Proof of Lemma 2.6

Let us first consider, on a probability space $(\Omega ,\mathbb {P})$ , a random variable $Y=(Y(i):i\in \mathbb {Z})$ taking values in $\mathbb {A}^{\mathbb {Z}}$ , and such that $\mathscr {L}(Y)=\mu $ . Let us fix some integers $i,j,k$ such that $j\le k$ and $i\in \{j,\ldots ,k,k+1\}$ . For each block $w\in \mathbb {A}^{\{j,\ldots ,k\}}$ and each $a\in \mathbb {A}$ , it is straightforward to check from the construction of $\tau $ that there exists a unique block $w'\in \mathbb {A}^{\{j,\ldots ,k,k+1\}}$ such that:

• • $w'(i)=a$ ;

• • $( w'(h)+w'(h+1):j\le h\le k)=w$ .

Let us denote this block $w'$ by $\tau _{a,i}^{-1}(w)$ . We then have the following equivalence:

$$ \begin{align*} \tau Y[j,k]=w \Longleftrightarrow \text{ there exists } a\in\mathbb{A},\ Y[j,k+1]=\tau_{a,i}^{-1}(w). \end{align*} $$

But by construction of $\mu $ , all the $|\mathbb {A}| $ events $(Y[j,k+1]=\tau _{a,i}^{-1}(w))$ , $a\in \mathbb {A}$ , have the same probability $|\mathbb {A}|^{-(k-j+2)}$ (and are of course disjoint). It follows that, for each $a\in \mathbb {A}$ ,

$$ \begin{align*} \mathbb{P}(Y(i)=a\,|\,\tau Y[j,k]=w) = \mathbb{P}(Y[j,k+1]=\tau_{a,i}^{-1}(w)\,|\,\tau Y[j,k]=w) = \frac{1}{|\mathbb{A}| }. \end{align*} $$

Since this holds for any block $w\in \mathbb {A}^{\{j,\ldots ,k\}}$ , this proves that $Y(i)$ is independent of $\tau Y[j,k]$ . Then, since this is true for each $j,k$ such that $i\in \{j,\ldots ,k,k+1\}$ , $Y(i)$ is independent of $\tau Y$ .

Let us apply this with $Y=X_n$ for some $n\le 0$ : we get that, for each $i_n\in \mathbb {Z}$ , $X_n(i_n)$ is independent of $\tau X_n=X_{n-1}$ . Now if we have an arbitrary sequence $(i_n)_{n\le 0}$ of integers, we note that for each $n\ge 0$ , the random variables $X_m(i_m)$ , $m\le n-1$ , are all ${\mathscr {F}_{n-1}^{\tau ^{-1}}}$ -measurable, hence $X_n(i_n)$ is independent of $\boldsymbol {\Sigma }(X_m(i_m):m\le n-1)$ .

Proof For a fixed m, we prove the result by induction on n. If $n=m$ the result is obvious. Assume now that, for some $m+1\le n\le 0$ , the result holds up to $n-1$ . Then $X_{n-1}[i_m,i_m+n-1-m]$ is measurable with respect to $\Sigma ( X_m(i_m),\ldots , X_{n-1}(i_{n-1}))$ . Remembering the notation $\tau _{a,i}^{-1}(w)$ from the proof of Lemma 2.6, and since by assumption we have $i_n\in \{i_m,\ldots ,i_m+n-m\}$ , we can then write

$$ \begin{align*} X_n[i_m,i_m+n-m] = \tau_{X_n(i_n),i_n}^{-1}(X_{n-1}[i_m,i_m+n-1-m]), \end{align*} $$

hence $X_n[i_m,i_m+n-m]$ is measurable with respect to $\Sigma ( X_m(i_m),\ldots , X_n(i_n))$ as claimed.

