n}(x), Q^{\prime}_{T, F_n}(x)) \\&\quad= \frac{1}{|F_n|} \sum_{f \in F_n} 1_{\{ y : Q(y) \neq Q'(y) \}}(T^f x) \xrightarrow{L^1(\mu)} \mu \{ y : Q(y) \neq Q'(y) \} < \frac{\epsilon}{4}, \end{align*} $$
so, in particular, for n sufficiently large, we have
$$ \begin{align*} \mu \{ x : d_{F_n}(Q_{T, F_n}(x), Q^{\prime}_{T, F_n}(x)) < \epsilon/2 \}> 1 - \frac{\epsilon}{4}. \end{align*} $$
Let Y denote the set
$\{ x : d_{F_n}(Q_{T, F_n}(x), Q^{\prime }_{T, F_n}(x)) < \epsilon /2 \}$
.
Recall that
$Q'$
is refined by
$P_{T, F_m}$
, so
$Q^{\prime }_{T, F_n}$
is refined by
$(P_{T, F_m})_{T, F_n} = P_{T, F_m F_n}$
. Let
$\ell = \ell (m,n)$
be the minimum number of
$P_{T, F_m F_n}$
cells required to cover a set of
$\mu $
-measure at least
$1-\epsilon /4$
, and let
$C_1, \ldots , C_{\ell }$
be such a collection of cells satisfying
$\mu ( \bigcup _i C_i ) \geq 1 - \epsilon /4$
. If any of the
$C_i$
do not meet the set Y, then drop them from the list. Because
$\mu (Y)> 1- \epsilon /4$
we can still assume after dropping that
$\mu ( \bigcup _i C_i )> 1 - \epsilon /2$
. Choose a set of representatives
$y_1, \ldots y_{\ell }$
with each
$y_i \in C_i \cap Y$
.
Now we claim that
$Y \cap \bigcup _i C_i \subseteq \bigcup _{i=1}^{\ell } B_{\operatorname {Ham}}(Q, T, F_n, y_i, \epsilon )$
. To see this, let
$x \in Y \cap \bigcup _i C_i$
. Then there is one index j such that x and
$y_j$
are in the same cell of
$P_{T, F_mF_n}$
. We can then estimate
$$ \begin{align*} d_{F_n}( Q_{T, F_n}(x), Q_{T, F_n}(y_j) ) &\leq d_{F_n}( Q_{T, F_n}(x), Q^{\prime}_{T, F_n}(x) ) + d_{F_n}( Q^{\prime}_{T, F_n}(x), Q^{\prime}_{T, F_n}(y_j) )\\&\quad + d_{F_n}( Q^{\prime}_{T, F_n}(y_j), Q_{T, F_n}(y_j) ) \\&< \frac{\epsilon}{2} + 0 + \frac{\epsilon}{2} = \epsilon. \end{align*} $$
The bounds for the first and third terms come from the fact that
$x, y_j \in Y$
. The second term is
$0$
because
$Q^{\prime }_{F_n}$
is refined by
$P_{T, F_m F_n}$
and
$y_j$
was chosen so that x and
$y_j$
are in the same
$P_{T, F_m F_n}$
-cell. Therefore,
$\operatorname {\mathrm {cov}}(Q, T, F_n, \mu ) \leq \ell (m,n)$
. Thus, the proof is complete once we find a fixed sequence
$(a_n)$
that is subexponential in
$|F_n|$
and eventually dominates
$\ell (m,n)$
for each fixed m.
Because T has zero entropy, Lemmas 2.5 and 2.6 imply that
$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{|F_n|} \log \ell(m,n) = \limsup_{n \to \infty} \frac{|F_m F_n|}{|F_n|} \cdot \frac{1}{|F_m F_n|} \log \ell(m,n) = 0 \quad \text{for each fixed } m. \end{align*} $$
Note also that because
$P_{T, F_{m+1}F_n}$
refines
$P_{T, F_m F_n}$
, we have
$\ell (m+1, n) \geq \ell (m,n)$
for all
$m,n$
. Therefore, we can apply Lemma 2.7 to the numbers
$b(m,n) = |F_n|^{-1} \log \ell (m,n)$
to produce a sequence
$(a_n')$
satisfying
$a_n' \to 0$
and
$a_n' \geq b(m,n)$
eventually for each fixed m. Then
$a_n := \exp (|F_n| a_n')$
is the desired sequence.
3 Cocycles and extensions
Let I be the unit interval
$[0,1]$
and let m be Lebesgue measure on I. Denote by
$\operatorname {\mathrm {Aut}}(I,m)$
the group of invertible m-preserving transformations of I. A cocycle on X is a family of measurable maps
$\alpha _g : X \to \operatorname {\mathrm {Aut}}(I,m)$
indexed by
$g \in G$
that satisfies the cocycle condition: for every
$g,h \in G$
and
$\mu $
-almost every x,
$\alpha _{hg}(x) = \alpha _h(T^g x) \circ \alpha _g(x)$
. A cocycle can equivalently be thought of as a measurable map
$\alpha : R \to \operatorname {\mathrm {Aut}}(I,m)$
, where
${R \subseteq X \times X}$
is the orbit equivalence relation induced by T (that is,
$(x,y) \in R$
if and only if
$y = T^g x$
for some
$g \in G$
). With this perspective, the cocycle condition takes the form
$\alpha (x,z) = \alpha (y,z) \circ \alpha (x,y)$
. A cocycle
$\alpha $
induces the skew product action
$T_{\alpha }$
of G on the larger space
$X \times I$
defined by
$$ \begin{align*} T_{\alpha}^g(x,t) := (T^g x, \alpha_g(x) (t)). \end{align*} $$
This action preserves the measure
$\mu \times m$
and is an extension of the original action
$(X,T,\mu )$
.
By a classical theorem of Rokhlin (see, for example, [Reference GlasnerGla03, Theorem 3.18]), any infinite-to-one ergodic extension of
$(X,\mu ,T)$
is isomorphic to
$T_{\alpha }$
for some cocycle
$\alpha $
. Therefore, by topologizing the space of all cocycles on X we can capture the notion of a ‘generic’ extension: a property is said to hold for a generic extension if it holds for a dense
$G_{\delta }$
set of cocycles. Denote the space of all cocycles on X by
$\operatorname {Co}(X)$
. Topologizing
$\operatorname {Co}(X)$
is done in a few stages.
