Proof. For convenience, we assume that the relative entropy is $1$ .

As in [Reference Glasner, Thouvenot and Weiss8], let $\mathcal {R} \subset \mathcal {X}$ be a finite relatively generating partition for $\mathbf {X}$ over $\mathbf {X}_0$ with entropy  $1$ (so that $\mathcal {R}$ is a Bernoulli partition independent of $\mathbf {X}_0$ ), and let $\mathcal {R}_0 \subset \mathcal {X}_0$ be a finite generator for $\mathbf {X}_0$ . Let $\mathcal {S}$ be the collection of Rokhlin cocycles with values in MPT $(I, \unicode{x3bb} )$ , where $\unicode{x3bb} $ is the normalized Lebesgue measure on the unit interval $I =[0,1]$ . Thus, an element $S \in \mathcal {S}$ is a measurable map $x \mapsto S_x\! \in $ MPT $(I, \unicode{x3bb} )$ , and we associate to it the skew product transformation

$$ \begin{align*} \hat{S}(x,u) = (Tx, S_x u)\quad (x \in X, u \in I). \end{align*} $$

Let $Y = X \times I$ and set $\mathbf {Y} = (Y, \mathcal {Y}, \mu \times \unicode{x3bb} )$ , with $\mathcal {Y} = \mathcal {X} \otimes \mathcal {C}$ .

Part I: By Theorem 4.1 of [Reference Glasner, Thouvenot and Weiss8], there is a dense $G_{\delta }$ subset $\mathcal {S}_0 \subset \mathcal {S}$ with $h(\hat {S}) = 1$ for every $S \in \mathcal {S}_0$ . We will first show that the collection of the elements $S \in \mathcal {S}_0$ for which the corresponding $\hat {S}$ is relatively Bernoulli over $\mathbf {X}_0$ forms a $G_{\delta }$ set.

As the inverse limit of relatively Bernoulli systems is relatively Bernoulli, see [Reference Thouvenot24, Proposition 7], to show that a transformation T on $(X, \mathcal {X}, \mu )$ is relatively Bernoulli over $\mathbf {X}_0$ , it suffices to show that for a refining sequence of partitions

$$ \begin{align*} \mathcal{P}_1 \prec \cdots \prec \mathcal{P}_n \prec \mathcal{P}_{n+1} \prec \cdots \end{align*} $$

such that the corresponding algebras $\hat {\mathcal {P}}_n$ satisfy $\bigvee _{n \in \mathbb {N}} \hat {\mathcal {P}}_n = \mathcal {X}$ , for each n, the process $(T, \mathcal {P}_n)$ is relatively very weak Bernoulli relative to $(T,\mathcal {R}_0)$ .

For each $n \in \mathbb {N}$ , let $\mathcal {Q}_n$ denote the dyadic partition of $[0,1]$ into intervals of size $1/2^n$ , and let

$$ \begin{align*} \mathcal{P}_n = \mathcal{R} \times \mathcal{Q}_n. \end{align*} $$

For any $S \in \mathcal {S}_0$ , the relative entropy of $\mathbf {Y} = \mathbf {X} \times [0,1]$ over $\mathbf {X}_0$ is also $1$ . Thus, for all n, we have

$$ \begin{align*} H\bigg(\mathcal{P}_n \mid \bigg(\bigvee\nolimits_{i= -\infty}^{-1}\hat{S}^{-i}\mathcal{P}_n\bigg) \vee \bigg(\bigvee\nolimits_{i= -\infty}^{\infty}\hat{S}^{-i}\mathcal{R}_0\bigg)\bigg) =1, \end{align*} $$

and for all $N \geq 1$ ,

$$ \begin{align*} H\bigg(\bigvee\nolimits_{i=0}^{N-1} \hat{S}^{-i}\mathcal{P}_n \mid \bigg(\bigvee\nolimits_{i= -\infty}^{-1}\hat{S}^{-i}\mathcal{P}_n\bigg) \vee \bigg(\bigvee\nolimits_{i= -\infty}^{\infty}\hat{S}^{-i}\mathcal{R}_0\bigg)\bigg) =N. \end{align*} $$

