2.1 A combinatorial lemma

Let A be a finite alphabet and $L\subseteq A^{m}$ . We say that L admits a full binary treeif there are sets $L_{i}\subseteq A^{i}$ for $0\leq i\leq m$ , such that $L_{0}$ consists of the empty word, $L_{m}=L$ , and each $a\in L_{i}$ extends in exactly two ways to $a',a"\in L_{i+1}$ . Functions or random variables $W_{1},\ldots ,W_{m}:\Omega \rightarrow A$ admit a full binary tree if the image of $(W_{1},\ldots ,W_{m})$ does.

If we assume in addition that $|B|$ is bounded, then the conditional distribution of $W_{1}^{m}$ on the atoms of $\mathcal {F}$ will tend to independence as $\varepsilon \rightarrow 0$ , and the conclusion is trivial. However, in general, there is no such implication. The example to have in mind is when $(X_{n})_{n=-\infty }^{\infty }$ is an ergodic process of positive entropy and $W_{i}$ are consecutive blocks of n variables, $W_{i}=X_{in}\cdots X_{(i+1)n-1}$ . If $\mathcal {F}$ is trivial, or a factor algebra that does not exhaust the entropy, then for small $\eta>0$ and every $m\in \mathbb {N}$ and $\varepsilon>0$ , the random variables $W_{1},\ldots ,W_{m}$ will satisfy the hypotheses of the lemma if n is large enough.

Proof. Let E denote the event $W_{i}\in B_{i}$ for all i. We claim that we can assume that $\mathbb {P}(E)=1$ . By condition (1), the entropy of each $W_{i}$ on $E^{c}$ is bounded by $\log |B|<(1/\eta )\log |B_{i}|$ , and by condition (2), $\mu (E^{c})\leq m\varepsilon $ . Thus, restricting W and $\mathcal {F}$ to E changes $H(W|\mathcal {F})$ by no more than $\varepsilon '\sum \log |B_{i}|$ for an $\varepsilon '$ that can be made arbitrarily small by decreasing $\varepsilon $ , so after restricting, condition (3) still holds for a slightly larger $\varepsilon $ .

Now proceed by induction on m. When $m=1$ , we must show that, for a $1-\eta $ fraction of $F\in \mathcal {F}$ , the variable $W_{1}$ is not almost surely (a.s.) constant on F, or equivalently that the conditional entropy of $W_{1}$ on F is positive. Since $W_{1}\in B_{1}$ , we have a pointwise upper bound $\log |B_{1}|$ on the entropy of $W_{1}$ on each atom of $\mathcal {F}$ , and these conditional entropies average to $H(W_{1}|\mathcal {F})$ , which by conditions (3) and (1) is at least $\log |B_{1}|-\varepsilon \log |B|>(1-\varepsilon /\eta )\log |B_{1}|$ . As $\varepsilon $ decreases, these upper (pointwise) and lower (average) bounds approach each other, so, for small enough $\varepsilon $ , on a fraction of atoms approaching full measure, we get a lower pointwise bound of the same magnitude.

For $m>1$ , our assumption $W_{i}\in B_{i}$ implies the trivial bounds $H(W_{1}|\mathcal {F})\leq \log |B_{1}|$ and $H(W_{2}^{m}|\mathcal {F}\lor \sigma (W_{1}))\leq \sum _{i=2}^{m}\log |B_{i}|$ . Using the chain rule for entropy, condition (3) becomes

$$ \begin{align*} \bigg|(\log|B_{1}|-H(W_{1}|\mathcal{F}))+\bigg(\sum_{i=2}^{m}\log|B_{i}|-H(W_{2}^{m}|\mathcal{F}\lor\sigma(W_{1}))\bigg)\bigg|<\varepsilon\log|B|. \end{align*} $$

Since both summands on the left are non-negative, the bound $\varepsilon \log |B|$ applies to both, and this is condition (3) for the sequence $W_{2}^{m}$ and algebra $\mathcal {F}\lor \sigma (W_{1})$ . Thus, for $\varepsilon $ small, we can apply the induction hypothesis to $W_{2}^{m}$ and $\mathcal {F}\lor \sigma (W_{1})$ with parameters $\eta ^{2}/4$ instead of $\eta $ (the decreasing $\eta $ does not invalidate condition (1)).

