Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing

Question

The following result is on page 26 of this paper by Ferenczi [PDF].

Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) \to \mu(A)\mu(B)$ as $n \to \infty$).

Proof. By using Rokhlin towers, we can find a set $E$ such that $\mu(E \cap T^{h_n} E) \ge \frac{1}{L} \mu(E)$, where $h_n$ is some length of a word $W_{n,r}$ and $L$ is the maximum of the length of substitution $\sigma_n$. [Length of the $\sigma_n$ refers to the length of the elements in the images of the $\sigma_n$.]

Context. This corollary comes after Ferenczi's formulation of a finite exact rank (no spacers) construction for minimal systems of sublinear complexity (language complexity function $p(n)$ bounded by $Cn$ for $n \ge 1$, for some $C$). Basically, there are words $\{\{W_{n,r}\}_{1 \le r \le d} \}_{n \in \mathbb{N}}$ that generate the language (every word of the language is in $W_{n,0}$ for sufficiently large $n$) and are such that each $W_{n,r}$ is a concatenation of words in $\{W_{n-1,r}\}_{1\le r\le d}$. Each concatenation relationship can be "encoded" in the form of substitutions $\sigma_i$ as in the following example. \begin{align*} W_{19,0} &= W_{18,9} W_{18,3} W_{18,4}\\ 0 &\mapsto 934 \end{align*}

Question. I am having trouble fleshing out Ferenczi's proof of the above corollary. I understand that if I find $E$ satisfying $\mu(E) \le \frac{1}{2L}$ as well as $\mu(E \cap T^{h_n} E) \ge \frac{1}{L} \mu(E)$ for a sequence $h_n \to \infty$, then we have $$\mu(E \cap T^{h_n} E)-\mu(E)^2 \ge \frac{1}{2L} \mu(E),$$ giving our desired contradiction.

However, I am at a loss for how to select $E$. The only reasonable candidate I can think of is the bases of all the Rokhlin towers at some stage $n$, i.e., $E=\bigcup_{r=1}^d [W_{n,r}]$, where $[W_{n,r}]$ denotes the cylinder associated with $W_{n,r}$. However, I don't see how to get the above inequality to work for a sequence $h_k$.

Answer 1

I know how to prove something similar that I learned from an argument of A. Katok in his 1980 paper Interval exchange transformations and some special flows are not mixing. But you need to assume that there is a uniform upper bound on the number of return words to any given factor (I am not sure if how it fits with linear complexity).

The proof is very nice (and the article well written, so you might have a look there for details). Consider a shift $X \subset A^\mathbb{Z}$ such that for any factor of $X$ it has at most $m$ return words. You consider a given word $w$ and all its return word $w_1$, $\ldots$, $w_s$ (here $s \leq m$ by hypothesis). That gives you a partition into Rokhlin towers of $X$. More precisely $$X = \bigcup_{i=1}^s \bigcup_{k=0}^{t_i-1} f^k(\Delta_i)$$ where $\Delta_i = [w_i w]$ is the bottom of a Rokhlin tower above $w$, $f$ is the shift and $t_i = |w_i|$. Then you consider a measurable set $A$ that is a finite union of elements of the partition $\{f^k [w_i w]: i=1,\ldots,s \ \text{and}\ k=0,\ldots,t_i-1\}$.

Now, for each of the $w_i$ you consider their return words $w_{ij}$ where $j=1,\ldots,s_i$ (as before $s_i \leq m$). Let $\Delta_{ij} = [w_{ij} w_i|$ and $t_{ij} = |w_{ij}|$. Then, by definition $f^{t_{ij}} \Delta_{ij} \subset \Delta_i$. Consequently, $$f^{t_{ij}} (\Delta_{ij}^n) \subset \Delta_i^n$$ for $n = 0, \ldots, t_i$ where $\Delta_i^n = f^n \Delta_i$. (this is the equation (5) in Katok's article).

From that and using the fact that you have Rokhlin towers, it is easy to deduce that $$A \subset \bigcup_{i=1}^s \bigcup_{j=1}^{s_i} f^{-t_{ij}}(A).$$ Then for at least one of the $t_{ij}$ you have $\mu(A \cap f^{t_{ij}} A) \geq 1/m^2 \mu(A)$. From there it is then easy to derive a contradiction.

Conclusion: I do not understand why the $L$ in Ferenczi article stands for the length of the substitution and not the number of letters. Secondly, it might be possible to adapt the proof using Rokhlin towers given by the desubstitution (for which the number of towers is clearly bounded from Ferenczi construction). And I am pretty sure that it is what he was thinking about.

Answer 2

I think Ferenczi has something very different in mind: take a ginormous Rokhlin tower: of height $M$ (MUCH bigger than $h_n$) and microscopic error set. Let $E$ consist of a union of levels of the tower: $\lceil h_n/2\rceil$ levels in $E$ followed by $\lfloor h_n/2\rfloor$ levels out of $E$ repeated until the levels of the tower are all assigned to $E$ or $E^c$. Then $T^{h_n}E$ is almost exactly the same as $E$ except for the levels that had the misfortune to be within $h_n$ of the top of the tower. So in particular, $\mu(E)$ and $\mu(E\cap T^{h_n}E)$ are both very close to $\frac 12$.