Therefore, we can find a suitably small $\delta>0$ such that for $k_0$ large enough,

$$ \begin{align*} H\bigg(\bigvee\nolimits_{i=0}^{N-1} \hat{S}^{-i}\mathcal{P}_n \mid \bigg(\bigvee\nolimits_{i= -k_0}^{-1}\hat{S}^{-i}\mathcal{P}_n\bigg) \vee \bigg(\bigvee\nolimits_{i= -k_0}^{k_0}\hat{S}^{-i}\mathcal{R}_0\bigg)\bigg) < N +\delta. \end{align*} $$

Now, conditioned on the partition

$$ \begin{align*} \bigg(\bigvee\nolimits_{i= -k_0}^{-1}\hat{S}^{-i}\mathcal{P}_n\bigg) \vee \bigg(\bigvee\nolimits_{i= -k_0}^{k_0}\hat{S}^{-i}\mathcal{R}_0\bigg), \end{align*} $$

the partition $\bigvee _{i=0}^{N-1}\hat {S}^{-i}\mathcal {P}_n$ will be $\eta $ -independent of

$$ \begin{align*} \bigg(\bigvee\nolimits_{i= -k}^{-k_0 -1}\hat{S}^{-i}\mathcal{P}_n\bigg) \vee \bigg(\bigvee\nolimits_{i= -k}^{-k_0 +1}\hat{S}^{-i}\mathcal{R}_0\bigg) \vee \bigg(\bigvee\nolimits_{i= k_0 +1}^{k} \hat{S}^{-i}\mathcal{R}_0\bigg) \end{align*} $$

for all $k \geq k_0$ for $\eta $ small enough (see Definition 5.1 in [Reference Glasner, Thouvenot and Weiss8] and the following discussion), so that the inequality (3) in §2 (with $\mathcal {P}_n$ replacing $\mathcal {R}$ ) for $k = k_0$ will imply (3) with $2\epsilon $ , for all $k> k_0$ .

Define the set $U(n, N_1, N_2, \epsilon , \delta )$ to consist of those $S \in \mathcal {S}_0$ that satisfy:

1. (1) $H(\bigvee _{i=0}^{N_1 -1} \hat {S}^{-i} \mathcal {P}_n \mid (\bigvee _{i=-N_2}^{-1} \hat {S}^{-i} \mathcal {P}_n) \vee (\bigvee _{i=-N_2}^{N_2} \hat {S}^{-i} \mathcal {R}_0 )) < N_1 + \delta $ ;

2. (2) $\bar {d}_{N_1} (\bigvee _{i=0}^{N_1 -1} \hat {S}^{-i} \mathcal {P}_n \upharpoonright A \cap B, \bigvee _{i=0}^{N_1 -1} \hat {S}^{-i} \mathcal {P}_n \upharpoonright A' \cap B ) < \epsilon ,$ for a set of atoms $A, A' \in \mathcal {G}, \ B \in \mathcal {G}_0,$ where $\mathcal {G} \subset \bigvee _{-N_2}^{-1}\hat {S}^{-i} \mathcal {P}_n, \ \mathcal {G}_0 \subset \bigvee _{-N_2}^{N_2} \hat {S}^{-i} \mathcal {R}_0$ and $(\mu \times \unicode{x3bb} )(\bigcup \{A\cap B : A \in \mathcal {G}, \ B \in \mathcal {G}_0\} )> 1 - \epsilon .$

Now the sets $U(n, N_1, N_2, \epsilon , \delta )$ are open (easy to check) and the $G_{\delta }$ set

$$ \begin{align*} \mathcal{S}_1 = \bigcap_{n, k, l} \bigcup_{N_1, N_2} U(n, N_1, N_2, 1/k, 1/l) \end{align*} $$

comprises exactly the elements $S \in \mathcal {S}_0$ for which the corresponding $\hat {S}$ is relatively Bernoulli over $\mathbf {X}_0$ . Thus, if $S \in \mathcal {S}_0$ is such that $\hat {S}$ is relatively Bernoulli, then for every $n, \epsilon , \delta $ , there are $N_1, N_2$ such that $S \in U(n, N_1, N_2, \epsilon , \delta )$ , and conversely, for every relatively Bernoulli $\hat {S}$ , the corresponding S is in $\mathcal {S}_1$ .

