2 Interval exchange transformations

In this section, we recall some notions of the theory of interval exchange transformations following [Reference Avila and Leguile8, Reference Sinai and Ulcigrai21, Reference Viana24].

We define interval exchange transformations as in [Reference Avila and Leguile8, Reference Viana24]. Let $\mathcal {A}$ be an alphabet on $d \geq 2$ symbols and let $I \subset \mathbb {R}$ be an interval having $0$ as left endpoint. In what follows, we denote $\mathbb {R}^{\mathcal {A}} \simeq \mathbb {R}^{d}$ and $\mathbb {R}_+^{\mathcal {A}} \simeq \mathbb {R}_+^{d}$ . We choose a partition $\{I_{\alpha }\}_{\alpha \in \mathcal {A}}$ of I into subintervals that we assume to be closed on the left and open on the right. An IET is a bijection of I defined by two data:

(1) a vector $\unicode{x3bb} =(\unicode{x3bb} _{\alpha })_{\alpha \in \mathcal {A}} \in \mathbb {R}_+^{\mathcal {A}}$ with coordinates corresponding to the lengths of the subintervals, that is, for all $\alpha \in \mathcal {A}$ , $\unicode{x3bb} _{\alpha } = |I_{\alpha }|$ . We write $I=I(\unicode{x3bb} )= [0,|\unicode{x3bb} |)$ , where $|\unicode{x3bb} |=\sum _{\alpha \in \mathcal {A}} \unicode{x3bb} _{\alpha }$ ;

(2) a pair $\pi =(\!\begin {smallmatrix} \pi _{0}\\ \pi _{1} \end {smallmatrix}\!)$ of bijections $\pi _{\varepsilon }:\mathcal {A}\rightarrow \{ 1,\ldots ,d \}$ , $\varepsilon =0,1$ , describing the ordering of the subintervals $I_{\alpha }$ before and after the application of the map. This is represented as

$$ \begin{align*} \pi=\left(\begin{array}{@{}llll@{}} \alpha_{1}^{0} & \alpha_{2}^{0} & \cdots & \alpha_{d}^{0}\\[4pt] \alpha_{1}^{1} & \alpha_{2}^{1} & \cdots & \alpha_{d}^{1} \end{array}\right)\!. \end{align*} $$

We call $\pi $ a permutation and identify it, at times, with its monodromy invariant $\tilde {\pi }= \pi _1 \circ \pi _0^{-1} :\{1,\ldots d\} \rightarrow \{1,\ldots d\} $ . We denote by $\mathfrak {S}({\mathcal {A}})$ the set of irreducible permutations, that is, $\pi \in \mathfrak {S}({\mathcal {A}})$ if and only if $\tilde {\pi }(\{1,\ldots ,k\})\neq \{1,\ldots ,k\}$ for $1\leq k < d$ .

Define a linear map $\Omega _{\pi }: \mathbb {R}^{\mathcal {A}} \rightarrow \mathbb {R}^{\mathcal {A}}$ by

(2.1)

Given a permutation $\pi \in \mathfrak {S}({\mathcal {A}})$ and $\unicode{x3bb} \in \mathbb {R}_+^{\mathcal {A}}$ , the interval exchange transformation associated with this data is the map $f_{\unicode{x3bb} ,\pi }$ that rearranges $I_\alpha $ according to $\pi $ , that is,

(2.2) $$ \begin{align} f_{\unicode{x3bb},\pi}(x)=x+ \upsilon_{\alpha} \end{align} $$

for any $x \in I_\alpha $ , where $\upsilon _{\alpha }=(\Omega _{\pi }(\unicode{x3bb} ))_{\alpha }$ .

We will assume throughout the rest of this paper that $(\unicode{x3bb} ,\pi )$ satisfies the infinite distinct orbit condition (IDOC), first introduced by Keane in [Reference Keane13]. Here, $(\unicode{x3bb} ,\pi )$ satisfies the IDOC if the orbits of the endpoints of the subintervals $\{I_{\alpha }\}_{\alpha \in \mathcal {A}}$ are as disjoint as possible,

for all $n \geq 1$ , and $\alpha , \beta \in \mathcal {A}$ with $\pi _0(\beta ) \neq 1$ . In particular, the IDOC implies minimality of $f_{\unicode{x3bb} ,\pi }$ , that is, every orbit is dense in the interval.