We conclude that on $1-\eta ^{2}/4$ of the atoms of $\mathcal {F}\lor \sigma (W_{1})$ , the restriction of $W_{2}^{m}$ admits a full binary tree. Thus, on $1-\eta /2$ of the atoms $F\in \mathcal {F}$ , at least $1-\eta /2$ of the atoms $G\in \sigma (W_{1})$ (with respect to the conditional measure on F) are such that $W_{2}^{m}$ admits a full binary tree on $F\cap G$ . Also, arguing as in the case for $m=1$ , on a $1-\eta /2$ fraction of atoms $F\in \mathcal {F}$ , the conditional distribution of $W_{1}$ has large entropy, and in particular does not take any single value with probability higher than $1-\eta /2$ . It follows that, with probability $1-\eta $ over $F\in \mathcal {F}$ , there are two atoms $G_{1},G_{2}\in \sigma (W_{1})$ such that $W_{2}^{m}$ admit a full binary tree on $F\cap G_{1}$ , $F\cap G_{2}$ , and therefore $W_{1}^{m}$ admits a full binary tree on F.

2.2 Proof of the theorem in the symbolic case

In this section, we prove that if $\Omega \subseteq A^{\mathbb {Z}}$ is a subshift of positive entropy, then it is not doubly recurrent. By the variational principle, we may fix an ergodic shift invariant measure on $\Omega $ with positive entropy. Taking $X_{n}:\Omega \rightarrow A$ to be the coordinate projections, $(X_{n})_{n=-\infty }^{\infty }$ becomes an A-valued stationary ergodic process of positive entropy, and our goal is to find two realizations $u,v$ of $(X_{n})$ such that $(u,v)$ is not recurrent in $A^{\mathbb {Z}}\times A^{\mathbb {Z}}$ .

For $n\in \mathbb {N}$ , define an n-interval to be a set of the form $I=[kn,(k+1)n)$ with $k\in \mathbb {Z}$ , and say that I is odd or even according to the parity of k. Suppressing the parameter n from the notation, let

$$ \begin{align*} Y_{i}=(X_{2in},\ldots,X_{(2i+1)n-1}),\quad Z_{i}=(X_{(2i+1)n},\ldots,X_{(2i+1)n-1}) \end{align*} $$

be the blocks of variables in even and in odd n-intervals, respectively. Then, $\mathbb {Z}$ decomposes into disjoint n-intervals which are alternately odd and even, and similarly

$$ \begin{align*} X_{-\infty}^{\infty}=(\ldots,Y_{-2},Z_{-2},Y_{-1},Z_{-1},Y_{0},Z_{0},Y_{1},Z_{1},\ldots). \end{align*} $$

Let $h>0$ denote the entropy of the process $(X_{n})$ , so that

$$ \begin{align*} \frac{1}{n}H(X_{1}^{n})=h+o(1) \end{align*} $$

(all error terms are asymptotic as $n\rightarrow \infty $ ). It is not hard to see that for large n, the processes $(Y_{i})$ and $(Z_{i})$ are roughly independent, in the sense that the entropy of each is $\tfrac 12(h+o(1))$ , and their joint entropy is h. We will not use this property directly, although it is implicit in the proof below. Set

$$ \begin{align*} E^{-} & =\bigcup\{\mathrm{even}\ {n}\text{-}\mathrm{intervals\ to\ the\ left\ of}\ {0}\},\\E^{+} & =\bigcup\{\mathrm{odd}\ {n}\text{-}\mathrm{intervals\ to\ the\ right\ of}\ {0}\}. \end{align*} $$