-
(1) Let
$\mathcal {B}(I)$
be the Borel sets in I and let
$(E_n)$
be a sequence in
$\mathcal {B}(I)$
that is dense in the
$m ( \cdot \,\triangle \, \cdot )$
metric. For example,
$(E_n)$
could be an enumeration of the family of all finite unions of intervals with rational endpoints. -
(2) The group
$\operatorname {\mathrm {Aut}}(I,m)$
is completely metrizable via the metric
$$ \begin{align*} d_A(\phi, \psi) = \frac12 \sum_{n \geq 1} 2^{-n} [m (\phi E_n \,\triangle\, \psi E_n) + m(\phi^{-1} E_n \,\triangle\, \psi^{-1} E_n)]. \end{align*} $$
Note that with this metric,
$\operatorname {\mathrm {Aut}}(I,m)$
has diameter at most
$1$
. See, for example, [Reference KechrisKec10, §1.1] -
(3) If
$\alpha _0, \beta _0$
are maps
$X \to \operatorname {\mathrm {Aut}}(I,m)$
, then define
$\operatorname {\mathrm {dist}}(\alpha _0, \beta _0) = \int d_A(\alpha _0(x), \beta _0(x))\, d\mu (x)$
. -
(4) The metric defined in the previous step induces a topology on
$\operatorname {\mathrm {Aut}}(I,m)^X$
. Therefore, because
$\operatorname {Co}(X)$
is just a certain (closed) subset of
$(\operatorname {\mathrm {Aut}}(I,m)^X)^G$
, it just inherits the product topology.
To summarize, if
$\alpha $
is a cocycle, then a basic open neighborhood
$\alpha $
is specified by two parameters: a finite subset
$F \subseteq G$
and
$\eta> 0$
. The
$(F,\eta )$
-neighborhood of
$\alpha $
is
${\{ \beta \in \operatorname {Co}(X): \operatorname {\mathrm {dist}}(\alpha _g, \beta _g) < \eta \text { for all } g \in F \}}$
. In practice, we always arrange things so that there is a set of x of measure at least
$ 1-\eta $
on which
$\alpha _g(x) = \beta _g(x)$
for all
$g \in F$
, which is sufficient to guarantee that
$\beta $
is in the
$(F,\eta )$
-neighborhood of
$\alpha $
.
Let
$\overline {Q}$
be the partition
$\{X \times [0,1/2], X \times (1/2, 1] \}$
of
$X \times I$
. We derive Theorem 1.1 from the following result about covering numbers of extensions, which is the main technical result of the paper.
Theorem 3.1. For any sequence
$(a_n)$
satisfying
$\limsup _{n \to \infty } ({1}/{|F_n|}) \log a_n = 0$
, there is a dense
$G_{\delta }$
set
$\mathcal {U} \subseteq \operatorname {Co}(X)$
such that for any
$\alpha \in \mathcal {U}$
,
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m)> a_n$
for infinitely many n.
Proof that Theorem 3.1 implies Theorem 1.1
Choose a sequence
$(a_n)$
as in Proposition 2.8 such that for any partition Q,
$\operatorname {\mathrm {cov}}(Q, T, F_n, \mu ) \leq a_n$
for sufficiently large n. Let
$\mathcal {U}$
be the dense
$G_{\delta }$
set of cocycles associated to
$(a_n)$
as guaranteed by Theorem 3.1 and let
${\alpha \in \mathcal {U}}$
, so we know that
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m)> a_n$
for infinitely many n. Now if
$\varphi : (X \times I, T_{\alpha }, \mu \times m) \to (X, T, \mu )$
were an isomorphism, then by Lemma 2.3,
$\varphi \overline {Q}$
would be a partition of X satisfying
$\operatorname {\mathrm {cov}}(\varphi \overline {Q}, T, F_n, \mu ) = \operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m)> a_n$
for infinitely many n, contradicting the conclusion of Proposition 2.8. Therefore, we have produced a dense
$G_{\delta }$
set of cocycles
$\alpha $
such that
$T_{\alpha } \not \simeq T$
, which implies Theorem 1.1.
To prove Theorem 3.1, we need to show roughly that
$\{ \alpha \in \operatorname {Co}(X) : \operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m) \text { is large} \}$
is both open and dense. We address the open part here and leave the density part until the next section. Let
$\pi $
be the partition
$\{[0,1/2), [1/2, 1]\}$
of I.
Lemma 3.2. If
$\beta ^{(n)}$
is a sequence of cocycles converging to
$\alpha $
, then for any finite
$F \subseteq G$
, we have
$$ \begin{align*} (\mu \times m) \{ (x,t) : \overline{Q}_{T_{\beta^{(n)}}, F}(x,t) = \overline{Q}_{T_{\alpha}, F}(x,t) \} \to 1 \quad \text{as } n \to \infty. \end{align*} $$
Proof. For the names
$\overline {Q}_{T_{\beta ^{(n)}}, F}(x,t)$
and
$\overline {Q}_{T_{\alpha }, F}(x,t)$
to be the same means that for every
$g \in F$
,
$$ \begin{align*} \overline{Q}( T^g x, \beta^{(n)}_g(x) t ) = \overline{Q} ( T^g x, \alpha_g(x) t ), \end{align*} $$
which is equivalent to
$$ \begin{align} \pi( \beta^{(n)}_g(x) t ) = \pi ( \alpha_g(x) t ). \end{align} $$
The idea is the following. For fixed g and x, if
$\alpha _g(x)$
and
$\beta ^{(n)}_g(x)$
are close in
$d_A$
, then (1) fails for only a small measure set of t. In addition, if
$\beta ^{(n)}$
is very close to
$\alpha $
in the cocycle topology, then
$\beta ^{(n)}_g(x)$
and
$\alpha _g(x)$
are close for all
$g \in F$
and most
$x \in X$
. Then, by Fubini’s theorem, we will get that the measure of the set of
$(x,t)$
failing (1) is small.
Here are the details. Fix
$\rho> 0$
; we show that the measure of the desired set is at least
$1-\rho $
for n sufficiently large. First, let
$\sigma $
be so small that for any
$\phi , \psi \in \operatorname {\mathrm {Aut}}(I,m)$
,
$$ \begin{align*} d_A(\phi, \psi) < \sigma \quad \text{implies} \quad m \{ t : \pi(\phi t) = \pi(\psi t) \}> 1-\rho/2. \end{align*} $$
This is possible because
$$ \begin{align*} \{ t : \pi(\phi t) \neq \pi(\psi t) \} \subseteq (\phi^{-1}[0,1/2) \,\triangle\, \psi^{-1}[0,1/2)) \cup (\phi^{-1}[1/2,1] \,\triangle\, \psi^{-1}[1/2,1]). \end{align*} $$
Then, from the definition of the cocycle topology, we have
$$ \begin{align*} \mu \{ x \in X : d_A ( \beta^{(n)}_g(x), \alpha_g(x) ) < \sigma \text{ for all } g \in F \} \to 1 \quad \text{as } n \to \infty. \end{align*} $$
Let n be large enough so that the above is larger than
$1-\rho /2$
. Then, by Fubini’s theorem, we have
$$ \begin{align*} &(\mu \times m) \{ (x,t) : \overline{Q}_{T_{\beta^{(n)}}, F}(x,t) = \overline{Q}_{T_{\alpha}, F}(x,t) \} \\&\quad= \int m \{ t : \pi( \beta^{(n)}_g(x) t ) = \pi ( \alpha_g(x) t ) \text{ for all } g \in F \} \,d\mu(x). \end{align*} $$
We have arranged things so that the integrand above is greater than
$1-\rho /2$
on a set of x of
$\mu $
-measure greater than
$ 1-\rho /2$
, so the integral is at least
$(1-\rho /2)(1-\rho /2)> 1-\rho $
as desired.