2.1 Rauzy induction

We define Rauzy induction as in [Reference Viana24]. Let $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}})$ . For $\varepsilon =0,1$ , denote by $\beta _{\varepsilon }$ the last symbol in the expression of $\pi _{\varepsilon }$ , that is,

$$ \begin{align*} \beta_{\varepsilon}=\pi_{\varepsilon}^{-1}(d)=\alpha_{d}^{\varepsilon}. \end{align*} $$

Assume the intervals $I_{\beta _{0}}$ and $I_{\beta _{1}}$ have different lengths. We say that $(\unicode{x3bb} ,\pi )$ is of type $0$ if $\unicode{x3bb} _{\beta _{0}}>\unicode{x3bb} _{\beta _{1}}$ and is of type $1$ if $\unicode{x3bb} _{\beta _{0}}<\unicode{x3bb} _{\beta _{1}}$ . The largest interval is called winner and the smallest is loser of $(\unicode{x3bb} ,\pi )$ . Let $I^{(1)}$ be interval obtained by removing the loser from $I(\unicode{x3bb} )$ :

The first return map of $f_{\unicode{x3bb} ,\pi }$ to the subinterval $I^{(1)}$ is again an IET, $f_{\unicode{x3bb} ^{(1)},\pi ^{(1)}}$ , where the parameters $(\unicode{x3bb} ^{(1)},\pi ^{(1)})$ are defined as follows. If $(\unicode{x3bb} ,\pi )$ is of type $0$ , then

(2.3) $$ \begin{align} \pi^{(1)}=\left(\begin{array}{@{}ll@{}} \pi_{0}^{(1)}\\ \pi_{1}^{(1)} \end{array}\right)=\left(\begin{array}{@{}llllllll@{}} \alpha_{1}^{0} & \cdots & \alpha_{k-1}^{0} & \alpha_{k}^{0} & \alpha_{k+1}^{0} & \cdots & \cdots & \beta_{0}\\[4pt] \alpha_{1}^{1} & \cdots & \alpha_{k-1}^{1} & \beta_{0} & \beta_{1} & \alpha_{k+1}^{1} & \cdots & \alpha_{d-1}^{1} \end{array}\right)\!, \end{align} $$

where $k \in \{1,\ldots ,d-1\}$ is defined by $\alpha _{k}^{1}=\beta _{0}$ , and $\unicode{x3bb} ^{(1)}=(\unicode{x3bb} _{\alpha }^{(1)})_{\alpha \in \mathcal {A}}$ , where

If $(\unicode{x3bb} ,\pi )$ is of type $1$ , then

(2.4) $$ \begin{align} \pi^{(1)}=\left(\begin{array}{@{}ll@{}} \pi_{0}^{(1)}\\ \pi_{1}^{(1)} \end{array}\right)=\left(\begin{array}{@{}llllllll@{}} \alpha_{1}^{0} & \cdots & \alpha_{k-1}^{0} & \beta_{1} & \beta_{0} & \alpha_{k+1}^{0} & \cdots & \alpha_{d-1}^{0}\\[4pt] \alpha_{1}^{1} & \cdots & \alpha_{k-1}^{1} & \alpha_{k}^{1} & \alpha_{k+1}^{1} & \cdots & \cdots & \beta_{1}\\ \end{array}\right)\!, \end{align} $$

where $k \in \{1,\ldots ,d-1\}$ is defined by $\alpha _{k}^{0}=\beta _{1}$ , and $\unicode{x3bb} ^{(1)}=(\unicode{x3bb} _{\alpha }^{(1)})_{\alpha \in \mathcal {A}}$ , where

(2.5)

The map $\mathcal {R}:\mathbb {R}_+^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}}) \rightarrow \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}})$ defined by $\mathcal {R}(\unicode{x3bb} ,\pi )= (\unicode{x3bb} ^{(1)},\pi ^{(1)})$ is called the Rauzy induction map.

The IDOC condition assures that the iterates $\mathcal {R}^n$ are defined for all $n\geq 0$ . We denote

(2.6) $$ \begin{align} \mathcal{R}^n(\unicode{x3bb},\pi)=(\unicode{x3bb}^{(n)},\pi^{(n)}), \end{align} $$

with $\pi ^{(n)}= (\!\!\begin {array}{ll} \pi _{0}^{(n)}\hspace {-5pt} &\pi _{1}^{(n)} \end {array}\!\!)^T$ . Furthermore, we denote by $\beta _{\varepsilon ,n}$ the last symbol in the expression of $\pi _{\varepsilon }^{(n)}$ , $\varepsilon (n)$ the type of $f_{\unicode{x3bb} ^{(n)},\pi ^{(n)}}$ , by $I^{(n)}$ its domain and by $\{I_{\alpha }^{(n)}\}_{\alpha \in \mathcal {A}}$ its partition in subintervals, for $n\geq 0$ . We also denote the translation vector of $f_{\unicode{x3bb} ^{(n)},\pi ^{(n)}}$ by $\upsilon ^{( n )} = \Omega _{\pi ^{( n )}}(\unicode{x3bb} ^{(n )})$ .