Lemma 2.3. As $n\to\infty$ , we have

$$ \begin{align*} H(X_{-n}^{n-1}|\{X_{i}\}_{i\in E^{-}\cup E^{+}})=H(Z_{-1},Y_{0}|Y_{-\infty }^{-1},Z_{0}^{\infty })=2n(h+o(1)). \end{align*} $$

In the lemma, we have conditioned on both past and future times, and it is important to note that, in general, this can lead to a sharp decrease in entropy, even when the density of the times is small. For example, if one takes the even blocks in both directions, it can happen that $H(Y_{0}|Y_{-\infty }^{\infty })=0$ for all n, as can be shown using a construction similar to that in [Reference Ornstein and Weiss6]. The validity of the lemma relies crucially on the asymmetry between the blocks that we condition on in the past and in the future.

Proof of the lemma

The inequality $\leq $ is trivial since

$$ \begin{align*} H(X_{-n}^{n-1}|Y_{-\infty}^{-1},Z_{0}^{\infty})\leq H(X_{-n}^{n-1})=2n(h+o(1)). \end{align*} $$

Thus, we need only prove $\geq $ . Note that $X_{-n}^{n-1}=(Z_{-1},Y_{0})$ and

$$ \begin{align*} H(Z_{-1},Y_{0}|Y_{-\infty}^{-1},Z_{0}^{\infty}) & =H(Y_{0}|Y_{-\infty}^{-1},Z_{0}^{\infty})+H(Z_{-1}|Y_{-\infty}^{0},Z_{0}^{\infty})\\ & \geq H(Y_{0}|Y_{-\infty}^{-1},Z_{-\infty}^{\infty})+H(Z_{-1}|Y_{-\infty}^{\infty},Z_{0}^{\infty}). \end{align*} $$

It suffices to show that each of the summands on the right-hand side is $n(h+o(1))$ . We prove this for the first summand, the second is similar.

Fix a large integer $\ell $ and note that

$$ \begin{align*} 4\ell n(h+o(1)) & =H(X_{-2\ell n},X_{-\ell n+1},\ldots,X_{2\ell n-1})\\ & =H(Y_{-\ell},Z_{-\ell},Y_{-\ell+1},Z_{-\ell+1},\ldots,Y_{\ell-1},Z_{\ell-1})\\ & =H(Z_{-\ell}^{\ell-1})+H(Y_{-\ell}^{\ell-1}|Z_{-\ell}^{\ell-1})\\ & =H(Z_{-\ell}^{\ell-1})+\sum_{i=-\ell}^{\ell-1}H(Y_{i}|Y_{-\ell}^{i-1},Z_{-\ell}^{\ell-1})\\ & =2\ell n(h+o(1))+\sum_{i=-\ell}^{\ell-1}H(Y_{0}|Y_{-\ell-i}^{-1},Z_{-\ell-i}^{\ell-i-1}). \end{align*} $$

Rearranging and dividing by $2\ell $ , we have

$$ \begin{align*} \frac{1}{2\ell}\sum_{i=-\ell}^{\ell-1}H(Y_{0}|Y_{-\ell-i}^{-1},Z_{-\ell-i}^{\ell-i-1})=n(h+o(1)). \end{align*} $$

However, by Martingale convergence and monotonicity of entropy under conditioning,

$$ \begin{align*} \lim_{k,m\rightarrow\infty}H(Y_{0}|Y_{-k}^{-1},Z_{-k}^{m})=\inf_{k,m}H(Y_{0}|Y_{-k}^{-1},Z_{-k}^{m})=H(Y_{0}|Y_{-\infty}^{-1},Z_{-\infty}^{\infty}). \end{align*} $$

Inserting this in the previous equation and letting $\ell \rightarrow \infty $ gives

$$ \begin{align*} H(Y_{0}|Y_{-\infty}^{-1},Z_{-\infty}^{\infty})=n(h+o(1)) \end{align*} $$

as required.