Proof of Lemma 2.7

The sequence $(i_n)$ defined in the statement of Lemma 2.7 obviously satisfies (5). Moreover, we have $i_m\to -\infty $ as $m\to -\infty $ , and for any fixed $n\le 0$ , $i_m+n-m\to +\infty $ as $m\to -\infty $ . By application of Lemma 2.8, it follows that $X_n$ is measurable with respect to $\boldsymbol {\Sigma }(X_m(i_m):m\le n)$ . But conversely, all the random variables $X_m(i_m)$ , $m\le n$ , are ${\mathscr {F}_{n}^{\tau ^{-1}}}$ -measurable. Hence these variables generate ${\mathscr {F}_{n}^{\tau ^{-1}}}$ .

Proof Since ${\mathscr {F}_{0}^{{\tau _{\varepsilon }}}}$ is generated by the countable family of random variables $(X_n(i):$ $n\leq 0,i\in \mathbb {Z} )$ , it is sufficient by Proposition 2.11 to check that, for each integer $k\ge 0$ , the random variable $Y_k:=(X_n[-k,k]:-k\le n\leq 0 )$ is I-cosy. In fact, we will see at the end of the proof that it is enough by stationarity of the process X to consider simpler targets, which are the random variables of the form $X_0[-k,k]$ , $(k\ge 0)$ .

So we fix an integer $k\ge 0$ , we consider the target $X_0[-k,k]$ , and we fix a real number $\delta>0$ . To check the I-cosiness of $X_0[-k,k]$ , we have to construct on some probability space $(\Omega ,\mathbb {P})$ two copies $X'$ and $X"$ of the process X, such that:

• • for some $n_0\le 0$ ,

  (7) $$ \begin{align} \mathscr{F}^{\prime}_{n_0}=\boldsymbol{\Sigma}(X^{\prime}_m:m\le n_0)\text{ and }\mathscr{F}^{\prime\prime}_{n_0}=\boldsymbol{\Sigma}(X^{\prime\prime}_m:m\le n_0)\text{ are independent} \end{align} $$
  (to ensure that the filtrations $\mathscr {F}'$ and $\mathscr {F}"$ generated by the negative parts of these process are independent in the distant past);

• • for each $n_0+1\le n\le 0$ ,

  (8) $$ \begin{align} \mathscr{L}(X^{\prime}_{n} | \mathscr{F}^{\prime}_{n-1}\vee\mathscr{F}^{\prime\prime}_{n-1}) = {\tau_{\varepsilon}}(X^{\prime}_{n-1}, \cdot) \text{ and }\mathscr{L}(X^{\prime\prime}_{n} | \mathscr{F}^{\prime}_{n-1}\vee\mathscr{F}^{\prime\prime}_{n-1}) = {\tau_{\varepsilon}}(X^{\prime\prime}_{n-1}, \cdot) \end{align} $$
  (to get the joint immersion of the filtrations $\mathscr {F}'$ and $\mathscr {F}"$ ); and

• • the copies $(X^{\prime }_0[-k,k])$ and $(X^{\prime \prime }_0[-k,k])$ of the target random variable satisfy

  (9) $$ \begin{align} \mathbb{P}(X^{\prime}_0[-k,k] = X^{\prime\prime}_0[-k,k])> 1-\delta. \end{align} $$

Here is how we will proceed. We fix some $n_0\le 0$ and consider a probability space $(\Omega ,\mathbb {P})$ in which we have two independent copies $(X^{\prime }_n:n\le n_0)$ and $(X^{\prime \prime }_n:n\le n_0)$ of $(X_n:n\le n_0)$ , and an independent family of i.i.d. random variables ${(U_n(i):n\ge n_0+1,i\in \mathbb {Z})}$ which are all uniformly distributed on the interval $[0,1]$ . These random variables will be used to construct inductively the error processes $(\xi ^{\prime }_n(i):n\ge n_0+1,i\in \mathbb {Z})$ and $(\xi ^{\prime \prime }_n(i):n\ge n_0+~1, i\in \mathbb {Z})$ and the rest of the Markov processes $(X^{\prime }_n:n\ge n_0+1)$ and ${(X^{\prime \prime }_n:n\ge n_0~+~1)}$ through the formula

$$ \begin{align*} X^{\prime}_{n}:=\tau X^{\prime}_{n-1} + \xi^{\prime}_{n}\quad\text{and}\quad X^{\prime\prime}_{n}:=\tau X^{\prime\prime}_{n-1} + \xi^{\prime\prime}_{n}. \end{align*} $$