2.3 Rauzy cocycle

We define Rauzy cocycle as in [Reference Avila and Leguile8]. A linear cocycle is a pair $(T,A)$ , where $T:\mathfrak {X} \rightarrow \mathfrak {X}$ and $A:\mathfrak {X} \rightarrow GL(p,\mathbb {R})$ , viewed as a linear skew-product $(x,v)\rightarrow (T(x),A(x)\cdot v)$ on $\mathfrak {X} \times \mathbb {R}^p$ , $p\in \mathbb {N}$ . Notice that $(T,A)^n=(T^n,A^{(n)})$ , where

$$ \begin{align*} A^{(n)}(x)=A(T^{n-1}(x))\cdot \cdots \cdot A(x), \quad n \geq 0. \end{align*} $$

In what follows, we denote $SL({\mathcal {A}},\mathbb {Z}) \simeq SL(d,\mathbb {Z})$ . Let $\| \cdot \|$ denote a matrix norm on $SL({\mathcal {A}},\mathbb {Z})$ . Let $\log ^+y= \max \{\log (y),0 \}$ for any $y>0$ . If $(\mathfrak {X}, \mu )$ is a probability space, $\mu $ is an ergodic measure for T and

$$ \begin{align*} \int_{\mathfrak{X}} \log^+\| A(x) \| \,d \mu(x) < + \infty,\end{align*} $$

we say $(T,A)$ is a measurable cocycle.

Denote by $E_{\alpha \beta }$ the elementary matrix $(\delta _{i\alpha }\delta _{j\beta })_{i\geq 1, j\leq d}$ . Let $\mathfrak {R}\subseteq \mathfrak {S}({\mathcal {A}})$ be a Rauzy class. To any given path $\gamma \in \Pi (\mathfrak {R})$ , we associate a matrix $B_{\gamma } \in SL({\mathcal {A}},\mathbb {Z})$ , defined inductively as follows.

1. (i) If $\gamma $ is an arrow labelled by $0$ , set $B_{\gamma }=\mathbf {1}_d+E_{\alpha (1) \alpha (0)}$ , with $\mathbf {1}_d$ denoting the $d\times d$ identity matrix.

2. (ii) If $\gamma $ is an arrow labelled by $1$ , set $B_{\gamma }=\mathbf {1}_d+E_{\alpha (0) \alpha (1)}$ .

3. (iii) If $\gamma =(\gamma _1,\ldots ,\gamma _n)$ , set $B_{\gamma }=B_{\gamma _n}\cdots B_{\gamma _1}$ .

We denote by $\gamma (\unicode{x3bb} ,\pi ) \in \Pi (\mathfrak {R}(\pi ))$ , the arrow in the Rauzy graph starting at $\pi $ labelled by the type of $(\unicode{x3bb} ,\pi )$ .

Define the function $B_R: \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R} \rightarrow SL({\mathcal {A}},\mathbb {Z})$ such that $B_R(\unicode{x3bb} ,\pi )=B_{\gamma (\unicode{x3bb} ,\pi )}$ . The Rauzy cocycle is the linear cocycle over the Rauzy induction $(\mathcal {R},B_R)$ on $\mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R} \times \mathbb {R}^{\mathcal {A}}$ . Note that $(\mathcal {R},B_R)^n=(\mathcal {R}^n,B_R^{(n)})$ , where

(2.7) $$ \begin{align} B_R^{(0)}(\unicode{x3bb},\pi)=\mathbf{1}_d, \quad B_R^{(n)}(\unicode{x3bb},\pi)= B(\unicode{x3bb}^{(n-1)},\pi^{(n-1)})\cdot\cdots \cdot B(\unicode{x3bb}^{(1)},\pi^{(1)})\cdot B(\unicode{x3bb},\pi) \end{align} $$

for $n\geq 1$ , with $(\unicode{x3bb} ^{(n)},\pi ^{(n)})$ as in equation (2.6). Note that we have

$$ \begin{align*} \unicode{x3bb}= (B_R^{(n)}(\unicode{x3bb},\pi) )^*\cdot \unicode{x3bb}^{(n)} \end{align*} $$

for all $n\geq 0$ , where $^*$ denotes the transpose operator.

We now stress an important property of the Rauzy cocycle (see [Reference Viana24]). For any $n\geq 0$ and $x \in I^{(n)}$ , let $r_{\unicode{x3bb} ,\pi }^n(x)$ denote the first return time of x by $f_{\unicode{x3bb} ,\pi }$ to $I^{(n)}$ . Note that $r_{\unicode{x3bb} ,\pi }^n(x)$ is constant on each $I^{(n)}_{\alpha }$ for any $\alpha \in \mathcal {A}$ . We write $r_{\unicode{x3bb} ,\pi }^n(I^{(n)}_{\alpha })$ to mean $r_{\unicode{x3bb} ,\pi }^n(x)$ for any $x \in I^{(n)}_{\alpha }$ .