Lemma 3.3. For any finite
$F \subseteq G$
and any
$L> 0$
, the set
$\{ \alpha \in \operatorname {Co}(X) : \operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F, \mu \times m)> L \}$
is open in
$\operatorname {Co}(X)$
.
Proof. Suppose
$\beta ^{(n)}$
is a sequence of cocycles converging to
$\alpha $
and satisfying
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\beta ^{(n)}}, F, \mu \times m) \leq L$
for all n. We show that
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F, \mu \times m) \leq L$
as well. The covering number
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\beta ^{(n)}}, F, \mu \times m)$
is a quantity which really depends only on the measure
$( \overline {Q}_{T_{\beta ^{(n)}} , F} )_* (\mu \times m) \in \operatorname {\mathrm {Prob}} ( \{0,1\}^F )$
, which we now call
$\nu _n$
for short. The assumption that
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\beta ^{(n)}}, F, \mu \times m) \leq L$
for all n says that for each n, there is a collection of L words
$w_1^{(n)}, \ldots , w_L^{(n)} \in \{0,1\}^F$
such that the Hamming balls of radius
$\epsilon $
centered at these words cover a set of
$\nu _n$
-measure at least
$1-\epsilon $
. As
$\{0,1\}^F$
is a finite set, there are only finitely many possibilities for the collection
$( w_1^{(n)}, \ldots , w_L^{(n)} )$
. Therefore, by passing to a subsequence and relabeling, we may assume that there is a fixed collection of words
$w_1, \ldots , w_L$
with the property that if we let
$B_i$
be the Hamming ball of radius
$\epsilon $
centered at
$w_i$
, then
$\nu _n ( \bigcup _{i=1}^L B_i ) \geq 1-\epsilon $
for every n.
Now, by Lemma 3.2, the map
$\overline {Q}_{T_{\beta ^{(n)}} , F}$
agrees with
$\overline {Q}_{T_{\alpha } , F}$
on a set of measure converging to
$1$
as
$n \to \infty $
. This implies that the measures
$\nu _n$
converge in the total variation norm on
$\operatorname {\mathrm {Prob}} ( \{0,1\}^F )$
to
$\nu := ( \overline {Q}_{T_{\alpha } , F} )_*(\mu \times m)$
. As
$\nu _n ( \bigcup _{i=1}^L B_i ) \geq 1-\epsilon $
for every n, we conclude that
$\nu ( \bigcup _{i=1}^L B_i ) \geq 1-\epsilon $
also, which implies that
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F, \mu \times m) \leq L$
as desired.
Define
$\mathcal {U}_N := \{\alpha \in \operatorname {Co}(X) : \operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m)> a_n \text { for some } n > N \}$
. By Lemma 3.3, each
$\mathcal {U}_N$
is a union of open sets and therefore open. In addition,
$\bigcap _{N} \mathcal {U}_N$
is exactly the set of
$\alpha \in \operatorname {Co}(X)$
satisfying
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m)> a_n$
infinitely often. Therefore, by the Baire category theorem, in order to prove Theorem 3.1 it suffices to prove
Proposition 3.4. For each N,
$\mathcal {U}_N$
is dense in
$\operatorname {Co}(X)$
.
The proof of this proposition is the content of the next section.
4 Proof of Proposition 3.4
4.1 Setup
Let N be fixed and let
$\alpha _0$
be an arbitrary cocycle. Consider a neighborhood of
$\alpha _0$
determined by a finite set
$F \subseteq G$
and
$\eta> 0$
. We can assume without loss of generality that
$\eta \ll \epsilon = 1/100$
. We produce a new cocycle
$\alpha \in \mathcal {U}_N$
such that there is a set
$X'$
of measure at least
$1-\eta $
on which
$\alpha _f(x) = (\alpha _0)_f(x)$
for all
$f \in F$
, implying that
$\alpha $
is in the
$(F,\eta )$
-neighborhood of
$\alpha _0$
. The construction of such an
$\alpha $
is based on the fact that the orbit equivalence relation R is hyperfinite.
Theorem 4.1. [Reference Ornstein and WeissOW80, Theorem 6]
There is an increasing sequence of equivalence relations
$R_n \subseteq X \times X$
such that:
-
• each
$R_n$
is measurable as a subset of
$X \times X$
; -
• every cell of every
$R_n$
is finite; and -
•
$\bigcup _n R_n$
agrees
$\mu $
-almost everywhere with R.
Fix such a sequence
$(R_n)$
and for
$x \in X$
, write
$R_n(x)$
to denote the cell of
$R_n$
that contains x.
Lemma 4.2. There exists an
$m_1$
such that
$\mu \{ x \in X : T^F x \subseteq R_{m_1}(x) \}> 1-\eta $
.
Proof. Almost every x satisfies
$T^G x = \bigcup _{m} R_m(x)$
, so, in particular, for
$\mu $
-almost every x, there is an
$m_x$
such that
$T^F x \subseteq R_m(x)$
for all
$m \geq m_x$
. Letting
$X_{\ell } = \{x \in X : m_x \leq \ell \}$
, we see that the sets
$X_{\ell }$
are increasing and exhaust almost all of X. Therefore, we can pick
$m_1$
so that
$\mu (X_{m_1})> 1-\eta $
.
Now we drop
$R_1, \ldots , R_{m_1 - 1}$
from the sequence and assume that
$m_1 = 1$
.
Lemma 4.3. There exists a K such that
$\mu \{x : |R_1(x)| \leq K \}> 1-\eta $
.
Proof. Every
$R_1$
-cell is finite, so if we define
$X_k = \{x \in X : |R_1(x)| \leq k \}$
, then the
$X_k$
are increasing and exhaust all of X. Thus, we pick K so that
$\mu (X_K)> 1-\eta $
.
Continue to use the notation
$X_K = \{x \in X : |R_1(x)| \leq K \}$
.