Part II: The collection $\mathcal {S}_1$ is non-empty. To see this, we first note that the Bernoulli system $\mathbf {X}_1$ admits a proper extension $\hat {\mathbf {X}}_1 \to \mathbf {X}_1$ which is also Bernoulli and has the same entropy. This follows e.g. by a deep result of Rudolph [Reference Rudolph20, Reference Rudolph21], who showed that every weakly mixing group extension of $\mathbf {X}_1$ is again a Bernoulli system. An explicit example of such an extension of the $2$ -shift is given by Adler and Shields [Reference Adler and Shields2]. Since $\hat {\mathbf {X}}_1$ is weakly mixing, the product system $\hat {\mathbf {X}} = \mathbf {X}_0 \times \hat {\mathbf {X}}_1$ is ergodic and $\hat {\mathbf {X}} \to \mathbf {X}_0$ is an element of $\mathcal {S}_1$ .

Now apply the relative Halmos theorem [Reference Glasner and Weiss9, Proposition 2.3] to deduce that the $G_{\delta }$ subset $\mathcal {S}_1$ is dense in $\mathcal {S}$ , as claimed.

Proof. By Austin’s weak Pinsker theorem [Reference Austin3], we can present $\mathbf {X}$ as a product system $\mathbf {X} = \mathbf {B} \times \mathbf {Z}$ , where $\mathbf {B}$ is a Bernoulli system with finite entropy. Thus, $\mathbf {X}$ is relatively Bernoulli over $\mathbf {Z}$ , and by Theorem 3.1, it follows that a generic extension $\hat {S}$ of $\mathbf {X}$ is relatively Bernoulli over $\mathbf {Z}$ . Therefore, for such $\hat {S}$ , the system $\mathbf {Y} = (X \times I, \mathcal {X} \times \mathcal {C}, \mu \times \unicode{x3bb} , \hat {S})$ is again of the form $\mathbf {Y} = \mathbf {B}' \times \mathbf {Z}$ with $\mathbf {B}'$ a Bernoulli system with the same entropy as that of $\mathbf {B}$ . By Ornstein’s theorem [Reference Ornstein17], $\mathbf {B} \cong \mathbf {B}'$ , whence also $\mathbf {X} \cong \mathbf {Y}$ , and our proof is complete.

Proof. For the $G_{\delta }$ part, we follow, almost verbatim, the proof of Theorem 3.1, where we now let $\mathcal {Q}_n$ denote the product dyadic partition of $I \times I$ into squares of size ${1}/{2^n} \times {1}/{2^n}$ and, with notation as in the proof of Theorem 3.1, we let $\mathcal {P}_n = \mathcal {R} \times \mathcal {Q}_n$ .

Thus, it only remains to show that the $G_{\delta }$ set $\mathcal {S}_1$ , comprising those $S \in \mathcal {S}_0$ for which $\check {S}$ is relatively Bernoulli on $W = X \times I \times I$ relative to $\mathbf {X}_0$ , is non-empty. Now, examples of skew products over a Bernoulli system with such properties are provided by Hoffman in [Reference Hoffman11]. The base Bernoulli transformation that Hoffman constructs for his example can be arranged to have arbitrarily small entropy by an appropriate choice of the parameters used in the construction in §4 (the skew product example is in §5 and the proof of Bernoullicity is in §5). Using such construction on $\mathbf {X}$ (where the cocycle is measurable with respect to the Bernoulli direct component of $\mathbf {X}$ ), we obtain our required extension of $\mathbf {X}$ . This completes our proof.

Proof. By the weak Pinsker theorem [Reference Austin3], we can present $\mathbf {X}$ as a product system ${\mathbf {X} = \mathbf {Z} \times \mathbf {B}}$ , where $\mathbf {B}$ is a Bernoulli system with finite entropy. Thus, $\mathbf {X}$ is relatively Bernoulli over $\mathbf {Z}$ , and by Theorem 3.4, it follows that a generic extension $\check {S}$ of $\mathbf {X}$ to $X \times I \times I$ is still relatively Bernoulli over $\mathbf {Z}$ . Thus, the extended system $\mathbf {W}$ on $W = X \times I \times I$ with $\check {S}$ action has the form $\mathbf {W} = \mathbf {Z} \times \mathbf {B}'$ with $\mathbf {B}'$ again a Bernoulli system.

Now, for the system $\mathbf {Y}$ , defined on $Y = X \times I$ by

$$ \begin{align*} \hat{S}(x,u) = (Tx, S_xu), \end{align*} $$

we have that the corresponding relative product system $\mathbf {Y} \underset {\mathbf {X}}{\times } \mathbf {Y}$ is isomorphic to $\mathbf {W}$ , which is a Bernoulli extension of $\mathbf {Z}$ and therefore, by Lemma 3.5, a relatively mixing extension of $\mathbf {Z}$ . A fortiori, $\mathbf {Y} \underset {\mathbf {X}}{\times } \mathbf {Y}$ is a relatively mixing extension of $\mathbf {X}$ and our proof is complete.