Each entry $(B_R^{(n)}(\unicode{x3bb} ,\pi ))_{\alpha ,\beta }$ of the matrix $B_R^{(n)}(\unicode{x3bb} ,\pi )$ counts the number of visits of $I_{\alpha }^{(n)}$ to the interval $I_\beta $ up to the $r_{\unicode{x3bb} ,\pi }^n(I^{(n)}_{\alpha })$ th iterate of $f_{\unicode{x3bb} ,\pi }$ . That is, for every $\alpha , \beta \in \mathcal {A}$ and every $n \geq 1$ ,

$$ \begin{align*} (B_R^{(n)}(\unicode{x3bb},\pi))_{\alpha,\beta}= \textrm{card} \{0\leq j < r^{n}_{\unicode{x3bb},\pi} (I_{\alpha}^{(n)}):f_{\unicode{x3bb},\pi}(I_{\alpha}^{(n)}) \subset I_\beta \}. \end{align*} $$

2.4 Projection of the Rauzy cocycle on $\mathbb {T}^{\mathcal {A}}$

Denote $\mathbb {Z}^{\mathcal {A}} \simeq \mathbb {Z}^{d}$ and let $\mathbb {T}^{\mathcal {A}} \simeq \mathbb {T}^{d}$ be the d-dimensional torus $\mathbb {R}^{\mathcal {A}} / 2\pi \mathbb {Z}^{\mathcal {A}}$ . Furthermore, let $p:\mathbb {R}^{\mathcal {A}} \rightarrow \mathbb {T}^{\mathcal {A}}$ be the natural projection,

$$ \begin{align*} p(v) = ( ( v )_{\alpha}\,\mod\! 2 \pi )_{\alpha \in \mathcal{A}} \quad \textrm{for all } v \in \mathbb{R}^{\mathcal{A}}.\end{align*} $$

We sometimes denote $p(v)= v \,\mod \! 2 \pi $ .

We introduce the projection of the Rauzy cocycle on $\mathbb {T}^{\mathcal {A}}$ as the application $B_{\mathbb {T}^{\mathcal {A}}} : \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R} \times \mathbb {T}^{\mathcal {A}} \rightarrow \mathbb {T}^{\mathcal {A}}$ such that $B_{\mathbb {T}^{\mathcal {A}}}(\unicode{x3bb} ,\pi )\cdot \theta = p(B_{R}(\unicode{x3bb} ,\pi ) \cdot v)$ for any $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R}$ , $n \geq 0$ and $\theta \in \mathbb {T}^{\mathcal {A}}$ , with $v \in p^{-1}(\theta )$ . Note that, as $B_{R}$ is an integral cocycle, for any $v, v' \in p^{-1}(\theta )$ , we have $p( B_{R}(\unicode{x3bb} ,\pi ) \cdot v) = p( B_{R}(\unicode{x3bb} ,\pi ) \cdot v')$ and thus the map $B_{\mathbb {T}^{\mathcal {A}}}$ is well defined. We also denote

(2.8) $$ \begin{align} B_{\mathbb{T}^{\mathcal{A}}}^{(n)}(\unicode{x3bb},\pi) \cdot \theta = B_{R}^{(n)}(\unicode{x3bb},\pi) \cdot v \,\mod\! 2\pi \end{align} $$

for any $n \geq 0$ and $\theta \in \mathbb {T}^{\mathcal {A}}$ , with $v \in p^{-1}(\theta )$ .

3 Breaking sequence

In this section, we define the breaking sequence, a sequence of curves associated to IET parameters $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}})$ and a rotational parameter $\theta \in \mathbb {T}^{\mathcal {A}}$ via the breaking operator, an operator acting on the space of piecewise linear curves. We then relate the dynamics of a breaking sequence and that of the underlying IET.

Given $\ell>0$ , we denote the class of continuous piecewise linear maps $\gamma :[0,\ell )\rightarrow \mathbb {C}$ such that all x in the domain of differentiability of $\gamma $ satisfy $|\gamma (x)'|=1$ by $\mathcal {PL}(\ell )$ . Note that for any $\gamma \in \mathcal {PL}(\ell )$ , its image $\gamma ([0,\ell ))$ has an arc length equal to $\ell $ .

We say that a sequence of mutually disjoint intervals $\{J_n\}$ is an ordered sequence of disjoint intervals if whenever $m<m'$ , we have $x<x'$ for all $x \in J_{m}$ and $x' \in J_{m'}$ .

We define an operator that acts on $\mathcal {PL}(\ell )$ , which, given a sequence of subintervals of its domain, takes a curve and rotates by a fixed angle the pieces corresponding to these subintervals. Visually and informally, the action of this operator is to rigidly ‘break’ the curve at these segments.

Consider a map $\gamma \in \mathcal {PL}(\ell )$ , a real number $\varphi \in [-\pi ,\pi )$ and $r in \mathbb {N}$ (here, r is arbitrary and is the number of disjoint intervals in J). Consider also an ordered sequence of disjoint intervals $J=\{J_k\}_{0\leq k \leq r-1}$ of equal length $\Delta \in (0,\ell / r)$ . We write $J_k=[y_k,y_k+\Delta )\subset \mathbb {R}$ , where $y_k+\Delta \leq y_{k+1}$ and $k \in \{0,\ldots ,r-1\}$ .