Returning to the proof of the theorem, split $[-n,n)$ into disjoint intervals $I_{1},\ldots ,I_{8}$ of length $n/4$ (we can assume $n/4\in \mathbb {N}$ ) and note that every sub-interval $J\subseteq [-n,n)$ of length $n/2$ contains one of the $I_{i}$ . Let

$$ \begin{align*} W_{i}=(X_{p})_{p\in I_{i}},\quad i=1,\ldots,8. \end{align*} $$

We would like to apply Corollary 2.2 to $W_{1},\ldots ,W_{8}$ and the $\sigma $ -algebra $\mathcal {F}=\sigma (Y_{-\infty }^{-1},Z_{0}^{\infty })$ , with suitable sets $B,B_{1},\ldots ,B_{8}$ . To set things up, apply the Shannon–MacMillan theorem to find a set $A'\subseteq A^{n/4}$ of size

$$ \begin{align*} |A'|=2^{(h+o(1))n/4}=|A|^{(c+o(1))n} \end{align*} $$

for $c=\log 2/(4\log |A|)>0$ and satisfying

$$ \begin{align*} \sum_{a\in A'}\mu(a)=1-o(1). \end{align*} $$

Set $\eta =c/2$ , $B=A^{n/4}$ , and $B_{1},\ldots ,B_{8}=A'$ in the hypothesis of Corollary 2.2. Also, by Lemma 2.3,

$$ \begin{align*} H(W_{1}^{8}|Y_{-\infty}^{-1},Z_{0}^{\infty})=H(X_{-n}^{n-1}|Y_{-\infty}^{-1},Z_{0}^{\infty}) & =2(h-o(1))n, \end{align*} $$

but we also have $H(W_{i})=(h-o(1))\cdot n/2$ , so

$$ \begin{align*} H(W_{1}^{8}|Y_{-\infty}^{-1},Z_{0}^{\infty})\leq\sum_{i=1}^{8}H(W_{i})\leq2(h+o(1))n. \end{align*} $$

Combining these bounds,

$$ \begin{align*} \bigg|H(W_{1}^{8})-\sum_{i=1}^{8}H(W_{i})\bigg|=o(n), \end{align*} $$

which, assuming n is large, completes the hypotheses of Corollary 2.2.

The corollary now provides us with samples $u,v$ for the process satisfying:

• • $u,v$ are realized on the same atom of $\mathcal {F}=\sigma (Y_{-\infty }^{-1},Z_{0}^{\infty })$ , and hence $u,v$ agree on every n-interval in $E^{-}\cup E^{+}$ ;

• • $u|_{I_{i}}\neq v|_{I_{i}}$ for $i=1,\ldots ,8$ , that is, for each i, there is $j\in I_{i}$ with $u_{j}\neq v_{j}$ .

To conclude the proof, we must show that $(u,v)$ is not recurrent. Indeed, let $(u',v')$ be a shift of $(u,v)$ by k, with $|k|>2n$ .

• • Since $E^{-},E^{+}$ consist of n-intervals separated by gaps of length n, after shifting $E^{-}\cup E^{+}$ by k, some n-interval $J\subseteq (E^{-}\cup E^{+})+k$ will intersect $[-n,n)$ in a set of size at least $n/2$ , implying that $u^{\prime }_{j}=v^{\prime }_{j}$ for all $j\in J\cap [-n,n)$ .

• • Every interval of length $n/2$ in $[-n,n]$ and, in particular, the interval $J\cap [-n,n)$ contains one of the $I_{i}$ . Thus, there is then a $j\in I_{i}\subseteq J$ such that $u_{j}\neq v_{j}$ .

It follows that $(u,v)$ and $(u',v')$ differ on at least one coordinate $j\in [-n,n)$ . This holds for all shifts $(u',v')$ of $(u,v)$ by $2n$ or more, so $(u,v)$ is not recurrent.

2.3 The general case

Let $(X,T)$ be a topological system and $\mu $ a T-invariant Borel probability measure of positive entropy. Our plan is as follows.