Proof. We first choose a strictly ergodic model $\mathbf {X} =(X, \mathcal {X}, \mu _0,T)$ for our system which is a subshift of $\{0,1\}^{\mathbb {Z}}$ . By the variational principle, this model will have zero topological entropy. (To see that such a model exists, see for example [Reference Denker, Grillenberger and Sigmund6], where this fact can be deduced from property (b) on pp. 281 and Theorem 29.2 on pp. 301.) Denote by $a_n$ the number of n-blocks in X so that $a_n$ is sub-exponential.

For $x_0 \in X$ and $\mathcal {Q}=\{Q_0,Q_1\}$ a partition of X, let

$$ \begin{align*} B_n(x_0,\epsilon) = \{x \in X : \bar{d}_n(Q_n(x), Q_n(x_0)) < \epsilon \}, \end{align*} $$

where for a point $x \in X$ and $n \geq 1$ , we write

$$ \begin{align*} Q_n(x) =\omega_0\omega_1\omega_2\ldots\omega_{n-1} \quad {\text{when}}\ x \in \bigcap_{i=0}^{n-1} T^{-i}(Q_{\omega_i}). \end{align*} $$

Lemma 4.4. For $\epsilon < {1}/{100}$ and $\delta < {1}/{100}$ , there is an N such that for all $n \geq N$ , if m is the minimal number such that there are points $x_1, x_2, \ldots , x_m$ with

$$ \begin{align*} \mu_0\bigg(\bigcup_{i=1}^m B_n(x_i,\epsilon)\bigg)> 1 -\delta, \end{align*} $$

then $m \leq a_{2n}$ .

Proof. Denote by $\mathcal {P} =\{P_1, P_2\}$ the partition of X according to the $0$ th coordinate. Given $\epsilon>0$ , there is some $k_0$ and a partition $\hat {\mathcal {Q}}$ measurable with respect to $\bigvee _{i=-k_0}^{k_0}T^i \mathcal {P}$ such that

$$ \begin{align*} d(\mathcal{Q}, \hat{\mathcal{Q}}) < \frac{\epsilon}{2}. \end{align*} $$

By ergodicity, there exists an N such that for $n \geq N$ , there is a set $A \subset X$ with $\mu _0(A)> 1 - \delta $ with

$$ \begin{align*} \bar{d}_n(Q_n(x), \hat{Q}_n(x)) < \epsilon \quad \text{for all } x \in A. \end{align*} $$

Let $\{\alpha _i\}_{i=1}^{\ell }$ be those atoms of $\bigvee _{i=-k_0}^{n + k_0}T^i \mathcal {P}$ such that $\alpha _i \cap A \not =\emptyset $ , so that $\ell \leq a_{n + 2k_0 +1}$ . Choose $x_i \in \alpha _i \cap A, \ 1\leq i \leq \ell $ . We claim that

$$ \begin{align*} A \subset \bigcup_{i=1}^{\ell} B_n(x_i,\epsilon). \end{align*} $$

For $x \in \bigcup _{i=1}^{\ell } \alpha _i$ , we denote by $i(x)$ that index such that $x \in \alpha _{i(x)}$ . Now, since x and $x_{i(x)}$ are in A, we have

$$ \begin{align*} \bar{d}_n(Q_n(x), \hat{Q}_n(x)) < \epsilon \quad {\text{and}} \quad \bar{d}_n(Q_n(x), \hat{Q}_n(x)) < \epsilon. \end{align*} $$

Since $x \in \alpha _{i(x)}$ , $\hat {Q}_n(x) = Q_n(x)$ . Therefore,

$$ \begin{align*} \bar{d}_n(Q_n(x), Q_n(x_{i(x)})) < 2\epsilon, \end{align*} $$

whence $x \in B_n(x_{i(x)}, \epsilon )$ . This proves our claim and we conclude that $m \leq \ell \leq a_{n +2k_0 +1}$ . Thus, for sufficiently large n, we indeed get $m \leq a_{2n}$ .

