The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2

Abstract

We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension $d-\frac{3}{2}$ [in the $(d-1)$-dimension space of d-IETs] for $d\ge 5$. For $d=4$ this was shown by Athreya–Chaika and for $d\in \{2,3\}$ the set is known to have dimension $d-2$. This provides lower bounds on the Hausdorff dimension of non-weakly mixing IETs and, with input from Al-Saqban et al. (Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension, 2017. arXiv:1711.10542), identifies the Hausdorff dimension of non-weakly mixing IETs with permutation $(d,d-1,\ldots ,2,1)$ when d is odd.

Appendix

We denote by A matrices of freedom on LHS. Recall from the proof of Proposition 9.3, the definition of a path of Rauzy induction $\gamma (x,m)$ being $\pi _s$ via $\pi '$ isolated. We wish to prove

Lemma 13.1

There exists $\rho >0$ and $m_0$ so that for any hyperplane H

$$\begin{aligned} \lambda _{d-1}(\{x \in \Delta \times \pi _L: \exists \gamma (x,m)\ \pi _s\ \text {via}\ \pi ' \text {isolated}\ m\le m_0\ \text {with}\ A(x,m) \Delta \cap H =\emptyset \})>\rho . \end{aligned}$$

With the same notations as above, via usual balanced estimates we have:

Corollary 13.2

Given $\zeta$ there exists $\rho _1$ such that if $M=M(x,r)$ a matrix of Rauzy induction so that

• $\frac{|C_i(M)}{|C_{i'}(M|}<\zeta$ for all $1\le i,i'\le d-2$,

• $\pi (R^ry)=\pi _L$.

and if H is any hyperplane contained in ${\text {span}}_{\Delta }(C_1(M),\ldots ,C_{d-2}(M))$ then

$$\begin{aligned}&\lambda _{d-3}(\{y\in {\text {span}}_{\Delta }(C_1(M),\ldots ,C_{d-2}(M)): \exists m\le m_0 \text { with } \gamma (R^ry,m)\ \pi _s\ \text {via}\ \pi '\ \text {isolated and } \\&\quad MA(R^ry,m)\Delta \cap H =\emptyset \})>\rho _1 \lambda _{d-3}({\text {span}}_{\Delta }(C_1(M),\ldots ,C_{d-2}(M))). \end{aligned}$$

In order to prove Lemma 13.1 we need the following simple lemma first.

Lemma 13.3

There is constant $\epsilon _0$ such that if a hyperplane in $\Delta$ intersects the radius $\frac{1}{2}$ ball about the $e_1$ vertex it cannot intersect the $\epsilon _0$ ball about every other vertex.

Proof

The proof is by contradiction. If the lemma is false there is a sequence of hyperplanes $H_n$ defined by $a_{1,n}x_1+a_{2,n}x_2+\cdots + a_{d,n}x_n=1$ which intersect the $\frac{1}{n}$ neighborhoods of $e_j$ for $j>1$ and the $\frac{1}{2}$ neighborhood of $e_1$. This forces the coefficients $a_{j,n}$ to be bounded. Passing to a subsequence and taking a limit we find that the limiting hyperplane H would contain $e_j$ for $j>1$, and intersect the $\frac{1}{2}$ neighborhood of $e_1$. Then H must be of form

$$\begin{aligned} a_1x_1+x_2+\cdots +x_d=1. \end{aligned}$$

But this is not a hyperplane subset of $\Delta$. $\square$

Proof of Lemma 13.1

Let $\epsilon _0$ be from the last lemma. It suffices to show that for any $i>1$ there exists a bounded length path of Rauzy induction whose corresponding subsimplices are contained in $B(e_i,\epsilon _0)$, and there exists a path whose corresponding subsimplex of Rauzy induction is contained in $B(e_1,\frac{1}{2})$.

In the second case consider the path where 1 wins $d-3$ consecutive times. It reaches $\pi '$ and then after 1 beats 2 it returns to $\pi _s$. So the first interval is longer than the sum of the other intervals, so its length is at least $\frac{1}{2}$.