We define the breaking operator $\mathfrak {Br}(\varphi ,J):\mathcal {PL}(\ell ) \rightarrow \mathcal {B}([0,\ell ),\mathbb {C})$ as

(3.1) $$ \begin{align} \mathfrak{Br}(\varphi,J)\cdot \gamma(x)= \begin{cases} \gamma(x), & x \in [0,y_0),\\ \gamma(x)\cdot e^{i\varphi}+\overline{\epsilon}_k(\varphi,J), & x \in [y_k,y_k+\Delta),\\ \gamma(x)+\underline{\epsilon}_k(\varphi,J), & x \in [y_{k}+\Delta,y_{k+1}),\\ \end{cases} \end{align} $$

for $k \in \{0,\ldots ,r-1\}$ , where $y_r=\ell $ ,

(3.2) $$ \begin{align} \overline{\epsilon}_0=\gamma(x_0)(1-e^{i \varphi}), \quad \overline{\epsilon_k}=\gamma(y_k)(1-e^{i \varphi})+\underline{\epsilon}_{k-1} \end{align} $$

and also

(3.3) $$ \begin{align} \underline{\epsilon}_0=(\gamma(y_0)-\gamma(y_0+\Delta))(1-e^{i \varphi}), \quad \underline{\epsilon}_k=\overline{\epsilon}_k-\gamma(y_k+\Delta)(1-e^{i \varphi}). \end{align} $$

We first need to show that for all $\ell>0$ and $\varphi \in [-\pi ,\pi )$ , $\mathfrak {Br}(\varphi ,J)$ maps $\mathcal {PL}(\ell )$ into a subset of $\mathcal {PL}(\ell )$ . We do this in our next lemma.

Proof. Let $\gamma \in \mathcal {PL}(\ell )$ . It is clear that $\mathfrak {Br}(\varphi ,J)\cdot \gamma $ is piecewise linear and continuous. In particular, it is semi-differentiable. Denote by $\partial _{-}$ and $\partial _{+}$ its left and right derivative, respectively.

Given $x \in (0,\ell )$ , we have

$$ \begin{align*}|\partial_{-} ( \mathfrak{Br}(\varphi,J)\cdot \gamma)(x)|=|\partial_{+} ( \mathfrak{Br}(\varphi,J)\cdot \gamma)(x)|=|\partial_{+}\gamma(x)|.\end{align*} $$

Since $\gamma \in \mathcal {PL}(\ell )$ , $|\partial _{+}\gamma '(x)|=1$ and hence, if $\mathfrak {Br}(\varphi ,J)\cdot \gamma $ is differentiable at x, we must have $|( \mathfrak {Br}(\varphi ,J)\cdot \gamma )'(x)|=1$ . This finishes our proof.

We will later need the estimate in the next lemma.

Lemma 3.2. Let $\ell>0$ , $\gamma \in \mathcal {PL}(\ell )$ , $\varphi \in [-\pi ,\pi )$ , $\Delta <\ell $ be a positive constant and J be an ordered sequence of disjoint intervals of length $\Delta $ . For all $k \in \mathbb {N}$ , we have

$$ \begin{align*} \max(|\overline{\epsilon}_k|,|\underline{\epsilon}_k|) \leq 2\ell \sin\bigg|\frac{\varphi}{2}\bigg|. \end{align*} $$

Proof. Let r be the number of subintervals in J and denote $J=\{[y_k,y_k+\Delta )\}_{0\leq k<r}$ .

By inserting equation (3.3) in equation (3.2), it is clear that for any $1\leq k<r$ , we have

$$ \begin{align*} \overline{\epsilon}_{k}= (\gamma(y_k) - \gamma(y_{k-1}+\Delta) )(1-e^{i\varphi}) + \overline{\epsilon}_{k-1}, \end{align*} $$

and applying the triangle inequality, we get for any $1\leq k<r$ that

$$ \begin{align*} |\overline{\epsilon}_{k}|\leq |1-e^{i\varphi}| \bigg[ |\gamma(y_k)| + \sum_{l=0}^{k-1}|\gamma(y_l)-\gamma(y_{l}+\Delta)| \bigg]. \end{align*} $$

As $|1{\kern-1pt}-{\kern-1pt}e^{i\varphi }| {\kern-1pt}={\kern-1pt} |\sin (\varphi /2)|$ , $y_k{\kern-1pt}\leq{\kern-1pt} \ell {\kern-1pt}-{\kern-1pt}(r{\kern-1pt}-{\kern-1pt}k)\Delta $ and $ |\gamma (y_l){\kern-1pt}-{\kern-1pt}\gamma (y_{l}+\Delta )| {\kern-1pt}\leq{\kern-1pt} \gamma ([y_k, y_k+\Delta ))\leq \Delta $ , we get as $r \Delta \leq \ell $ ,

$$ \begin{align*} |\overline{\epsilon}_k| \leq 2\ell \sin\bigg|\frac{\varphi}{2}\bigg|. \end{align*} $$

It is also clear from equations (3.2) and (3.3) applying the triangle inequality that for any $1\leq k<r$ , we have

$$ \begin{align*} |\underline{\epsilon}_{k}|\leq |1-e^{i\varphi}| \sum_{l=0}^{k}|\gamma(y_l)-\gamma(y_{l}+\Delta)| \end{align*} $$

and, in a similar way as before, we can prove that $ |\underline {\epsilon }_{k}|\leq k \Delta \sin |\varphi /2| \leq \ell \sin |{\varphi }/{2}|.$

Let $\mathcal {J}$ be a collection of mutually disjoint intervals. We say an ordered sequence of intervals $\{J_n\}$ is an ordering of $\mathcal {J}$ if for all $J \in \mathcal {J}$ , there is a unique m such that $J_m=J$ . Note that if $\mathcal {J}$ is a finite collection, then it has a unique ordering.