We will show that a generic extension of T to $(Y, \mu ) = (X \times [0,1],\mu _0 \times \unicode{x3bb} )$ , with $\unicode{x3bb} $ Lebesgue measure on $[0,1]$ , is not isomorphic to $\mathbf {X}$ . To do this, we will show that for a generic extension $\hat {S}$ , the partition $\mathcal {Q}$ of Y, defined by splitting $X \times [0,1]$ into $\{Q_0, Q_1\} = \{X \times [0,\tfrac 12], X \times [\tfrac 12,1]\}$ , will not satisfy the conclusion of this lemma.

Notation.

• • $\mathcal {S}$ is the Polish space comprising the measurable Rohklin cocycles $x \mapsto S_x \in \textrm {MPT}([0,1], \unicode{x3bb} )$ .

• • For $S \in \mathcal {S}$ , let $\hat {S}(x,u) =(Tx,S_xu)$ .

• • $Q_n^{\hat {S}}(y) =\omega _0\omega _1\omega _2\ldots \omega _{n-1}$ , where $y \in \bigcap _{i=0}^{n-1} \hat {S}^{-i}(Q_{\omega _i})$ .

• •

  $$ \begin{align*} C(\hat{S}, n, \epsilon, \delta) = \min\bigg\{&k : \text{there exists } y_1,y_2, \ldots,y_k \in Y, \\[-2pt] &{\text {such that}}\ \mu\bigg(\bigcup_{i=1}^k B_n^{\hat{S}}(y_i,\epsilon)\bigg)> 1 -\delta\bigg\}. \end{align*} $$

Define now

$$ \begin{align*} \mathcal{U}(N, \epsilon, \delta) = \{S \in \mathcal{S}: \text{there exists } n \geq N\ {\text{such that}}\ C(\hat S,n, \epsilon, \delta)> 2 a_{2n}\}. \end{align*} $$

This is an open subset of $\mathcal {S}$ (see e.g. [Reference Glasner, Thouvenot and Weiss8] for similar claims). We will show that, for sufficiently small $\epsilon $ and $\delta $ , it is dense in $\mathcal {S}$ .

First, consider the case $S_0 = {\textrm {id}}$ . Let $\eta>0$ be given and choose M so that $1/M < \eta $ . Now build a Rohklin tower for T, with base $B_0$ and heights $mM> N$ and $mM + 1$ for a suitable m, filling all of X (for this version of the Rokhlin lemma, see [Reference Weiss26, p. 32]). Let $B = B_0 \times [0,1]$ be the base of the corresponding tower in $(Y,\mu , \hat {S})$ . We modify $S_0 ={\textrm {id}}$ only on the levels $T^{jM-1}B_0$ for $1 \leq j \leq m$ , so that the new S will be within $\eta $ of $S_0$ . The Q-M names of the points in $T^{jM-1}B$ are constant for all $0 \leq j < m$ . We modify $S_0$ on the levels $T^{jM-1}B$ so that we see all possible $0$ - $1$ names for the M-blocks as we move up the tower with equal measure. A similar procedure is described as independent cutting and stacking and is explained in detail in §I.10.d in Shields’ book [Reference Shields23].

From this lemma, it follows that to achieve even $\tfrac 12$ as $\mu (\bigcup _{i=1}^L B_{mM}(y_i,\epsilon ))$ , we must have $L \cdot 2^{m(- 1/2 + H(2\epsilon , 1 -2\epsilon ))}> \tfrac 12$ , and hence

$$ \begin{align*} L \geq \tfrac{1}{2} \cdot 2^{m( 1/2 - H(2\epsilon, 1 -2\epsilon))}. \end{align*} $$

Since $a_n$ is sub-exponential, this lower bound certainly exceeds $a_{2mM}$ if m is sufficiently large. This shows that this modified S is an element of $\mathcal {U}(N,\epsilon ,\delta )$ .

A similar construction can be carried out for any $S \in \mathcal {S}$ . The main point that needs to be checked is that for small $\epsilon $ , no $B_M^{\hat {S}}(y,\epsilon )$ -ball can have measure greater than $\tfrac 12 + \epsilon $ .