We will use the auxiliary process $Z:=X'-X"$ , and we note that for $n\ge n_0+1$ we have

(10) $$ \begin{align} Z_n=\tau Z_{n-1}+\xi^{\prime}_n-\xi^{\prime\prime}_n. \end{align} $$

Rewriting (9) with this notation, we want to achieve the coupling in such a way that, provided $|n_0|$ is large enough,

(11) $$ \begin{align} \mathbb{P}(Z_0[-k,k] = (0_{\mathbb{A}},\ldots,0_{\mathbb{A}}))> 1-\delta. \end{align} $$

We explain now how we construct the error processes inductively. Assuming that, for some $n\ge n_0+1$ , we have already defined the processes $X'$ and $X"$ up to time $n-1$ , we choose for each $i\in \mathbb {Z}$ two random maps $g^{\prime }_{n,i}$ and $g^{\prime \prime }_{n_i}$ from $[0,1]$ to $\mathbb {A}$ and set $\xi ^{\prime }_{n}(i):=g^{\prime }_{n,i}(U_n(i))$ and $\xi ^{\prime \prime }_{n}(i):=g^{\prime \prime }_{n,i}(U_n(i))$ . These maps are random in the sense that they depend on the realizations of the processes up to time $n-1$ . However, they have to comply with the required law for the errors, which we ensure by choosing them among the family of maps $(g_a)_{a\in \mathbb {A}}$ defined as follows. Recalling the definition (3) of $\tilde {\varepsilon }$ and the formulation (4) of the law of the error, we cut the interval $[0,1]$ into $|\mathbb {A}| +1$ subintervals: the first $|\mathbb {A}|$ of them are of the same length $\tilde {\varepsilon }/|\mathbb {A}|$ , and we denote them by $J_b$ , $b\in \mathbb {A}$ (they correspond to the uniform measure part of the error); the last one is $[\tilde {\varepsilon },1]$ . Then, for each $a\in \mathbb {A}$ , we define the map $g_a:[0,1]\to \mathbb {A}$ by setting

$$ \begin{align*} g_a(u):= \begin{cases} a+b &\text{ if }u\in J_b\ (b\in\mathbb{A}),\\ {0_{\mathbb{A}}} &\text{ if }u\in [\varepsilon,1]. \end{cases} \end{align*} $$

By choosing $g^{\prime }_{n,i}$ and $g^{\prime \prime }_{n_i}$ in the set $\{g_a: a\in \mathbb {A}\}$ , we get the correct conditional law of the errors $\xi ^{\prime }_n$ and $\xi ^{\prime \prime }_n$ knowing $\mathscr {F}^{\prime }_{n-1}\vee \mathscr {F}^{\prime \prime }_{n-1}$ , and we have (8). It remains to explain how we make this choice for our purposes.

Figure 2

Choice of $g^{\prime }_{n,i}$ and $g^{\prime \prime }_{n,i}$ when we apply the strategy $S_a$ .