In the first case $(i>1)$ starting at $\pi _L$ have $d-2$ beat 1 then $2, \ldots , i-1$. Then have i beat $d-2, d-3,\ldots i+1$ for $\frac{4}{\epsilon _0}$ consecutive times. Then have $d-2$ beat i, $i+1\ldots 2$ then have it beat $1, \ldots , d-3$. This implies that $x_{d-2}>2(x_1+\cdots x_{i-1}+x_{i+1}\ldots +x_{d-3})$. So,

$$\begin{aligned} x_i> & {} \frac{4}{\epsilon _0}(x_{d-2}-(x_1+\cdots +x_{i-1})>\frac{4}{\epsilon _0}\frac{1}{2} x_{d-2}\\> & {} \frac{4}{\epsilon _0}\frac{1}{4}(x_1+\cdots +x_{i-1}+x_{i+1}+\cdots +x_{d-2}), \end{aligned}$$

establishing that the hyperplane intersects $B(e_i, {\epsilon _0})$. $\square$

1.1 Measure of parallel planes

Let $\mathcal {D}$ be a compact, convex set in $\mathbb {R}^k$. Let $P_0$ be a plane and $\mathcal {P}$ be the set of planes parallel to $P_0$ that intersect $\mathcal {D}$. The orthocomplement of $P_0$ is a copy of $\mathbb {R}^{k-2}$ and has Lebesgue measure. We identify $\mathcal {P}$ with a (convex) set $E\subset \mathbb {R}^{k-2}$, by identifying a point in $\mathcal {P}$ with the point in the orthocomplement it intersects. This induces a measure $\nu$ on $\mathcal {P}$. Let

$$\begin{aligned} A=\max \{\lambda _2(P \cap \mathcal {D}):P \in \mathcal {P}\}. \end{aligned}$$

Proposition 13.4

There exists a constant C, depending only on dimension, such that for all $\epsilon >0$,

$$\begin{aligned} \nu (\{P\in \mathcal {P}:\lambda _2(P \cap \mathcal {D})<\epsilon A\})\le C\sqrt{\epsilon }\nu (\mathcal {P}). \end{aligned}$$

We first prove

Lemma 13.5

Let $f:E\rightarrow [0,\infty )$ by $f(e)=\lambda _2(P_e\cap \mathcal {D})$ where $P_e$ is the element of $\mathcal {P}$ that contains e. Then on each line, f can be represented as $u+v$ where u is a concave function and v is the square of a concave function.

Proof

Let $e_0,e_1 \in E$, identify $e_0$ with 0, $e_1$ with 1 and $(1-t)e_0+te_1$ with t. Consider a direction w in $P_0$ and parametrize the lines parallel to w in each $P \in \mathcal {P}$ by the orthogonal direction to w in $P_0$. So on each P we have a family of lines $\ell _x$ parallel to w, for $x\in \mathbb {R}$. Let h(t, x) be the length of the line $\ell _x$ on $P_{(1-t)e_0+te_1}$ intersected with $\mathcal {D}$. We assume our parametrization of orthogonal direction to w is chosen so that

$$\begin{aligned} \inf \{x:h(1,x)>0\}=\inf \{x:h(0,x)>0\} \end{aligned}$$

and that these infimum are 0. Let us assume that

$$\begin{aligned} b=\sup \{x:h(1,x)>0\}\ge \sup \{x:h(0,x)>0\}=c. \end{aligned}$$

Now since the simplex is convex, for all x with $h(0,x)>0$ and for all $0\le t\le 1$

$$\begin{aligned} h(t,x)\ge (1-t)h(0,x)+th(1,x). \end{aligned}$$

So we have that

$$\begin{aligned} u(t)=\int _0^ch(t,x)dx \end{aligned}$$

is a concave function. Now we wish to show that if we set

$$\begin{aligned} v(t)=\int _c^bh(t,x)dx \end{aligned}$$

then $\sqrt{v}$ is concave. To do that it suffices to show that for $0\le s\le 1$

$$\begin{aligned} v(s)\ge s^2 v(1) \end{aligned}$$

(67)

(because $v(0)=0$). Let $\ell _x(0)$ be the line $\ell _x$ on $P_{e_0}$ and $\ell _x(1)$ be the line $\ell _x$ on $P_{e_1}$. Let $\mathcal {E}$ be the convex hull of $\cup _{x\in [c,b]}\ell _x(1) \cap \mathcal {D}$ and any point in $\ell _c(0)\cap \mathcal {D}$. Note that $\ell _c(0)\ne \emptyset$ because $\mathcal {D}$ is closed. Since $\mathcal {D}$ is convex, $\mathcal {E}\subset \mathcal {D}$ and so

$$\begin{aligned} v(s)\ge \lambda _2(P_{(1-s)e_0+se_1}\cap \mathcal {E}). \end{aligned}$$

(68)

Now $\mathcal {E}$ is a convex cone so

$$\begin{aligned} \lambda _2(P_{(1-a)e_0+se_1}\cap \mathcal {E})=s^2\lambda _2(\mathcal {E}\cap P_{e_1}). \end{aligned}$$

(69)

Combining (68) and (69) verifies the sufficient condition (67), completing the proof. $\square$

Lemma 13.6

Let $\ell \subset \mathbb {R}^{1}$ be a segment If $g:\ell \rightarrow [0,\infty )$ is concave then for all $\epsilon >0$, $\lambda (\{x \in \ell :g(x)<\epsilon \max g\})<\epsilon \lambda (\ell )$.