Lemma 4.4. For all n sufficiently large,
$\mu \{x \in X : {|(T^{F_n} x) \cap X_K|}/{|F_n|}> 1-2\eta \} > 1-\eta $
.
Proof. We have
$|(T^{F_n} x) \cap X_K| = \sum _{f \in F_n} 1_{X_K}(T^f x)$
. By the mean ergodic theorem [Reference GlasnerGla03, Theorem 3.33], we get
$$ \begin{align*} \frac{|(T^{F_n} x) \cap X_K|}{|F_n|} \to \mu(X_K)> 1-\eta \quad \text{in probability as } n \to \infty. \end{align*} $$
Therefore, in particular,
$\mu \{ x \in X : {|(T^{F_n} x) \cap X_K|}/{|F_n|}> 1-2\eta \} \to 1$
as
$n \to \infty $
, so this measure is greater than
$1-\eta $
for all n sufficiently large.
From now on, let n be a fixed number that is large enough so that the above lemma holds,
$n> N$
, and
$\tfrac 12 \exp ( {1}/{8K^2} \cdot |F_n| )> a_n$
. This is possible because
$(a_n)$
is assumed to be subexponential in
$|F_n|$
. The relevance of the final condition will appear at the end.
Lemma 4.5. There is an
$m_2$
such that
$\mu \{x \in X : T^{F_n} x \subseteq R_{m_2}(x) \}> 1-\eta $
.
Proof. The proof follows the same lines as Lemma 4.2.
Again, drop
$R_2, \ldots , R_{m_2-1}$
from the sequence of equivalence relations and assume
$m_2 = 2$
.
4.2 Construction of the perturbed cocycle
Let
$(R_n)$
be the relabeled sequence of equivalence relations from the previous section. The following measure-theoretic fact is well known. Recall that two partitions P and
$P'$
of I are said to be independent with respect to m if
$m(E \cap E') = m(E)m(E')$
for any
$E \in P$
,
$E' \in P'$
.
Lemma 4.6. Let P and
$P'$
be two finite partitions of I. Then there exists a
$\varphi \in \operatorname {\mathrm {Aut}}(I,m)$
such that P and
$\varphi ^{-1}P'$
are independent with respect to m.
Proposition 4.7. For any
$\alpha _0 \in \operatorname {Co}(X)$
, there is an
$\alpha \in \operatorname {Co}(X)$
such that:
-
(1)
$\alpha _g(x) = (\alpha _0)_g(x)$
whenever
$(x, T^g x) \in R_1$
; and -
(2) for
$\mu $
-almost every x, the following holds. If C is an
$R_1$
-cell contained in
$R_2(x)$
, consider the map
$Y_C: t \mapsto \overline {Q}_{T_{\alpha }, \{g : T^g x \in C \}}(x,t)$
as a random variable on the underlying space
$(I,m)$
. Then as C ranges over all such
$R_1$
-cells, the random variables
$Y_C$
are independent.
Proof. We give here only a sketch of the proof and leave the full details to Appendix A. It is more convenient to adopt the perspective of a cocycle as a map
$\alpha : R \to \operatorname {\mathrm {Aut}}(I,m)$
satisfying the condition
$\alpha (x,z) = \alpha (y,z) \circ \alpha (x,y)$
.
Step 1. For
$(x,y) \in R_1$
, let
$\alpha (x,y) = \alpha _0(x,y)$
.
Step 2. Fix an
$R_2$
-cell
$\overline {C}$
. Enumerate by
$\{C_1, \ldots , C_k\}$
all of the
$R_1$
-cells contained in
$\overline {C}$
and choose from each a representative
$x_i \in C_i$
.
Step 3. Let
$\pi $
denote the partition
$\{[0,1/2), [1/2, 1]\}$
of I. Define
$\alpha (x_1, x_2)$
to be an element of
$\operatorname {\mathrm {Aut}}(I,m)$
such that
$$ \begin{align*} \bigvee_{y \in C_1} \alpha(x_1, y)^{-1} \pi \quad \text{and} \quad \alpha(x_1, x_2)^{-1} \bigg(\bigvee_{y \in C_2} \alpha(x_2, y)^{-1} \pi \bigg) \end{align*} $$
are independent. These expressions are well defined because
$\alpha $
has already been defined on
$R_1$
and we use Lemma 4.6 to guarantee that such an element of
$\operatorname {\mathrm {Aut}}(I,m)$
exists.
Step 4. There is now a unique way to extend the definition of
$\alpha $
to
$(C_1 \cup C_2) \times (C_1 \cup C_2)$
that is consistent with the cocycle condition. For arbitrary
$y_1 \in C_1, y_2 \in C_2$
, define
$$ \begin{align*} \alpha(y_1, y_2) &= \alpha(x_2, y_2) \circ \alpha(x_1, x_2) \circ \alpha(y_1, x_1) \quad \text{and}\\ \alpha(y_2, y_1) &= \alpha(y_1, y_2)^{-1}. \end{align*} $$
The middle term in the first equation was defined in the previous step and the outer two terms were defined in step 1.
Step 5. Extend the definition of
$\alpha $
to the rest of the
$C_i$
inductively, making each cell independent of all the previous ones. Suppose
$\alpha $
has been defined on
$( C_1 \cup \dots \cup C_j ) \times ( C_1 \cup \dots \cup C_j )$
. Using Lemma 4.6 again, define
$\alpha (x_1, x_{j+1})$
to be an element of
$\operatorname {\mathrm {Aut}}(I,m)$
such that
$$ \begin{align*} \bigvee_{y \in C_1 \cup \dots \cup C_j} \alpha(x_1, y)^{-1}\pi \quad \text{and} \quad \alpha(x_1,x_{j+1})^{-1} \bigg( \bigvee_{y \in C_{j+1}} \alpha(x_{j+1}, y)^{-1}\pi \bigg) \end{align*} $$
are independent. Then, just as in step 4, there is a unique way to extend the definition of
$\alpha $
to all of
$(C_1 \cup \dots \cup C_{j+1}) \times (C_1 \cup \dots \cup C_{j+1})$
. At the end of this process,
$\alpha $
has been defined on all of
$\overline {C} \times \overline {C}$
. This was done for an arbitrary
$R_2$
-cell
$\overline {C}$
, so now
$\alpha $
is defined on
$R_2$
.
Step 6. For each
$N \geq 2$
, extend the definition of
$\alpha $
from
$R_N$
to
$R_{N+1}$
with the same procedure, but there is no need to set up any independence. Instead, every time there is a choice for how to define
$\alpha $
between two of the cell representatives, just take it to be the identity. This defines
$\alpha $
on
$\bigcup _{N \geq 1} R_N$
, which is equal mod
$\mu $
to the full orbit equivalence relation, so
$\alpha $
is a well-defined cocycle.