Lemma 4.6. For any $\hat {S}$ and all $y_0$ ,

$$ \begin{align*} \mu(B_M^{\hat{S}}(y_0,\epsilon)) \leq \tfrac12 + \epsilon. \end{align*} $$

Proof. Let $Q_M^{\hat {S}}(y_0) = \omega _0\omega _1\ldots \omega _{M-1}$ . Then,

$$ \begin{align*} \bar{d}_M(Q_M^{\hat{S}}(y), Q_M^{\hat{S}}(y_0)) = \frac1M \sum_{i=0}^{M-1} \mathbf{1}_{Q_{\omega_i}}(\hat{S}^iy_0)(1 - \mathbf{1}_{Q_{\omega_i}}(\hat{S}^iy)), \end{align*} $$

and

$$ \begin{align*} \int_Y \bar{d}_M(Q_M^{\hat{S}}(y), Q_M^{\hat{S}}(y_0)) \, d\mu = \tfrac12. \end{align*} $$

Since $\bar {d}_M \leq 1$ , the measure of the set where $ \bar {d}_M(Q_M^{\hat {S}}(y), Q_M^{\hat {S}}(y_0)) \leq \epsilon $ cannot exceed $\tfrac 12 + \epsilon $ .

This lemma, which is formulated for the measure $\mu $ on the entire space Y, in fact holds as well for any level $L_j = \hat {S}^{jM}B$ in the tower, when we replace $\mu $ by the measure $\mu $ restricted to $L_j$ . This is so because the partition $\{Q_0, Q_1\}$ intersects each level of the tower in relative measure $\tfrac 12$ and $\hat {S}$ is measure preserving.

We now mimic the proof outlined for $S_0 = {\textrm {id}}$ and, given $S \in \mathcal {S}$ , using an independent cutting and stacking, we change $\hat {S}$ as follows. For the level $L_j = \hat {S}^{jM}B$ , consider the partition

$$ \begin{align*} \mathcal{R}_j = \bigvee\nolimits_{i=0}^{M-1} \hat{S}^{-i}(\mathcal{Q} \cap \hat{S}^{jM +i}B). \end{align*} $$

We change the transformation $\hat {S}$ at the transition from level $jM-1$ to level $jM$ , so that these partitions $\mathcal {R}_j$ will become independent.

We want to estimate the size of an $mM$ - $\epsilon $ ball around a point $y_0 \in B$ . If $y \in B$ belongs to this ball, there is a set $A \subset \{0,1,2,\ldots ,nM-1\}$ with $|A| \leq \epsilon \, mM$ where the $mM$ -names of y and $y_0$ differ. We need now a simple lemma.

Lemma 4.7. Let $A \subset \{0,1,\ldots ,mM-1\}$ such that $|A| \leq \epsilon \, mM$ . Denote $I_j = \{jM, jM+1,\ldots , jM+M -1\}, \ 0 \leq j < m-1$ . Let $J \subset \{0,1,\ldots ,m-1\}$ be the set of $\ell $ such that

$$ \begin{align*} |I_{\ell} \cap A| < \sqrt{\epsilon} M. \end{align*} $$

Then, $|J|> (1-\sqrt {\epsilon }) m$ .

Proof. Let $K = \{0,1,\ldots ,mM-1\} \setminus J$ . Then,

$$ \begin{align*} \epsilon \, mM \geq | \bigcup_{k \in K} I_k \cap A| \geq M \sqrt{\epsilon} |K|. \end{align*} $$

Thus, $|K| \leq \sqrt {\epsilon } m$ , whence $|J|> (1-\sqrt {\epsilon }) m$ .

Next, using Lemma 4.6 for each level of the form $T^{jM}B_0$ , we will estimate the size of an $mM$ - $\epsilon $ ball. So fix a point $y_0 \in B$ . If $y \in B_{mM}(y_0,\epsilon )$ , then by Lemma 4.7, there is a set of indices $J_y \subset \{1,2,\ldots ,m\}$ such that:

1. (1) $|J_y| \geq (1 -\sqrt {\epsilon })m$ ;

2. (2) for each $j \in J_y$ , $\hat {S}^{jM}y \in B_M(\hat {S}^{jM}y_0, \sqrt {\epsilon })$ .