First, we pass to a measure-theoretic factor $(X,\mu )\rightarrow (Y,\nu )$ , where Y is a subshift over a countable alphabet (for concreteness, $Y\subseteq \mathbb {N}^{\mathbb {Z}}$ ) and the entropy of Y is positive and finite. This factor will be defined using a partition $\mathcal {C}=\{C_{1},C_{2},\ldots \}$ of X (modulo $\mu $ ) into closed sets that are separated from each other in the sense that, for every n, no pair of atoms $C_{i},C_{j}$ with $i,j<n$ can be simultaneously $\delta _{n}$ -close to a third atom.

Next, pass to a further factor $(Y,\nu )\rightarrow (Z,\theta )$ , chosen so that Y has a finite generator relative to Z and, conditioned on Z, the relative entropy is positive. This factor is obtained by merging a large finite number of symbols $1,\ldots ,r$ in the alphabet of Y.

We can now run a relative version of the argument from the symbolic case on Y, conditioned on Z. This yields a pair of points $u,v\in Y$ that lie above the same point in Z and such that $(u,v)$ is not recurrent in $Y\times Y$ . More precisely, the lack of recurrence arises because every large enough shift $(u',v')$ of $(u,v)$ admits an index $j\in [-n,n)$ at which $u',v'$ display a common symbol k, while $u,v$ display distinct symbols from among $1,\ldots ,r$ .

Finally, taking preimages $x,y\in X$ of $u,v\in Y$ , respectively, we find that any large enough shift $(x',y')$ of $(x,y)$ can be brought, after another bounded shift, into an atom $C_{k}$ , whereas the corresponding shifts of $x,y$ lie in distinct atoms from among $C_{1},\ldots ,C_{r}$ . The separation properties of $\mathcal {C}$ now ensure that $(x',y')$ is at least $\delta _{r}$ -far from $(x,y)$ , which is non-recurrence.

We now give this argument in detail.

2.3.1 Step 1: constructing the factor $X\rightarrow Y$

We inductively construct a sequence of disjoint closed sets $C_{1},C_{2},\ldots \subseteq X$ that exhaust the measure $\mu $ .

Begin with a finite measurable partition $\mathcal {A}=\{A_{1},\ldots ,A_{n_{1}}\}$ of positive entropy and $\varepsilon>0$ , and choose closed sets $C_{i}\subseteq A_{i}$ such that $\mu (\bigcup _{i=1}^{n_{1}}C_{i})>1-\varepsilon _{1}$ .

Set $n_{k}=2^{k-1}n_{1}$ and suppose that after k steps, we have defined disjoint closed sets $C_{1},\ldots ,C_{n_{k}}$ . Let $\varepsilon _{k+1}$ be given. For $i=1,\ldots ,n_{k}$ , let $A_{i}^{k},\subseteq X$ denote the measurable sets that form the ‘Voronoi anuli’ of the sets $C_{1},\ldots ,C_{n_{k}}$ :

$$ \begin{align*} A_{i}^{k}=\left\{ x\in X\left|\begin{array}{@{}c@{}} d(x,C_{i})<d(x,C_{j})\text{ for }j<i\text{ and }\\ d(x,C_{i})\leq d(x,C_{j})\text{ for }j>i \end{array}\right.\!\right\} \setminus C_{i}. \end{align*} $$

The sets $A_{1}^{k},\ldots ,A_{n_{k}}^{k}$ are measurable, pairwise disjoint, and, together with $C_{1},\ldots ,C_{n_{k}}$ , they form a partition of X. Now choose closed sets $C_{n_{k}+i}\subseteq A_{i}^{k}$ , $i=1,\ldots ,n_{k}$ , satisfying $\mu (\bigcup _{i=1}^{n_{k+1}}C_{i})>1-\varepsilon _{k+1}$ . We have defined $C_{1},\ldots ,C_{n_{k+1}}$ .

Let $\mathcal {C}=\{C_{1},C_{2},\ldots \}$ denote the resulting family of sets. Observe the following.