Now we verify the two claimed properties of
$\alpha $
. Property (1) is immediate from step 1 of the construction. To check property (2), fix x and let
$C_j$
be any of the
$R_1$
-cells contained in
$R_2(x)$
. Note that the name
$\overline {Q}_{T_{\alpha }, \{g : T^g x \in C_j\}}(x,t)$
records the data
$\overline {Q}(T^g_{\alpha }(x,t)) = \overline {Q}(T^g x, \alpha _g(x) t) = \pi ( \alpha _g(x) t)$
for all g such that
$T^g x \in C_j$
, which, by switching to the other notation, is the same data as
$\pi (\alpha (x, y) t)$
for
$y \in C_j$
. Thus, the set of t for which
$\overline {Q}_{T_{\alpha }, \{g : T^g x \in C_j\}}(x,t)$
is equal to a particular word is given by a corresponding particular cell of the partition
$\bigvee _{y \in C_j} \alpha (x,y)^{-1} \pi = \alpha (x, x_1)^{-1} ( \bigvee _{y \in C_j} \alpha (x_1, y)^{-1} \pi )$
. The construction of
$\alpha $
was defined exactly so that the partitions
$\bigvee _{y \in C_j} \alpha (x_1, y)^{-1} \pi $
are all independent and the names
$\overline {Q}_{T_{\alpha }, \{g : T^g x \in _j\}}(x,t)$
are determined by these independent partitions pulled back by the fixed m-preserving map
$\alpha (x, x_1)$
, so they are also independent.
The reason this is only a sketch is because it is not clear that the construction described here can be done in a way so that the resulting
$\alpha $
is a measurable function. To do it properly requires a slightly different approach; see Appendix A for full details.
Letting
$\widetilde {X} = \{x \in X : T^F x \subseteq R_1(x) \}$
, this construction guarantees that
$\alpha _f(x) = (\alpha _0)_f(x)$
for all
$f \in F, x \in \widetilde {X}$
. By Lemma 4.2,
$\mu (\widetilde {X})> 1-\eta $
, so this shows that
$\alpha $
is in the
$(F,\eta )$
-neighborhood of
$\alpha _0$
.
4.3 Estimating the size of Hamming balls
Let
$\alpha $
be the cocycle constructed in the previous section. We estimate the
$(\mu \times m)$
-measure of
$(\overline {Q}, T_{\alpha }, F_n)$
-Hamming balls in order to get a lower bound for the covering number. The following formulation of Hoeffding’s inequality will be quite useful [Reference VershyninVer18, Theorem 2.2.6].
Theorem 4.8. Let
$Y_1, \ldots , Y_{\ell }$
be independent random variables such that each
${Y_i \in [0,K]}$
almost surely. Let
$a = \mathbb {E} [ \sum Y_i ]$
. Then for any
$t>0$
,
$$ \begin{align*} \mathbb{P}\bigg( \sum_{i=1}^{\ell} Y_i < a - t \bigg) \leq \exp\bigg( - \frac{2t^2}{K^2 \ell} \bigg). \end{align*} $$
Let
$X_0 = \{x \in X : {|(T^{F_n} x) \cap X_K|}/{|F_n|}> 1-2\eta \text { and } T^{F_n} x \subseteq R_2(x) \}$
. By Lemmas 4.4 and 4.5,
$\mu (X_0)> 1 - 2\eta $
. In addition, write
$\mu \times m = \int m_x \,d\mu (x)$
, where
${m_x = \delta _x \times m}$
.
Proposition 4.9. For any
$(x,t) \in X_0 \times I$
,
$$ \begin{align} m_x( B_{\operatorname{Ham}}(\overline{Q}, T_{\alpha}, F_n, (x,t),\epsilon) ) \leq \exp\bigg( - \frac{1}{8K^2} \cdot |F_n| \bigg). \end{align} $$
Proof. Let
$\mathcal {C}$
be the collection of
$R_1$
-cells C that meet
$T^{F_n} x$
and satisfy
$|C| \leq K$
. For each
$C \in \mathcal {C}$
, let
$F_C = \{f \in F_n : T^f x \in C \}$
. Define
$$ \begin{align*} Y(t') = |F_n| \cdot d_{F_n} ( \overline{Q}_{T_{\alpha}, F_n}(x,t), \overline{Q}_{T_{\alpha}, F_n}(x,t') ) = \sum_{f \in F_n} 1_{\overline{Q}(T_{\alpha}^f(x,t)) \neq \overline{Q}(T_{\alpha}^f(x,t'))}, \end{align*} $$
and for each
$C \in \mathcal {C}$
, define
$$ \begin{align*} Y_C(t') = \sum_{f \in F_C} 1_{\overline{Q}(T_{\alpha}^f(x,t)) \neq \overline{Q}(T_{\alpha}^f(x,t'))}. \end{align*} $$
Then we have
$$ \begin{align*} Y(t') \geq \sum_{C \in \mathcal{C}} Y_C(t'), \end{align*} $$
so to get an upper bound for
$m_x ( B_{\operatorname {Ham}}(\overline {Q}, T_{\alpha }, F_n, (x,t),\epsilon ) ) = m \{ t' : Y(t') < \epsilon |F_n| \}$
, it is sufficient to control
$m \{ t' : \sum _{C \in \mathcal {C}} Y_C(t') < \epsilon |F_n| \}$
.
View each
$Y_C(t')$
as a random variable on the underlying probability space
$(I,m)$
. Our construction of the cocycle
$\alpha $
guarantees that the collection of names
$\overline {Q}_{T_{\alpha }, F_C}(x,t')$
as C ranges over all of the
$R_1$
-cells contained in
$R_2(x)$
is an independent collection. Therefore, in particular, the
$Y_C$
for
$C \in \mathcal {C}$
are independent (the assumption that
$x \in X_0$
guarantees that all
$C \in \mathcal {C}$
are contained in
$R_2(x)$
).