Given a site $(n,i)$ , for each $a\in \mathbb {A}$ we define the strategy $S_a$ as the choice $(g^{\prime }_{n,i},g^{\prime \prime }_{n,i}):=(g_a,g_{{0_{\mathbb {A}}}})$ (see Figure 2). In this way, when we apply the strategy $S_a$ , we obtain

$$ \begin{align*} \xi^{\prime}_n(i)-\xi^{\prime\prime}_n(i)=\begin{cases} a &\text { if }U_n(i)\in[0,\tilde{\varepsilon}),\\ {0_{\mathbb{A}}} &\text{ otherwise.} \end{cases} \end{align*} $$

The special case $a={0_{\mathbb {A}}}$ gives $\xi ^{\prime \prime }_n(i)-\xi ^{\prime }_n(i)={0_{\mathbb {A}}}$ with probability $1$ , and thus the choice of the strategy $S_{0_{\mathbb {A}}}$ on a given site $(n,i)$ ensures that, locally, the evolution of the Z process is given by the deterministic action of the cellular automaton $\tau $ (remember (10)).

The set of sites $(n,i)$ for which we have to define $g^{\prime }_{n,i}$ and $g^{\prime \prime }_{n,i}$ is partitioned into ‘diagonals’ ${\mathcal {D}} (j)$ , $j\in \mathbb {Z}$ , where

$$ \begin{align*} {\mathcal{D}} (j) := \{ (n,i) / n\geq n_0+1,\ i=j-n\} \end{align*} $$

(see Figure 3). We observe that, for $i<j\in \mathbb {Z}$ , any error added on some site of ${\mathcal {D}} (j)$ has no influence on $Z_0(i)$ .

Figure 3

The choice of the local strategies to construct the real-time coupling.

On all the diagonals ${\mathcal {D}} (j)$ , $j<-k$ or $j>k$ , we systematically choose the strategy $S_{0_{\mathbb {A}}}$ , so that Z follows the deterministic evolution of the cellular automaton $\tau $ in these areas. It remains to explain the choice of the strategies on the sites of the diagonals ${\mathcal {D}} (j)$ , $-k\leq j\leq k$ .

Consider an integer n, $n_0<n\leq 0$ , and assume that we know the Z process up to time $n-1$ . We compute $\tau ^{|n-1|}(Z_{n-1})$ , and according to what we get we make the following choices.

• • If $\tau ^{|n-1|}(Z_{n-1})[-k,k]=({0_{\mathbb {A}}},\ldots ,{0_{\mathbb {A}}})$ , we can easily ensure that $Z_0[-k,k]=({0_{\mathbb {A}}},\ldots ,{0_{\mathbb {A}}})$ by choosing the strategy $S_{{0_{\mathbb {A}}}}$ for all the following sites. In this case we set $j_{n-1}:=k+1$ .

• • Otherwise, we define $j_{n-1}$ as the smallest integer $j\in \{-k,\ldots ,k\}$ such that

  $$ \begin{align*} \tau^{|n-1|}(Z_{n-1})(j)\neq {0_{\mathbb{A}}}, \end{align*} $$
  and we apply the strategy $S_{{0_{\mathbb {A}}}}$ at all sites $(n,i)$ except when $(n,i)$ falls on $\mathcal {D}_{j_{n-1}}$ (that is, when $i=j_{n-1}-n$ ), where we apply $S_{-a_{n-1}}$ with $a_{n-1}:= \tau ^{|n-1|}(Z_{n-1})(j_{n-1})$ .

With this method, if $j_{n-1}\in \{-k,\ldots ,k\}$ , the only i on line n for which $Z_{n,i}$ is not determined is $i=j_{n-1}-n$ , and the value of $Z_{n,j_{n-1}-n}$ now depends only on the random variable $U_n(j_{n-1}-n)$ :

• • If $U_n(j_{n-1}-n) \in [0,\tilde {\varepsilon })$ , the difference $\xi ^{\prime }_n(j_{n-1}-n)-\xi ^{\prime \prime }_n(j_{n-1}-n)$ takes the value $-a_{n-1}$ . This exactly compensates the deterministic action of $\tau $ for the target site $(0,j_{n-1})$ , which ensures that, at the next step, we get $\tau ^{|n|}(Z_{n})(j_{n-1})={0_{\mathbb {A}}}$ , and thus $j_n\ge j_{n-1}~+~1$ .