The number of possible sets that satisfy item (1) is bounded by $2^{mH(\sqrt {\epsilon }, 1- \sqrt {\epsilon })}$ . By Lemma 4.6 and by the independence, for such a fixed $J_y$ , the measure of the set of points that satisfy item (2) is at most

$$ \begin{align*} \big(\tfrac12 + 2\sqrt{\epsilon}\big)^{m(1-\sqrt{\epsilon})}. \end{align*} $$

Write $(\tfrac 12 + 2\sqrt {\epsilon })^{1-\sqrt {\epsilon }} = 2^{-c}$ , where $c \geq c_0>0$ for all sufficiently small $\epsilon $ . Then,

$$ \begin{align*} 2^{-cm} \cdot 2^{m H(2\epsilon, 1 -2\epsilon)} = 2^{m(-c + H(2\epsilon, 1 -2\epsilon))} \leq 2^{-({m}/{2}) c_0}, \end{align*} $$

for $H(2\epsilon , 1 -2\epsilon ) \leq \tfrac 12 c_0$ . We now see that the measure of the ball $B_{mM}(y_0,\epsilon )$ is bounded by $2^{-({m}/{2}) c_0}$ .

This was done for $y_0 \in B$ and as in the proof of Lemma 4.5, we obtain the suitable estimations for any y in the tower over B. We conclude the argument as in the case $S= {\textrm {id}}$ and again it follows that the resultant modified S is an element of $\mathcal {U}(N,\epsilon ,\delta )$ .

Finally, for fixed sufficiently small $\epsilon $ and $\delta $ , setting

$$ \begin{align*} \mathcal{E} = \bigcap_{N=1}^{\infty} \, \mathcal{U}(N, \epsilon, \delta), \end{align*} $$

we obtain the required dense $G_{\delta }$ subset of $\mathcal {S}$ , where for each $S \in \mathcal {E}$ , the corresponding $\hat {S}$ is not isomorphic to T. In fact, if $\hat {S}$ would be isomorphic to T, then the isomorphism would take the partition $\mathcal {Q}$ of Y to a partition $\tilde {\mathcal {Q}}$ of X. Applying Lemma 4.4 to $\tilde {\mathcal {Q}}$ , we see that there is some N such that for all $n \geq N$ , the conclusion of the lemma holds. However, since $S \in \mathcal {E}$ , this is a contradiction.

Proof. By the assumption, there is a finite partition $\mathcal {P}$ of X such that $\{T^{(1)}_g \mathcal {P} \}_{g \in G_1}$ are independent, $\bigvee _{g \in G_1}T^{(1)}_g \mathcal {P}$ is independent of $\mathcal {X}_0$ , and together with $\mathcal {X}_0$ spans $\mathcal {X}$ . These properties are equivalent to having the relative entropy of $\{T^{(1)}_g \mathcal {P} \}_{g \in G_1}$ being equal to $H(\mathcal {P})$ , and having $\{T^{(1)}_g \mathcal {P} \}_{g \in G_1}$ separating points relative to $X_0$ . By the first fact above, these properties persist for $\{T^{(2)}_g \mathcal {P} \}_{g \in G_2}$ and thus, using the same cocycle, the $G_2$ -extension is also relatively Bernoulli.

Proof. By [Reference Ornstein and Weiss18, Reference Ornstein and Weiss19], there is a measure-preserving transformation $T_0 : X_0 \to X_0$ such that orbits of $T_0$ coincide with G-orbits on $X_0$ , and such that $T_0$ has zero entropy. The G-factor map $\mathbf {X} = \mathbf {X}_0 \times \mathbf {X}_1\to \mathbf {X}_0$ is given by a constant cocycle whose constant value is the Bernoulli action on the Bernoulli factor $\mathbf {X}_1$ . We use this cocycle, now viewed as a cocycle on the equivalence relation defined by $T_0$ , to define an extension $T : X \to X$ . By [Reference Rudolph and Weiss22], the relative entropy of such a generic T over $T_0$ is the same as that of the G-action $\mathbf {X}$ over $\mathbf {X}_0$ . By Lemma 5.2, the extension of $\mathbb {Z}$ -systems $\pi : T \to T_0$ is again relatively Bernoulli. Applying Theorem 3.1 to $\pi $ , we conclude that a dense $G_{\delta }$ subset $\mathcal {S}_1(\mathbb {Z})$ of extensions of T is such that each $\hat {S} \in \mathcal {S}_1(\mathbb {Z})$ is relatively Bernoulli over $T_0$ . Finally, applying Lemma 5.2 again, we conclude that the corresponding set of extensions $\mathcal {S}_1(G)$ is a dense $G_{\delta }$ subset of $\mathcal {S}(G)$ and that for each $S \in \mathcal {S}_1(G)$ , the corresponding G-system is relatively Bernoulli over $\mathbf {X}_0$ .