1. (1) If $\varepsilon _{k}\rightarrow 0$ , then $\mathcal {C}$ is a partition of X up to $\mu $ -null sets.

2. (2) If $\varepsilon _{k}\rightarrow 0$ quickly enough, $H_{\mu }(\mathcal {C})<\infty $ . In particular, $h_{\mu }(T,\mathcal {C})<\infty $ . This is true because $n_{k}=2^{n}n_{1}$ partition elements added at stage k of the construction contribute at most $\varepsilon _{k}\cdot k\log n_{1}$ to the total entropy, so taking $\varepsilon _{k}=1/k^{3}$ , for example, gives $H_{\mu }(\mathcal {C})\leq \sum _{k}({1}/{k^{2}})\log n_{1}<\infty $ .

3. (3) If $\varepsilon _{k}\rightarrow 0$ quickly enough and $\varepsilon _{1}$ is small enough, then $h_{\mu }(T,\mathcal {C})>0$ . This is because for any $\varepsilon _{2},\varepsilon _{3},\ldots $ such that $H_{\mu }(\mathcal {C})<\infty $ , the Rohlin distance $H_{\mu }(\mathcal {A}|\mathcal {C})+H_{\mu }(\mathcal {C}|\mathcal {A})$ tends to zero as $\varepsilon _{1}\rightarrow 0$ , and hence $h_{\mu }(T,\mathcal {C})\rightarrow h_{\mu }(T,\mathcal {A})>0$ .

4. (4) For every $\ell \neq m$ ,

   $$ \begin{align*} \delta_{m,\ell}=\inf_{i\in\mathbb{N}}\{ \max\{d(C_{\ell},C_{i}),d(C_{m},C_{i})\}\}>0. \end{align*} $$
   Indeed, suppose $1\leq \ell ,m<n_{k}$ . Because the sets $C_{1},\ldots ,C_{n_{k}}$ are closed and disjoint, the infimum above is positive as i ranges over $\{1,\ldots ,n_{k}\}$ . For $i>n_{k}$ , each $C_{i}$ is contained in the Voronoi cell of exactly one of the previously defined sets and, in particular, either $d(C_{i},C_{\ell })\geq \tfrac 12d(C_{\ell },C_{m})$ or $d(C_{i},C_{m})\geq \tfrac 12d(C_{\ell },C_{m})$ . Thus, the infimum is positive over all i.

Assume that $\varepsilon _{k}$ have been chosen so that properties (1)–(3) above are satisfied and let $\pi :X\rightarrow \mathbb {N}^{\mathbb {Z}}$ be the associated measurable map,

$$ \begin{align*} \pi(x)_{i}=m\Longleftrightarrow\, T^{i}x\in C_{m}. \end{align*} $$

Let $\nu =\pi \mu $ be the push-forward measure, which is invariant under the shift S on $\mathbb {N}^{\mathbb {Z}}$ . Then,

$$ \begin{align*} h_{\nu}(S)=h_{\mu}(T,\mathcal{C})\in(0,\infty). \end{align*} $$

2.3.2 Step 2: constructing the factor $Y\rightarrow Z$

Let $\mathcal {D}$ denote the cylinder partition of $\mathbb {N}^{\mathbb {Z}}$ and let $\mathcal {D}^{(r)}$ denote the partition of Y obtained by merging the first r symbols into a single atom. Since $H_{\nu }(\mathcal {D})<\infty $ ,

$$ \begin{align*} H_{\nu}(\mathcal{D}^{(r)})\rightarrow0\quad\text{as }r\rightarrow\infty \end{align*} $$

and since $h_{\nu }(S)>0$ , we can choose r large enough that $H_{\nu }(D^{(r)})<h_{\nu }(S)$ , and hence

$$ \begin{align*} h_{\nu}(S,\mathcal{D}^{(r)})<h_{\nu}(S). \end{align*} $$

Consider the factor algebra $\mathcal {E}=\bigvee _{i\in \mathbb {Z}}S^{-i}\mathcal {D}^{(r)}$ . Evidently,

$$ \begin{align*} h_{\nu}(S|\mathcal{E})=h_{\nu}(S)-h_{\nu}(S|_{\mathcal{E}})>0. \end{align*} $$