We also have that each
$Y_C \in [0,K]$
and the expectation of the sum is
$$ \begin{align*} a &:= \sum_{C \in \mathcal{C}} \int Y_C(t') \,dm(t') = \sum_{C \in \mathcal{C}} \sum_{f \in F_C} \int 1_{\overline{Q}(T_{\alpha}^f(x,t)) \neq \overline{Q}(T_{\alpha}^f(x,t'))} \,dm(t') = \sum_{C \in \mathcal{C}} \frac{1}{2} |F_C| \\&= \frac12 \sum_{C \in \mathcal{C}}|C \cap (T^{F_n} x)|> \frac12 (1-2\eta)|F_n|, \end{align*} $$
where the final inequality is true because
$x \in X_0$
. Thus, we can apply Theorem 4.8 with
$t = a - \epsilon |F_n|$
to conclude
$$ \begin{align*} m \bigg\{ t' : \sum_{C \in \mathcal{C}} Y_C(t') < \epsilon|F_n| \bigg\} &\leq \exp\bigg( \frac{-2t^2}{K^2 |\mathcal{C}|} \bigg) \leq \exp \bigg( \frac{-2(1/2 - \eta - \epsilon)^2 |F_n|^2}{K^2 |F_n|} \bigg) \\&\leq \exp\bigg( - \frac{1}{8K^2} \cdot |F_n| \bigg). \end{align*} $$
The final inequality holds because
$\epsilon = 1/100$
and
$\eta \ll \epsilon $
is small enough so that
$1/2 - \eta - \epsilon> 1/4$
.
Corollary 4.10. Let
$y \in X_0$
. If B is any
$(\overline {Q}, T_{\alpha }, F_n)$
-Hamming ball of radius
$\epsilon $
, then
$m_y(B) \leq \exp (( -{1}/{8K^2}) \cdot |F_n| )$
.
Proof. If B does not meet the fiber above y, then obviously
$m_y(B) = 0$
. Thus, assume
$(y,s) \in B$
for some
$s \in I$
. Then applying the triangle inequality in the space
$(\{0,1\}^{F_n}, d_{F_n})$
shows that
$B \subseteq B_{\operatorname {Ham}}(\overline {Q}, T_{\alpha }, F_n, (y,s), 2\epsilon )$
. Now apply Proposition 4.9 with
$2\epsilon $
in place of
$\epsilon $
. The proof goes through exactly the same and we get the same constant
$1/8K^2$
in the final estimate because
$\epsilon $
and
$\eta $
are small enough so that
$1/2-\eta - 2\epsilon $
is still greater than
$ 1/4$
.
Corollary 4.11. We have
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m) \geq \tfrac 12 \exp (({1}/{8K^2}) \cdot |F_n|)$
.
Proof. Let
$\{ B_i \}_{i=1}^{\ell }$
be a collection of
$(\overline {Q}, T_{\alpha }, F_n)$
-Hamming balls of radius
$\epsilon $
such that
$$ \begin{align*} (\mu \times m) \bigg( \bigcup B_i \bigg)> 1-\epsilon. \end{align*} $$
Then
$$ \begin{align*} &1-\epsilon < (\mu \times m) \bigg( \bigcup B_i \bigg) \\&\quad= (\mu \times m) \bigg( \bigcup B_i \cap (X_0 \times I) \bigg) + (\mu \times m) \bigg( \bigcup B_i \cap (X_0^c \times I) \bigg) \\&\quad< \sum_{i=1}^{\ell} (\mu \times m)(B_i \cap (X_0 \times I)) + (\mu \times m)(X_0^c \times I) \\&\quad< \sum_{i=1}^{\ell} \int_{y\in X_0} m_y(B_i) \,d\mu(y) + 2\eta \\&\quad< \ell \cdot \exp\bigg( -\frac{1}{8K^2} \cdot |F_n| \bigg) + 2\eta, \end{align*} $$
implying that
$\ell> (1-\epsilon -2\eta ) \exp (( {1}/{8K^2}) \cdot |F_n| ) > (1/2) \exp (( {1}/{8K^2}) \cdot |F_n| )$
.
Our choice of n at the beginning now guarantees that
$\operatorname {\mathrm {cov}}(\overline {Q}, T_{\alpha }, F_n, \mu \times m) \geq \tfrac 12 \exp ({1}/{8K^2} \cdot |F_n|)> a_n$
, showing that
$\alpha \in \mathcal {U}_N$
as desired. This completes the proof of Proposition 3.4.
Acknowledgements
I am grateful to Tim Austin for originally suggesting this project and for endless guidance. I also thank Benjy Weiss and an anonymous referee for pointing out some referencing inaccuracies in earlier versions. This project was partially supported by NSF grant DMS-1855694.
A Appendix. Measurability of the perturbed cocycle
In this section, we give a more careful proof of Proposition 4.7 that addresses the issue of measurability. We need to use at some point the following measurable selector theorem [Reference FremlinFre06, Proposition 433F].
Theorem A.1. Let
$(\Omega _1, \mathcal {F}_1)$
and
$(\Omega _2, \mathcal {F}_2)$
be standard Borel spaces. Let
$\mathbb {P}$
be a probability measure on
$(\Omega _1, \mathcal {F}_1)$
and suppose that
$f: \Omega _2 \to \Omega _1$
is measurable and surjective. Then there exists a measurable selector
$g: \Omega _1 \to \Omega _2$
which is defined
$\mathbb {P}$
-almost everywhere (meaning
$g(\omega ) \in f^{-1}(\omega )$
for
$\mathbb {P}$
-almost every
$\omega \in \Omega _1$
).
Given
$x \in X$
, there is a natural bijection between
$T^G x$
and G because T is a free action. We can also identify subsets: if
$E \subseteq T^G x$
, then we write
$\widetilde {E} := \{g \in G : T^g x \in E \}$
. Note that this set depends on the ‘base point’ x. If x and y are two points in the same G-orbit, then the set
$\widetilde {E}$
based at x is a translate of the same set based at y. It will always be clear from context what the intended base point is.
Definition A.2. A pattern in G is a pair
$(H,\mathscr {C})$
, where H is a finite subset of G and
$\mathscr {C}$
is a partition of H.
Definition A.3. For
$x \in X$
, define
$\operatorname {\mathrm {pat}}_n(x)$
to be the pattern
$(H,\mathscr {C})$
, where
$H = \widetilde {R_n(x)}$
and
$\mathscr {C}$
is the partition of H into the sets
$\widetilde {C}$
where C ranges over all of the
$R_{n-1}$
-cells contained in
$R_n(x)$
.
Lemma A.4. The pattern
$\operatorname {\mathrm {pat}}_n(x)$
is a measurable function of x.
Proof. Because there are only countably many possible patterns, it is enough to fix a pattern
$(H, \mathscr {C})$
and show that
$\{x : \operatorname {\mathrm {pat}}_n(x) = (H, \mathscr {C}) \}$
is measurable. Enumerate
${\mathscr {C} = \{C_1, \ldots , C_k \}}$
. Saying that
$\operatorname {\mathrm {pat}}_n(x) = (H,\mathscr {C})$
is the same as saying that
$T^H x = R_n(x)$
and each
$T^{C_i}x$
is a cell of
$R_{n-1}$
. We can express the set of x satisfying this as
$$ \begin{align*} &\bigg( \bigcap_{i=1}^{k} \bigcap_{g,h \in C_i} \!\{x : (T^g x, T^h x) \in R_{n-1} \} \quad \cap \quad \bigcap_{(g, h) \in G^2 \setminus \bigcup (C_i \times C_i)} \!\{x : (T^g x, T^h x) \not\in R_{n-1} \} \bigg) \\&\quad\cap\,\bigg( \bigcap_{g \in H} \{x : (x, T^g x) \in R_n \} \quad \cap \quad \bigcap_{g \not\in H} \{x : (x,T^g x) \not\in R_n \} \bigg). \end{align*} $$
Because each
$R_n$
is a measurable set and each
$T^g$
is a measurable map, this whole thing is measurable.