Proof

Let p be a point that maximizes g. For any $q\in \Omega$, let $\ell (q,p)$ be the line connecting q and p with unit speed parametrization of $\ell (q,p)$. Let its length be r. If $g(q)<\epsilon g(p)$ then since g concave, $q \in [0,\epsilon r)$. The lemma follows. $\square$

We now prove Proposition 13.4.

Proof

We fix a point $p \in \mathcal {D}$ with $\lambda _2(P_p\cap \mathcal {D})=A$, where $P_p$ is the plane in $\mathcal {P}$ going through p. We compute

$$\begin{aligned} \nu (\{P\in \mathcal {P}:\lambda _2(P \cap \mathcal {D})<\epsilon A\}) \end{aligned}$$

via integrating with respect to polar coordinates. Indeed, let E be the $k-2$ dimensional subspace of $\mathbb {R}^{k}$ containing the directions orthogonal to the directions in the planes of $\mathcal {P}$. Let $S^{k-3}$ denote the unit sphere in E, $\ell _{\theta }$ denote the line in direction $\theta$ through p for each $\theta \in S^{k-3}$. Let $v_\theta$ denote $\ell _\theta \cap \mathcal {D}$. Let $\tau _\theta$ be the set of q at least half of the way from p to one of the endpoints of $v_\theta$. It is the union of two line segments, $\tau _{\theta ,1}$ and $\tau _{\theta ,2}$. Let $\sigma _{\theta }$ be the set of points on $v_\theta$ between a quarter and $\frac{3}{4}$ of the way from p to the endpoints of $v_\theta$. Let $\sigma _{\theta ,1}=\sigma _\theta \cap \tau _{\theta ,1}$ and $\sigma _{\theta ,2}=\sigma _\theta \cap \tau _{\theta ,2}$. Observe that if $\epsilon$ is small enough then for any $\theta ,i$ we have that if $q \in \sigma _{\theta ,i}$ then $\lambda _2(P_q\cap \mathcal {D})>\epsilon A$. Let $\hat{\phi }$ be normalized Lebesgue measure on $S^{k-3}$. Let $P_q$ denote the plane in $\mathcal {P}$ through q. By Lemma 13.5 the area of $P_q\cap \mathcal {D}$ as a function of $q\in \ell _\theta$ is the sum of a concave and the square of a concave function. Therefore by Lemma 13.6

$$\begin{aligned} |\{q\in \tau _{\theta ,i}:\text {area} (P_q\cap \mathcal {D})<\epsilon A\}|\le 2\epsilon |\tau _{\theta ,i}|. \end{aligned}$$

(70)

Observe that for $q \in \sigma _{\theta ,i}$, $q' \in \tau _{\theta ,i}$ we have that $\frac{d(q',p)^{k-3}}{d(q,p)^{k-3}}<2^{k-3}$. From this we obtain that for any measurable sets $B,B'$ using polar coordinates

$$\begin{aligned} \frac{\int _{S^{k-3}}\int _{\sigma _{\theta ,1}\cup \sigma _{\theta ,2}}d(q,p)^{k-3}\chi _B(q)dqd \hat{\phi }(\theta )}{\int _{S^{k-3}}\int _{\tau _{\theta ,1}\cup \tau _{\theta ,2}}d(q,p)^{k-3}\chi _{B'}(q)dqd \hat{\phi }(\theta )}\ge 2^{-(k-3)} \frac{\int _{S^{k-3}}\int _{\sigma _{\theta ,1}\cup \sigma _{\theta ,2}}\chi _B(q)dqd \hat{\phi }(\theta )}{\int _{S^{k-3}}\int _{\tau _{\theta ,1}\cup \tau _{\theta ,2}}\chi _{B'}(q)dqd \hat{\phi }(\theta )}. \end{aligned}$$

Considering $B=\mathcal {D}$ and $B'=\{q\in E: \lambda _2(P_q\cap \mathcal {D})<\epsilon A\}$, we obtain the Proposition from (70) applied to each line. $\square$

Proposition 13.7

Let $A'=\max \{\text {diam}(P\cap \mathcal {D})\}$. There exists a constant C, depending only on dimension, such that for all $\epsilon >0$,

$$\begin{aligned} \nu (\{P\in \mathcal {P}:\text {diam}(P \cap \mathcal {D})<\epsilon A'\})\le C{\epsilon }\nu (\mathcal {P}). \end{aligned}$$

Noticing that on each line in E, the function $f(e)=diam(P_e\cap \mathcal {D})$ is a concave function we obtain the proposition analogously to Proposition 13.4.