Let $\mathcal {B}$ denote the partition of $\mathbb {N}^{\mathbb {Z}}$ obtained by identifying all symbols $r+1,r+2,\ldots $ , and observe that $\mathcal {D}=\mathcal {B}\lor \mathcal {D}^{(r)}$ , so $\mathcal {B}$ generates relative to $\mathcal {E}$ , and hence

$$ \begin{align*} h_{\nu}(S,\mathcal{B}|\mathcal{E})=h_{\nu}(S|\mathcal{E})>0. \end{align*} $$

2.3.4 Step 4: lifting $u,v$ to X

Let $x\in \pi ^{-1}(u)$ and $y\in \pi ^{-1}(v)$ . To conclude the argument, we shall show that $(x,y)$ is non-recurrent for $T\times T$ . Indeed, let

$$ \begin{align*} \delta=\min_{1\leq\ell<m\leq n}\delta_{m,\ell}>0. \end{align*} $$

Fix $i\in \mathbb {Z}$ with $|i|>n$ , and write $x'=T^{i}x$ and $y'=T^{i}y$ , so $\pi x'=u'$ and $\pi y'=v'$ . By assumption, there exists $j\in \{-n,\ldots ,n\}$ such that $\ell =u_{j}$ , $m=v_{j}$ satisfy $1{\kern-1pt}\leq{\kern-1pt} \ell {\kern-1pt}\neq{\kern-1pt} m{\kern-1pt}\leq{\kern-1pt} n$ , and there is a $k\in \mathbb {N}$ such that $u^{\prime }_{j}=v^{\prime }_{j}=k$ . In other words, $T^{j}x\in C_{\ell }$ , $T^{j}y\in C_{m}$ , and $T^{j}x',T^{j}y'\in C_{k}$ . Thus, using the max-metric $d_{\infty }$ on $X\times X$ ,

$$ \begin{align*} \delta & \leq\delta_{\ell,m}\\ & \leq\max\{d(C_{\ell},C_{k}),d(C_{k},C_{m})\}\\ & \leq\max\{d(T^{j}x,T^{j}x'),d(T^{j}y',T^{j}y)\}\\ & =d_{\infty}(T^{j}(x,y),T^{j}(x',y')), \end{align*} $$

where, for brevity, in the last line, we wrote T for $T\times T$ . Since $-n\leq j\leq n$ and T is a homeomorphism, this implies

$$ \begin{align*} d_{\infty}(T^{j}(x,y),T^{j}(x',y'))>\delta' \end{align*} $$

for some $\delta '>0$ depending only on $\delta ,n$ . Since i was arbitrary and $(x',y')=T^{i}(x,y)$ , this proves non-recurrence of $(x,y)$ .

3.1 The original argument

In the original proof, one begins with a measure-preserving system $(X,\mathcal {B},\mu ,T)$ with $X=\{1,\ldots ,a\}^{\mathbb {Z}}$ , T the shift, and $h_{\mu }(T)\in (0,\infty )$ . Let $\mathcal {P}=\{P_{1},\ldots ,P_{a}\}$ be the partition according to the symbol at time zero. Let $E\subseteq X$ be a null set; we wish to find $x,y\in X\setminus E$ with $\overline {d}(x,y)=0$ .