For each pattern
$(H,\mathscr {C})$
, let
$X_{H,\mathscr {C}}^{(n)} = \{x \in X : \operatorname {\mathrm {pat}}_n(x) = (H, \mathscr {C}) \}$
. We define our cocycle
$\alpha $
inductively on the equivalence relations
$R_n$
. For each n, the sets
$X_{H,\mathscr {C}}^{(n)}$
partition X into countably many measurable sets, so it will be enough to define
$\alpha $
measurably on each
$X_{H,\mathscr {C}}^{(n)}$
. At this point, fix a pattern
$(H,\mathscr {C})$
, fix
$n = 2$
, and write
$X_{H, \mathscr {C}}$
instead of
$X_{H,\mathscr {C}}^{(2)}$
. Define
$$ \begin{align*} \Omega_2^{H,\mathscr{C}} &= \{ \psi : H \times H \to \operatorname{\mathrm{Aut}}(I,m) : \psi(h_1, h_3)\\ &= \psi(h_2, h_3) \circ \psi(h_1, h_2) \text{ for all } h_1,h_2,h_3 \in H \}, \\ \Omega_1^{H,\mathscr{C}} &= \bigg\{ \sigma : \bigcup_{C \in \mathscr{C}} C \times C \to \operatorname{\mathrm{Aut}}(I,m) : \sigma(g_1, g_3)\\ & = \sigma(g_2, g_3) \circ \sigma(g_1, g_2) \text{ for all } g_1,g_2,g_3 \in G \bigg\}, \\ \Omega_2^{H,\mathscr{C},\text{ ind}} &= \{ \psi \in \Omega_2^{H,\mathscr{C}} : \psi \ \text{is } (H,\mathscr{C})\text{-independent} \}, \end{align*} $$
where
$\psi \in \Omega _2^{H,\mathscr {C}}$
is said to be
$(H,\mathscr {C})$
-independent if for any fixed
$h_0 \in H$
, the partitions
$$ \begin{align*} \bigvee_{h \in C} \psi(h_0, h)^{-1} \pi \end{align*} $$
as C ranges over
$\mathscr {C}$
are independent with respect to m.
Proposition A.5. For every
$\sigma \in \Omega _1^{H,\mathscr {C}}$
, there is some
$\psi \in \Omega _2^{H,\mathscr {C},\text {ind}}$
that extends
$\sigma $
.
Proof. The idea is exactly the same as the construction described in steps 3–5 in the sketched proof of Proposition 4.7, but we write it out here also for completeness.
Enumerate
$\mathscr {C} = \{C_1, \ldots , C_k \}$
and for each i fix an element
$g_i \in C_i$
. First, obviously we define
$\psi = \sigma $
on each
$C_i \times C_i$
. Next, define
$\psi (g_1, g_2)$
to be an element of
$\operatorname {\mathrm {Aut}}(I,m)$
such that
$$ \begin{align*} \bigvee_{g \in C_1} \sigma(g_1, g)^{-1} \pi \quad \text{and} \quad \psi(g_1, g_2)^{-1} \bigg(\bigvee_{g \in C_2} \sigma(g_2, g)^{-1} \pi \bigg) \end{align*} $$
are independent. Then, define
$\psi $
on all of
$(C_1 \cup C_2) \times (C_1 \cup C_2)$
by setting
$$ \begin{align*} \psi(h_1, h_2) &= \sigma(g_2, h_2) \circ \psi(g_1, g_2) \circ \sigma(h_1, g_1) \quad \text{and} \\ \psi(h_2, h_1) &= \psi(h_1, h_2)^{-1} \end{align*} $$
for any
$h_1 \in C_1, h_2 \in C_2$
. Continue this definition inductively, making each new step independent of all the steps that came before it. If
$\psi $
has been defined on
$( C_1 \cup \dots \cup C_j ) \times ( C_1 \cup \dots \cup C_j )$
, then define
$\psi (g_1, g_{j+1})$
to be an element of
$\operatorname {\mathrm {Aut}}(I,m)$
such that
$$ \begin{align*} \bigvee_{g \in C_1 \cup \dots \cup C_j} \psi(g_1, g)^{-1}\pi \quad \text{and} \quad \psi(g_1,g_{j+1})^{-1} \bigg( \bigvee_{g' \in C_{j+1}} \sigma(g_{j+1}, g')^{-1}\pi \bigg) \end{align*} $$
are independent. Then extend the definition of
$\psi $
to all of
$(C_1 \cup \dots \cup C_{j+1}) \times (C_1 \cup \dots \cup C_{j+1})$
in the exact same way.
At the end of this process,
$\psi $
has been defined on
$(C_1 \cup \cdots \cup C_k) \times (C_1 \cup \cdots \cup C_k) = H \times H$
, and it satisfies the cocycle condition by construction. To verify that it also satisfies the independence condition, note that the construction has guaranteed that
$$ \begin{align*} \bigvee_{h \in C} \psi(g_1, h)^{-1}\pi \end{align*} $$
are independent partitions as C ranges over
$\mathscr {C}$
. To get the same conclusion for an arbitrary base point
$h_0$
, pull everything back by the fixed map
$\psi (h_0, g_1)$
. Because this map is measure preserving, pulling back all of the partitions by it preserves their independence.
Now we would like to take this information about cocycles defined on patterns and use it to produce cocycles defined on the actual space X. Define the map
$\sigma ^{H,\mathscr {C}}: X_{H,\mathscr {C}} \to \Omega _1^{H,\mathscr {C}}$
by
$\sigma ^{H,\mathscr {C}}_x(g_1, g_2) := \alpha _0(T^{g_1}x, T^{g_2} x)$
. Note that this is a measurable map because
$\alpha _0$
is a measurable cocycle.