• • Otherwise, we have $\xi ^{\prime }_n(j_{n-1}-n)-\xi ^{\prime \prime }_n(j_{n-1}-n)=0$ , so that $\tau ^{|n|}(Z_{n})(j_{n-1})=a_{n-1}$ and $j_n=j_{n-1}$ .

We construct the real-time coupling inductively by applying the above method for $n_0+1\le n\le 0$ . How we build the coupling for times $n\geq 0$ is irrelevant; we can, for example, choose always the strategy $S_{{0_{\mathbb {A}}}}$ for these positive times. Now we have to prove that (11) is achieved for $|n_0|$ large enough.

We inductively define random times $T(-k),\ldots ,T(k)$ . We start with $T(-k)$ , which only depends on the random variables $U_n(i)$ , $(n,i)\in {\mathcal {D}} (-k)$ :

$$ \begin{align*} T(-k):=\min\{n \geq n_0+1 :\, U_n(-k-n) \in [0,\tilde{\varepsilon}) \}. \end{align*} $$

(With the usual convention that $\min \emptyset =+\infty $ , but observe that $T(-k)<\infty $ with probability 1.) The random variable $T(-k)-n_0$ then follows a geometric law of parameter $\tilde {\varepsilon }$ . And by construction of the coupling, for each $n_0+1\le n\le 0$ we have

$$ \begin{align*} T(-k)\leq n \Longrightarrow j_n> -k. \end{align*} $$

Then, if we have already defined $T(j)$ for some $-k\le j\le k-1$ , we set

$$ \begin{align*} T(j+1):=\min\{n\geq T(j)+1 :\, U_n(j+1-n) \in [0,\tilde{\varepsilon}) \}. \end{align*} $$

Note that $T(j+1)$ only depends on $T(j)$ and on the random variables $U_n(i)$ , $(n,i)\in {\mathcal {D}} (j+1)$ . And by construction of the coupling, we get by induction that, for each $j\in \{-k,\ldots ,k\}$ and each $n\leq 0$ ,

$$ \begin{align*}T(j)\leq n \Longrightarrow j_n> j.\end{align*} $$

In particular,

(12) $$ \begin{align} T(k)\leq 0 \Longrightarrow j_0> k \Longrightarrow Z_0[-k,k]=({0_{\mathbb{A}}},\ldots,{0_{\mathbb{A}}}). \end{align} $$

For $j\in \{-k,\ldots ,k-1\}$ , the difference $T(j+1) -T(j)$ is distributed according to a geometric law of parameter $\tilde {\varepsilon }$ , and is independent of $T(-k),\ldots ,T(j)$ . Hence $T(k)-n_0$ is distributed as the sum of $2k+1$ independent geometric random variables of parameter $\tilde {\varepsilon }$ . Therefore, as $\tilde {\varepsilon }$ and k are fixed, we get

(13) $$ \begin{align} \mathbb{P}(T(k)\leq 0) {\xrightarrow[{n_0}\to{-\infty}]{}} 1. \end{align} $$

By (13) and (12), we get that (11) is satisfied for $|n_0|$ large enough.

This proves that for each $k\ge 0$ , the target $X_0[-k,k]$ is I-cosy. Now, the same argument works also to prove the I-cosiness of $X_{-k}[-2k,2k]$ for each $k\ge 0$ . This gives us a real-time coupling up to time $-k$ for which

$$ \begin{align*} \mathbb{P}(Z_{-k}[-2k,2k]=({0_{\mathbb{A}}},\ldots,{0_{\mathbb{A}}}))>1-\delta. \end{align*} $$

Note that

$$ \begin{align*} Z_{-k}[-2k,2k]=({0_{\mathbb{A}}},\ldots,{0_{\mathbb{A}}}) \Longrightarrow \text{for all } 0\le \ell\le k,\ \tau^\ell Z_k[-k,k]=({0_{\mathbb{A}}},\ldots,{0_{\mathbb{A}}}). \end{align*} $$

So we can continue this coupling by always using the strategy $S_{{0_{\mathbb {A}}}}$ on lines $-k+1,\ldots ,0$ to prove the I-cosiness of $Y_k=(X_n[-k,k]:-k\le n\leq 0 )$ .

