In this [paper by Jon Chaika and Howard Masur](https://arxiv.org/pdf/1410.1576.pdf) it is remarked at the end of page 1 that for an interval exchange transformation $T$ with $n$-intervals, one can bound the number of invariant measures generic with respect to $T$ and $T^{-1}$ by $n$. Generic in this situation means that there exists a point in the interval which equidistributes under the action of $T$ and $T^{-1}$ with respect to the measure. Furthermore, the remark states that one can use a "standard Rokhlin tower argument" to deduce the aforementioned fact.


It's not clear to me how the Rokhlin lemma can be used to prove this fact. If I had to guess where the bound on the number of generic measures is coming from, I would assume each generic measure was the limit in $\mathbb{P}(\mathbb{R}^n)$ of the rows of the cocycle associated to Rauzy induction on the IET. However, I can't quite seem to make this idea work, and it's not clear to me at all where Rokhlin's lemma plays a role.


It is also unclear to me how one would use the fact that one requires the measure to generic for $T$ and $T^{-1}$: is there a relation between the Rauzy cocyles associated to $T$ and $T^{-1}$?


Here are the questions I would like some input on, summarized in a list form.


1. Why is the number of generic (with respect to $T$ and $T^{-1}$) measures on an $n$-interval IET $T$ bounded above by $n$?
2. Is it true that the generic measures correspond to the limits of the rows of the Rauzy cocycle? Where does Rokhlin's lemma enter the picture?
3. Why is it important that the measure be generic with respect to $T$ and $T^{-1}$?



Bounding the number of generic measures on an interval exchange transformation
I ended up asking Jon Chaika this question, and he gave a rough sketch, in which I filled in the details, so if there are any mistakes, they are probably mine.


Suppose that there were $n+1$ measures $\{\mu\_1, \ldots, \mu\_{n+1}\}$ that were generic with respect to $T$ and $T^{-1}$, and let $\{x\_1, \ldots, x\_{n+1}\}$ be their generic points. Pick a subset $U$ that gets assigned different measure by each $\mu\_i$ (it's possible that no single set will do, in which case we pick finite collection of subsets, but the idea is the same). Let $2 \varepsilon$ be the smallest difference in the size of the sets: we have that $2 \varepsilon > 0$.


Since each $x\_i$ is $\mu\_i$-generic with respect to both $T$ and $T^{-1}$, there exists a large enough time $t\_i \in \mathbb{N}$ such that the average time the orbit of $x$ under both $T$ and $T^{-1}$ spend in $U$ for time $t \geq t\_i$ is within $\varepsilon$ distance of $\mu\_i(U)$.
\begin{align\*}
\left|\frac{\#(\{T^{j}x\_i\}\_{j=0}^{t} \cap U)}{t} - \mu\_i(U) \right| &\leq \varepsilon \\
\left|\frac{\#(\{T^{-j}x\_i\}\_{j=0}^{t} \cap U)}{t} - \mu\_i(U) \right| &\leq \varepsilon 
\end{align\*}


Now pick a very small sub-interval of our original IET $T$, and consider the first return map to this smaller sub-interval. By making the subinterval sufficiently small, we can ensure that the first return time for any point in the subinterval is greater than $2 \max(t\_i)$. Furthermore, since our original IET had $n$ intervals, the induced IET will also have $n$ intervals.


Label the $n$ intervals of the induced IET as $\{J\_1, \ldots, J\_n\}$ and let the return times of each of these intervals be $\{r\_1, \ldots, r\_n\}$. We make the following observations about the orbits of $J\_i$:
$$T^aJ\_c \cap T^b J\_d = \emptyset$$
where $0 \leq a \leq r\_c$, $0 \leq b \leq r\_d$ and $a \neq b$ and $c \neq d$. This means that in the time period we're interested in, the orbits are disjoint. But in the very same time period, the orbits fill out $[0,1]$.
$$[0, 1] = \bigcup\_{i=1}^{n} \bigcup\_{a=0}^{r\_i - 1} T^a J\_i$$


By the pigeonhole principle, at least $2$ of the $n+1$ generic points must be in the same $J\_i$ orbit. Without loss of generality, let's assume $x\_1$ and $x\_2$ lie in the orbit of $J\_1$. More specifically, we have the following:
\begin{align\*}
x\_1 &\in T^{s\_1}J\_1 \\
x\_2 &\in T^{s\_2}J\_1
\end{align\*}
Depending on whether $s\_1 < \frac{r\_1}{2}$ or $s\_1 > \frac{r\_1}{2}$, and similarly for $s\_2$, the analysis splits into four cases.


* Case 1 ($s\_1 < \frac{r\_1}{2}$ and $s\_2 < \frac{r\_1}{2}$): In this case, we consider the average number of times $T^i x\_1$ and $T^j x\_2$ intersect $U$ as $i$ goes from $s\_1$ to $r\_1$, and $j$ goes from $s\_2$ to $r\_1$. Since both these orbits cover the entirety of $\left[\frac{r\_1}{2}, r\_1\right]$, the orbit averages must be really close, specifically within $\frac{3\varepsilon}{2}$ distance, but since $r\_1$ is much larger than $t\_1$ and $t\_2$, the orbit averages must also be close to the space averages $\mu\_1(U)$ and $\mu\_2(U)$, which are at least $2\varepsilon$ distance, leading to a contradiction.
* Case 2 ($s\_1 > \frac{r\_1}{2}$ and $s\_2 > \frac{r\_1}{2}$): Dealing with this case is a matter of replacing $T$ with $T^{-1}$, which is where genericity with respect to $T^{-1}$ comes into play.


The other two cases are also dealt with similarly, by picking $T$ or $T^{-1}$ depending on whether $s\_i < \frac{r\_1}{2}$ or $s\_i > \frac{r\_1}{2}$.


This answers questions 1 and 3, but not 2. Also, there's a more general version of this result for systems with linear block growth (see Cyr and Kra's paper: <https://arxiv.org/pdf/1505.02748.pdf>).