Choose an open set $U\supseteq E$ of measure small enough that the partition

$$ \begin{align*} \mathcal{Q}_{0}=\{X\setminus U,U\cap P_{1},\ldots,U\cap P_{a}\} \end{align*} $$

generates a factor $X_{0}$ of X of entropy less than $h_{\mu }(T)$ . Now one constructs a tower of measure-theoretic factors $X_{n}$ between X and $X_{0}$ ,

$$ \begin{align*} X\rightarrow\cdots\rightarrow X_{2}\rightarrow X_{1}\rightarrow X_{0} \end{align*} $$

such that $X_{n}$ is generated by a finite partition $\mathcal {Q}_{n}$ that refines $\mathcal {Q}_{n-1}$ , all the factors have entropy strictly less than $h_{\mu }(X)$ , and there are numbers $N_{1},N_{2}\cdots \in \mathbb {N}$ , such that:

Assuming this, for each n, use the fact that $X\rightarrow X_{n}$ has positive relative entropy to choose a pair $x^{(n)}\neq y^{(n)}$ that lie above the same point in $X_{n}$ . This means that $x^{(n)},y^{(n)}$ project to the same point in $X_{0}$ as well. By shifting the points if necessary, we can assume $x_{0}^{(n)}\neq y_{0}^{(n)}$ , and hence $x^{(n)},y^{(n)}\in X\setminus U$ , for otherwise, they would lie in different atoms of $\mathcal {Q}_{0}$ and hence project to different points in $X_{0}$ .

Now pass to a subsequence so that $x^{(n)}\rightarrow x$ and $y^{(n)}\rightarrow y$ . We still have $x_{0}\neq y_{0}$ so $x\neq y$ , and since $x^{(n)},y^{(n)}\in X\setminus U$ and U is open, also $x,y\in X\setminus U$ . However, every common shift $\widehat {x},\widehat {y}$ of $x,y$ are limits of corresponding shifts $\widehat {x}^{(n)},\widehat {y}^{(n)}$ of $x^{(n)},y^{(n)}$ , and $\widehat {x}^{(n)},\widehat {y}^{(n)}$ still project to the same point in $X_{n}$ , so they agree on the central block $[-N_{n},N_{n}]$ except for a $1/n$ -fraction of the coordinates. This means that the same is true for $\widehat {x},\widehat {y}$ . What this argument shows is that on every block of length $2N_{n}$ , the points $x,y$ agree on a $1-({1}/{n})$ fraction of the coordinates. Thus, $\overline {d}(x,y)<1/n$ for every n, and hence ${\overline {d}(x,y)=0}$ .

To construct the factor $X_{n}$ (the partition $\mathcal {Q}_{n}$ ), for a large $L_{n}\in \mathbb {N}$ , one wants to encode enough information about the $(\mathcal {P},L_{n})$ -names so as to reveal a $1-1/n$ fraction of the coordinates, but omit enough information that the entropy remains below $h_{\mu }(T)$ . To achieve this, one partitions the typical $(\mathcal {P},L_{n})$ -names into families of names that are close in the hamming distance (that is, differ in a small fraction of their coordinates), and combinatorial estimates show that one can do so in a way that replacing $(\mathcal {P},L_{n})$ -names by the name of the corresponding family gives the desired result.

3.2 The infinite-entropy case

The reason this proof fails when $h_{\mu }(T)=\infty $ is that the system does not have a finite generator (indexed by a finite alphabet); and if we use an infinite alphabet, we lose compactness and cannot pass to the limit $x,y$ of $x^{(n)},y^{(n)}$ .

Our observation is that the entire argument can be carried out relative to a factor Z of X, provided that the relative entropy is positive and finite. Indeed, a measure-theoretic factor Z of finite relative entropy can be obtained using standard techniques or, in a more heavy-handed manner, by appealing to the weak Pinsker property [Reference Austin1]. One can then realize the factor $X\rightarrow Z$ topologically with $X=Z\times \{1,a\}^{\mathbb {Z}}$ and the factor being a projection to the first coordinate. From this point, the entire argument proceeds as before. One should note that the construction of the extension $X_{n}$ of $X_{n-1}$ in Ornstein and Weiss’s original paper already involves a relative argument, considering typical names in X relative to $X_{n-1}$ , and this can be carried out just as well when $X_{n-1}$ contains the factor Z. We leave the details to the interested reader.