Proof Let $\mu $ be a Q-invariant probability measure on ${\mathcal {X}}$ , and let $\nu $ be any probability measure on ${\mathcal {X}}$ . We want to prove that, for each continuous function $f:{\mathcal {X}}\to \mathbb {R}$ and each $\varepsilon>0$ , for each n large enough we have

$$ \begin{align*} \bigg|\int_{{\mathcal{X}}} f\, d(\nu Q^n) - \int_{{\mathcal{X}}} f\, d\mu\bigg| \le\ \varepsilon. \end{align*} $$

Given f and $\varepsilon $ , by uniform continuity of f on the compact $\mathcal {X}$ , we get $\delta>0$ such that,

$$ \begin{align*} \text{for all } x,y\in{\mathcal{X}},\quad d(x,y)\leq \delta\Longrightarrow | f(x)-f(y) | < \varepsilon/2. \end{align*} $$

We can also assume that $\delta \|f\|_\infty <\varepsilon /4$ .

By uniform cosiness, we can take $M>0$ such that, given any integer $n_0\leq -M$ , there exists a family of probability measures $(m_{x',x"})_{x',x"\in {\mathcal {X}}}$ associated to this $\delta $ as in the statement of the definition of uniform cosiness. We then define the probability measure $m_{\mu ,\nu }$ on $({\mathcal {X}}^{\{n_0,\ldots ,0\}})^2$ by

$$ \begin{align*} m_{\mu,\nu} := \int_{{\mathcal{X}}} \int_{{\mathcal{X}}} m_{x',x"}\, d\mu(x')\, d\nu(x"). \end{align*} $$

Under $m_{\mu ,\nu }$ , the processes $(X^{\prime }_n)_{n_0\leq n\leq 0}$ and $(X^{\prime \prime }_n)_{n_0\leq n\leq 0}$ are two Markov chains with transitions given by Q, with respective initial laws $\mathscr {L}(X^{\prime }_{n_0})=\mu $ and $\mathscr {L}(X^{\prime \prime }_{n_0})=\nu $ . We then have $\mathscr {L} (X^{\prime }_{0})=\mu {Q}^{|n_0|}\mu = \mu $ by invariance of $\mu $ , and $ \mathscr {L} (X^{\prime \prime }_{0})={\nu Q}^{|n_0|}$ . The uniform cosiness yields

$$ \begin{align*} m_{\mu,\nu}(d(X^{\prime}_0,X^{\prime\prime}_0)>\delta) = \int_{{\mathcal{X}}} \int_{{\mathcal{X}}} \underbrace{m_{x',x"}(d(X^{\prime}_0,X^{\prime\prime}_0)>\delta)}_{<\delta}\, d\mu(x')\, d\nu(x") < \delta. \end{align*} $$

We then have

$$ \begin{align*} \bigg|\int_{{\mathcal{X}}} f\, d(\nu{Q}^{|n_0|}) - \int_{{\mathcal{X}}} f\, d\mu\bigg| & \leq \mathbb{E}_{m_{\mu,\nu}}[ | f(X^{\prime\prime}_0)-f(X^{\prime}_0) | ] \\ & \leq \int_{d(X^{\prime}_0,X^{\prime\prime}_0)>\delta} | f(X^{\prime\prime}_0)-f(X^{\prime}_0) | \,dm_{\mu,\nu} + \varepsilon/2 \\ & \le 2\delta\|f\|_\infty + \varepsilon/2 \le \varepsilon.\\[-38pt] \end{align*} $$

Proof Let $(\Omega ,\mathbb {P},T)$ be the dynamical system where the factor filtration is defined, and let $X=(X_n)_{n\le 0}$ be a process generating $\mathscr {F}$ , where the coordinates $X_n$ are independent T-processes: $X_n=(X_n(i))_{i\in \mathbb {Z}}=(X_n(0)\circ T^i)_{i\in \mathbb {Z}}$ .