By Theorem A.1 applied to the measure
$\mathbb {P} = (\sigma ^{H, \mathscr {C}})_* (\mu (\cdot \,|\, X_{H,\mathscr {C}})) \in \operatorname {\mathrm {Prob}}(\Omega _1^{H,\mathscr {C}})$
, we get a measurable map
$E^{H,\mathscr {C}}: \Omega _1^{H,\mathscr {C}} \to \Omega _2^{H,\mathscr {C},\text {ind}}$
defined
$\mathbb {P}$
-almost everywhere such that
$E^{H,\mathscr {C}}(\sigma )$
extends
$\sigma $
. Denote the composition
$E^{H,\mathscr {C}} \circ \sigma ^{H,\mathscr {C}}$
by
$\psi ^{H, \mathscr {C}}$
and write the image of x under this map as
$\psi ^{H, \mathscr {C}}_x$
. To summarize, for every pattern
$(H,\mathscr {C})$
, there is a measurable map
$\psi ^{H,\mathscr {C}} : X_{H,\mathscr {C}} \to \Omega _2^{H,\mathscr {C},\text {ind}}$
defined
$\mu $
-almost everywhere with the property that
$\psi ^{H, \mathscr {C}}_x$
extends
$\sigma ^{H,\mathscr {C}}_x$
.
It is now natural to define our desired cocycle
$\alpha $
on the equivalence relation
$R_2$
by the formula
$\alpha (x, T^g x) := \psi ^{\operatorname {\mathrm {pat}}_2(x)}_x(e, g)$
. It is then immediate to verify the two properties of
$\alpha $
claimed in the statement of Proposition 4.7. The fact that
$\alpha $
agrees with
$\alpha _0$
on
$R_1$
follows from the fact that
$\psi _{H,\mathscr {C}}$
extends
$\sigma ^{H,\mathscr {C}}$
and the claimed independence property of
$\alpha $
translates directly from the independence property that the
$\psi ^{H,\mathscr {C}}_x$
were constructed to have (see also the discussion after step 6 in the sketched proof of Proposition 4.7). In addition,
$\alpha $
is measurable because for each fixed g, the map
$x \mapsto \alpha (x, T^g x)$
is simply a composition of other maps already determined to be measurable. The only problem is that
$\alpha $
, when defined in this way, need not satisfy the cocycle condition. To see why, observe that the cocycle condition
$\alpha (x, T^h x) = \alpha (T^g x, T^h x) \circ \alpha (x, T^g x)$
is equivalent to the condition
$$ \begin{align} \psi_x^{\operatorname{\mathrm{pat}}_2(x)}(e,h) = \psi_{T^g x}^{\operatorname{\mathrm{pat}}_2(T^g x)}(e, hg^{-1}) \circ \psi_x^{\operatorname{\mathrm{pat}}_2(x)}(e,g). \end{align} $$
However, in defining the maps
$\psi ^{H,\mathscr {C}}$
, we have simply applied Theorem A.1 arbitrarily to each pattern separately, so
$\psi ^{\operatorname {\mathrm {pat}}_2(x)}$
and
$\psi ^{\operatorname {\mathrm {pat}}_2(T^g x)}$
have nothing to do with each other. However, we can fix this problem with a little extra work, and once we do, we will have defined
$\alpha : R_2 \to \operatorname {\mathrm {Aut}}(I,m)$
with all of the desired properties.
Start by declaring two patterns as equivalent if they are translates of each other, and fix a choice of one pattern from each equivalence class. As there are only countably many patterns in total, there is no need to worry about how to make this choice. For each representative pattern
$(H_0, \mathscr {C}_0)$
, apply Theorem A.1 arbitrarily to get a map
$\psi ^{H_0, \mathscr {C}_0}$
. This does not cause any problems because two patterns that are not translates of each other cannot appear in the same orbit (this follows from the easy fact that
$\operatorname {\mathrm {pat}}_2(T^g x) = g^{-1} \cdot \operatorname {\mathrm {pat}}_2(x)$
), so it does not matter that their
$\psi $
maps are not coordinated with each other. For convenience, let us denote the representative of the equivalence class of
$\operatorname {\mathrm {pat}}_2(x)$
by
$\operatorname {\mathrm {rp}}(x)$
. Now, for every
$x \in X$
, let
$g^*(x)$
be the unique element of G with the property that
$\operatorname {\mathrm {pat}}_2(T^{g^*(x)} x) = \operatorname {\mathrm {rp}}(x)$
. Note that the maps
$g^*$
and
$\operatorname {\mathrm {rp}}$
are both constant on each subset
$X_{H,\mathscr {C}}$
and are therefore measurable.
Now for an arbitrary pattern
$(H,\mathscr {C})$
and
$x \in X_{H,\mathscr {C}}$
, we define the map
$\psi ^{H,\mathscr {C}}$
by
$$ \begin{align*} \psi_x^{H,\mathscr{C}}(g,h) := \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)} ( g\cdot g^*(x)^{-1}, h\cdot g^*(x)^{-1} ). \end{align*} $$
Note that this is still just a composition of measurable functions, so
$\psi ^{H,\mathscr {C}}$
is measurable. All that remains is to verify that this definition satisfies (A.1). The right-hand side of (A.1) is
$$ \begin{align*} &\psi_{T^{g^*(T^g x)} T^g x}^{\operatorname{\mathrm{rp}}(T^gx)} ( e g^*(T^g x)^{-1}, hg^{-1} g^*(T^g x)^{-1} ) \circ \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)} ( e g^*(x)^{-1}, g g^*(x)^{-1} ) \\&\quad= \psi_{T^{g^*(x)g^{-1}}T^g x}^{\operatorname{\mathrm{rp}}(x)} ( (g^*(x)g^{-1})^{-1}, hg^{-1} (g^*(x)g^{-1})^{-1} ) \circ \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)} (g^*(x)^{-1}, g g^*(x)^{-1} ) \\&\quad= \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)} ( g g^*(x)^{-1} , hg^*(x)^{-1} ) \circ \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)} (g^*(x)^{-1}, g g^*(x)^{-1} ) \\&\quad= \psi_{T^{g^*(x)} x}^{\operatorname{\mathrm{rp}}(x)}( g^*(x)^{-1}, hg^*(x)^{-1} ), \end{align*} $$
which is, by definition, equal to the left-hand side of (A.1) as desired.
This, together with the discussion surrounding (A.1), shows that if we construct the maps
$\psi ^{H,\mathscr {C}}$
in this way, then making the definition
$\alpha (x, T^g x) = \psi _x^{\operatorname {\mathrm {pat}}_2(x)}(e,g)$
gives us a true measurable cocycle with all of the desired properties. Finally, to extend the definition of
$\alpha $
to
$R_n$
with
$n \geq 3$
, repeat the exact same process, except it is even easier because there is no need to force any independence. The maps
$\psi ^{H,\mathscr {C}}$
only need to be measurable selections into the space
$\Omega _2^{H,\mathscr {C}}$
, and then everything else proceeds in exactly the same way.
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