Given $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}})$ , consider the collection of sets

(3.4) $$ \begin{align} \mathcal{J}(n)=\{ f_{\unicode{x3bb},\pi}^k(I^{(n-1)}\backslash I^{(n)}) \}_{0\leq k<r(n-1)}, \end{align} $$

where $r(n-1) = r_{\unicode{x3bb} ,\pi }^{n-1}(I_{\beta _{0,n-1}}^{(n-1)})$ and

$$ \begin{align*} \beta_{\varepsilon,m}=( \pi_{\varepsilon}^{(m)} )^{-1}(d). \end{align*} $$

It is clear that for all $n\geq 1$ , $r(n-1)$ is equal to the smallest $k\geq 1$ such that $f_{\unicode{x3bb} ,\pi }^k(I^{(n-1)}\backslash I^{(n)})\subset I^{(n)}$ .

We denote by $J^{(n)}=\{ J_k^{(n)} \}_{0\leq k <r(n-1)}$ the ordering of $\mathcal {J}(n)$ for all $n \geq 1$ .

Recall the projection of the Rauzy cocycle in $\mathbb {T}^{\mathcal {A}}$ . Given $\theta \in \mathbb {T}^{\mathcal {A}}$ , let

(3.5) $$ \begin{align} \theta^{(0)}=\theta, \quad \theta^{(n)}=B_{\mathbb{T}^{\mathcal{A}}}^{(n)}(\unicode{x3bb},\pi)\cdot \theta. \end{align} $$

We define the breaking sequence as a sequence of piecewise linear curves $\{\gamma _{\theta }^{(n)}(x)\} \in \mathcal {PL}(\ell )$ such that

(3.6) $$ \begin{align} \begin{array}{l} \gamma_{\theta}^{(0)}(x)= x ,\\ \gamma_{\theta}^{(n)}(x)= \mathfrak{Br}( \theta_{\beta_{1,n-1}}^{(n-1)}, J^{(n)}) \cdot \gamma_{\theta}^{(n-1)}(x), \end{array} \end{align} $$

for all $x \in [0,|\unicode{x3bb} |)$ and $n\geq 1$ .

Each map is a parametrization of a curve and is obtained by successively applying the breaking operator with angles $\theta _{\beta _{1,n-1}}^{(n-1)}$ and segments $J^{(n)}$ . Note that the number of these segments will increase, while their lengths will decrease as $n\rightarrow +\infty $ . In this way, this sequence of curves is related both to the IET $f_{\unicode{x3bb} ,\pi }$ and to a PWI with rotation vector $\theta $ .

Denote by $\Theta _{\unicode{x3bb} ,\pi }$ the set of all $\theta \in \mathbb {T}^{\mathcal {A}}$ such that for all $n \geq 0$ , $\gamma _{\theta }^{(n)}:I \rightarrow \mathbb {C}$ is an injective map. Throughout the rest of his paper, we will consider $\gamma ^{(n)} = \gamma _{\theta }^{(n)}$ with $\theta \in \Theta _{\unicode{x3bb} ,\pi }$ .

The monodromy invariant of the permutation $\pi ^{(m)}$ is the bijection ${\tilde \pi }^{(m)}\!:\{ 1,\cdots\hspace{-1pt}, d \}\!\rightarrow \{ 1,\cdots\hspace{-1pt}, d \}$ such that ${\tilde \pi }^{(m)}= \pi _1^{(m)} \circ (\pi _0^{(m)})^{-1}$ . We denote its inverse by ${\hat \pi }^{(m)}= \pi _0^{(m)} \circ (\pi _1^{(m)})^{-1}.$ We write

(3.7)

for $\varepsilon =0,1$ , where $x_{0,j}^{(m)}$ denotes the jth endpoint of the partition associated to $f_{\unicode{x3bb} ^{(m)}, \pi ^{(m)} }$ , this is $\{I_{\alpha }^{(m)}\}_{\alpha \in \mathcal {A}}$ , and $x_{1,j}^{(m)}$ denotes the jth endpoint of the image of this partition by $f_{\unicode{x3bb} ^{(m)}, \pi ^{(m)} }$ . Furthermore, we denote their image by $\gamma ^{(n)}$ as $\gamma _{\varepsilon , j}^{n,m} = \gamma ^{(n)} ( x_{\varepsilon ,j}^{(m)} ).$