We have to check the dynamical I-cosiness with respect to $(\mathscr {F},T)$ of a target random variable Y which is $\mathscr {F}_0$ -measurable. It is enough to consider the case where Y is measurable with respect to $\boldsymbol {\Sigma }(X_{n_0+1},\ldots ,X_0)$ for some $n_0\le -1$ .

On the product dynamical system $(\Omega \times \Omega ,\mathbb {P}\otimes \mathbb {P},T\times T)$ , we have two independent copies $X^{(1)}$ and $X^{(2)}$ of X, whose coordinates are independent $(T\times T)$ processes. Let us now consider the two copies $X'$ and $X"$ of X defined by $X':=X^{(1)}$ , and

$$ \begin{align*} X^{\prime\prime}_n := \begin{cases} X^{(2)}_n & \text{ if }n\le n_0,\\ X^{(1)}_n & \text{ if }n\ge n_0+1.\ \end{cases} \end{align*} $$

Define $\mathscr {F}'$ (respectively, $\mathscr {F}"$ ) as the filtration generated by $X'$ (respectively, $X"$ ). Then $((\mathscr {F}',T\times T),(\mathscr {F}",T\times T))$ is an $n_0$ -independent dynamical coupling of $(\mathscr {F},T)$ . Moreover, since by construction we have $X^{\prime }_n=X^{\prime \prime }_n$ for $n\ge n_0+1$ , the corresponding copies of $Y'$ and $Y"$ of Y satisfy $Y'=Y"$ . This concludes the proof.

Proof Since cosiness is preserved by isomorphism, we can assume without loss of generality that $(\mathscr {F},T)$ is immersed in some dynamically I-cosy factor filtration $(\mathscr {G},T)$ . Let $X=(X_n)_{n\le 0}$ be a process generating $\mathscr {F}$ , where the coordinates $X_n$ are T-processes. Let Y be an $\mathscr {F}_0$ -measurable random variable, taking values in the Polish metric space $(E,d)$ , and let $\delta>0$ . Since $\mathscr {F}_0\subset \mathscr {G}_0$ , and since $(\mathscr {G},T)$ is dynamically I-cosy, there exists a dynamical real-time coupling $((\mathscr {G}',S),(\mathscr {G}",S))$ of $(\mathscr {G},T)$ , independent in the distant past, such that the corresponding copies $Y'$ and $Y"$ of Y satisfy $d(Y',Y")<\delta $ with probability at least $1-\delta $ . But the generating process X is $\mathscr {G}_0$ -measurable, so we can consider its copies $X'$ and $X"$ in $\mathscr {G}'$ and $\mathscr {G}"$ , respectively. Let $\mathscr {F}'$ (respectively, $\mathscr {F}"$ ) be the filtration generated by $X'$ (respectively, $X"$ ). Since these copies are provided by a dynamical isomorphism, the coordinates of $X'$ and $X"$ are S-processes, hence we get two dynamically isomorphic copies $(\mathscr {F}',S)$ and $(\mathscr {F}",S)$ of the factor filtration $(\mathscr {F},T)$ . Since $\mathscr {G}'$ and $\mathscr {G}"$ are jointly immersed, and $\mathscr {F}'$ (respectively, $\mathscr {F}"$ ) is immersed in $\mathscr {G}'$ (respectively, $\mathscr {G}"$ ), $\mathscr {F}'$ and $\mathscr {F}"$ are jointly immersed. Hence we get a dynamical real-time coupling $((\mathscr {F}',S),(\mathscr {F}",S))$ of the factor filtration $(\mathscr {F},T)$ . Moreover this coupling is independent in the distant past because this is the case for $((\mathscr {G}',S),(\mathscr {G}",S))$ . But $Y'$ and $Y"$ are respectively $\mathscr {F}^{\prime }_0$ - and $\mathscr {F}^{\prime \prime }_0$ -measurable, as Y is $\mathscr {F}_0$ -measurable, so they are also the corresponding copies of Y in the dynamical real-time coupling $((\mathscr {F}',S),(\mathscr {F}",S))$ .

Proof The strategy is to show that the factor filtration $({\mathscr {F}_{}^{\tau ^{-1}}},\sigma )$ , and more precisely the target $X_0(0)$ , is not dynamically I-cosy. For this, we consider a real-time dynamical coupling of this factor filtration in a dynamical system $(\Omega ,\mathbb {P},T)$ , so that we have two copies $X'=(X^{\prime }_n)_{n\leq 0}$ and $X"=(X^{\prime \prime }_n)_{n\leq 0}$ of the process $(X_n)$ , where for each $n\leq 0$ ,

• • $X^{\prime }_n=(X^{\prime }_n(i))_{i\in \mathbb {Z}}=\tau ^{|n|} X^{\prime }_0$ ,

• • $X^{\prime \prime }_n=(X^{\prime \prime }_n(i))_{i\in \mathbb {Z}}=\tau ^{|n|} X^{\prime \prime }_0$ ,

and for each $i\in \mathbb {Z}$ ,

• • $X^{\prime }_n(i)=X^{\prime }_n(0)\circ T^i$ ,

• • $X^{\prime \prime }_n(i)=X^{\prime \prime }_n(0)\circ T^i$ .

For each $n\leq 0$ , set $Z_n:=X^{\prime }_n-X^{\prime \prime }_n$ . Then, for each n, $Z_n=(Z_n(i))_{i\in \mathbb {Z}}=(Z_n(0)\circ T^i)_{i\in \mathbb {Z}}$ is a stationary T-process taking values in $\mathbb {A}$ . And since $\tau $ is an endomorphism of the group $\mathbb {A}^{\mathbb {Z}}$ , we also have $Z_n=\tau ^{|n|}Z_0$ for each $n\leq 0$ .

Assume that this coupling is $n_0$ -independent for some $n_0\leq 0$ ; in other words, $X^{\prime }_{n_0}$ and $X^{\prime \prime }_{n_0}$ are independent. Then the probability distribution of $Z_{n_0}$ is nothing more than $\mu =\mathcal {U}_{\mathbb {A}}^{\otimes \mathbb {Z}}$ . Therefore, the Kolmogorov–Sinaï entropy of this T-process is

$$ \begin{align*} h(Z_{n_0},T)=\log|\mathbb{A}|. \end{align*} $$

But we have

$$ \begin{align*} Z_{n_0}=\tau^{|n_0|}Z_0, \end{align*} $$

so that the T-process $Z_{n_0}$ is a factor of the T-process $Z_{0}$ . In particular, since the Kolmogorov–Sinaï entropy cannot increase when we pass to a factor, and observing that $Z_0$ is a T process taking values in $\mathbb {A}$

$$ \begin{align*} \log|\mathbb{A}| = h(Z_{n_0},T)\leq h(Z_{0},T)\leq \log|\mathbb{A}|. \end{align*} $$

However the only T-process taking values in $\mathbb {A}$ and whose entropy is $\log |\mathbb {A}|$ is the uniform Bernoulli process. It follows that the probability distribution of $Z_0$ is necessarily also $\mu $ . Therefore,

$$ \begin{align*} \mathbb{P}(X^{\prime}_0(0)=X^{\prime\prime}_0(0)) = \mathbb{P}(Z_0(0)=0) = \frac{1}{|\mathbb{A}|}, \end{align*} $$

which prevents the dynamical I-cosiness of the target $X_0(0)$ .