We may now define points $\xi _j^{n,m} \in \mathbb {C}$ recursively as follows:

(3.8) $$ \begin{align} \begin{array}{ll} \xi_d^{n,m} = \gamma_{0,d}^{n,m}, \\\xi_j^{n,m}= e^{i \theta^{(m)}_{ (\pi_1^{(m)})^{-1} (j+1) }} ( \gamma^{n,m}_{0, {\hat \pi}^{(m)}(j+1) -1 } - \gamma^{n,m}_{0, {\hat \pi}^{(m)}(j+1) } )+\xi_{j+1}^{n,m}. \end{array} \end{align} $$

For all $\alpha \in {\mathcal A}$ , $n \in {\mathbb N}$ , $0\leq m \leq n$ and $z \in \mathbb {C}$ , we define a map,

(3.9) $$ \begin{align} {\hat T}_{\alpha}^{(n,m)} (z)= e^{ i \theta_{\alpha}^{(m)} } ( z - \gamma_{0, \pi_0^{(m)} (\alpha) }^{n,m} ) + \xi_{ \pi_1^{(m)} (\alpha) }^{n,m}. \end{align} $$

The isometries $\hat {T}_{\alpha }^{(n,m)}$ act on the segments $\gamma ^{(n)}(I_{\alpha }^{(m)})$ by rearranging their order according to the permutation $\pi ^{(m)}$ , via rotations by angles $\theta _{\alpha }^{(m)}$ . The right endpoint $\gamma _{0,\hat {\pi }^{(m)}(d)}^{n,m}$ of the segment $\gamma ^{(n)}(I_{\beta _{1,m}}^{(m)})$ is mapped to the right endpoint $\xi _{d}^{n,m}$ of $\gamma ^{(n)}(I^{(m)})$ . For $j<d$ , the right endpoint $\gamma _{0,\hat {\pi }^{(m)}(j)}^{n,m}$ of $\gamma ^{(n)}(I_{\hat {\pi }^{(m)}(j)}^{(m)})$ is mapped to the left endpoint $\xi _{j}^{n,m}$ of the image by $\hat {T}_{\hat {\pi }^{(m)}(j+1)}^{(n,m)}$ of $\gamma ^{(n)}(I_{\hat {\pi }^{(m)}(j+1)}^{(m)})$ . In this way, the union over $\alpha \in \mathcal {A}$ of all $\hat {T}_{\alpha }^{(n,m)}(\gamma ^{(n)}(I_{\alpha }^{(m)}))$ is a continuous curve which a priori may not coincide with $\gamma ^{(n)}(I^{(m)})$ .

We also define inductively a map $T_{\alpha }^{(n,m)}$ as follows:

(3.10) $$ \begin{align} T_{\alpha}^{(n,n)} (z)= {\hat T}_{\alpha}^{(n,n)} (z). \end{align} $$

For $z \in \mathbb {C}$ , $0 < m \leq n$ , if $\varepsilon (m-1) = 0$ , then

(3.11) $$ \begin{align} T_{\alpha}^{(n,m-1)} (z)= \begin{cases} T_{\alpha}^{(n,m)} (z), & \alpha \neq \beta_{1,m-1} ,\\ ( T_{ \beta_{0,m-1} }^{(n,m)} )^{-1} \circ T_{\alpha}^{(n,m)} (z), & \alpha = \beta_{1,m-1}; \end{cases} \end{align} $$

if $\varepsilon (m-1) = 1$ , then

(3.12) $$ \begin{align} T_{\alpha}^{(n,m-1)} (z)= \begin{cases} T_{\alpha}^{(n,m)} (z) ,& \alpha \neq \beta_{0,m-1}, \\ T_{\alpha}^{(n,m)} \circ ( T_{ \beta_{1,m-1} }^{(n,m)} )^{-1}(z) ,& \alpha = \beta_{0,m-1}. \end{cases} \end{align} $$

Finally, we define a map $T^{(n,m)} : \gamma ^{(n)} ( I^{(m)} ) \rightarrow {\mathbb C}$ by

$$ \begin{align*} T^{(n,m)} (z) = T_{\alpha}^{(n,m)} (z), \quad z \in \gamma^{(n)} ( I_{\alpha}^{(m)} ). \end{align*} $$

To understand the rationale behind the inductive procedure used to define $T^{(n,m)}$ , consider first the map $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)},\alpha }:\mathbb {R}\rightarrow \mathbb {R}$ such that $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)},\alpha }(x)= x+\upsilon _{\alpha }^{(m)}$ . If $\theta =0$ , by the definition of breaking sequence, note that $\gamma _{\theta }^{(n)}(x)=x$ for all $x \in I$ and $n\geq 0$ . Consequently, we have $\gamma _{\varepsilon , j }^{n,m}=x_{\varepsilon , j }^{(m)}$ , $\xi _{j }^{n,m}=x_{1, j }^{(m)}$ and thus, for all $z \in \mathbb {C}$ , we have

$$ \begin{align*} \hat{T}_{\alpha}^{(n,m)}(z) = f_{\unicode{x3bb}^{(m)},\pi^{(m)},\alpha}(\mathrm{Re}(z)) + i \mathrm{Im}(z) .\end{align*} $$

For $0<m\leq n$ and $\varepsilon (m-1)=0$ , equation (3.11) gives

$$ \begin{align*} f_{\unicode{x3bb}^{(m-1)},\pi^{(m-1)},\alpha} (\mathrm{Re}(z))= \begin{cases} f_{\unicode{x3bb}^{(m)},\pi^{(m)},\alpha} (\mathrm{Re}(z)), & \alpha \neq \beta_{1,m-1} ,\\ f_{\unicode{x3bb}^{(m)},\pi^{(m)},\beta_{0,m-1}}^{-1} \circ f_{\unicode{x3bb}^{(m)},\pi^{(m)},\alpha} (\mathrm{Re}(z)), & \alpha = \beta_{1,m-1}, \end{cases} \end{align*} $$

and as $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}(x)= f_{\unicode{x3bb} ^{(m)},\pi ^{(m)},\alpha }(x)$ , when $x \in I_{\alpha }^{(m)}$ , these identities can be easily verified to be equivalent to Rauzy induction in this case. An analogous set of identities can also be obtained for the case $\varepsilon (m-1)=1$ . Also note that for this example, we have $\hat {T}_{\alpha }^{(n,m)}=T_{\alpha }^{(n,m)}$ . This is no coincidence and indeed, later, we will prove that this identity holds in general. In this way, equations (3.11) and (3.12) are a generalization of Rauzy induction and hence $\{T^{(n,m)} \}_{n \geq 0}$ is a sequence of maps defined on $\gamma ^{(n)}(I^{(m)})$ which preserves this inductive structure.

For the remainder of this section, we state and prove several lemmas which serve as technical tools for our next section where we explore the relation among $\hat {T}_{\alpha }^{(n,m)}$ , $T_{\alpha }^{(n,m)}$ , $\gamma ^{(n)}$ and $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ . The following lemma gives useful expressions for compositions of $\hat {T}_{\alpha }^{(n,m)}$ that are related to the inductive procedure used to define $T_{\alpha }^{(n,m)}$ .

Lemma 3.3. For all $n\geq 1$ , $0< m \leq n$ and $z \in \mathbb {C}$ , if $\varepsilon (m-1)=0$ , then

$$ \begin{align*} ( {\hat T}_{ \beta_{0,m-1} }^{(n,m)} )^{-1} \circ {\hat T}_{\beta_{1,m-1}}^{(n,m)} (z) = e^{i\theta_{\beta_{1,m-1}}^{(m-1)}}( z - \gamma_{0,{\hat \pi}^{(m-1)}(d)-1}^{n,m-1} ) + \gamma_{1,d-1}^{n,m-1}, \end{align*} $$

and if $\varepsilon (m-1)=1$ , then

$$ \begin{align*} {\hat T}_{ \beta_{0,m-1} }^{(n,m)} \circ ( {\hat T}_{\beta_{1,m-1}}^{(n,m)} )^{-1}(z) = e^{i\theta_{\beta_{0,m-1}}^{(m-1)}}( z - \gamma_{0,d-1}^{n,m-1} ) + \xi_{{\tilde \pi}^{(m-1)}(d)-1}^{n,m}. \end{align*} $$

Proof. Assume first $\varepsilon (m-1)=0$ . It is clear that $\pi _{0}^{(m-1)}= \pi _{0}^{(m)}$ , $\pi _{1}^{(m)}(\beta _{1,m-1})= \pi _{1}^{(m)}(\beta _{0,m})+1$ and we get

(3.13) $$ \begin{align} \xi_{\pi_{1}^{(m)}(\beta_{1,m-1})}^{n,m} - \xi_{\pi_{1}^{(m)}(\beta_{0,m-1})}^{n,m} = e^{i\theta_{\beta_{1,m-1}}^{(m)}} ( \gamma_{0, {\hat \pi}^{(m-1)}(d)}^{n,m} - \gamma_{0, {\hat \pi}^{(m-1)}(d)-1}^{n,m} ) \end{align} $$

directly from the definition of $\xi _{j}^{n,m}$ with $j=\pi _{1}^{(m)}(\beta _{0,m})$ .

From equation (3.5), we can write

(3.14) $$ \begin{align} \theta_{\alpha}^{(m)}= \begin{cases} \theta_{\alpha}^{(m-1)}, & \alpha \neq \beta_{1,m-1},\\ \theta_{\beta_{1,m-1}}^{(m-1)}+ \theta_{\beta_{0,m-1}}^{(m-1)}, & \alpha = \beta_{1,m-1}. \end{cases} \end{align} $$

Now, since for any $j <d$ we have $\gamma _{0,j}^{n,m-1}=\gamma _{0,j}^{n,m}$ , from the above relations using equation (3.9), we prove our lemma in this case.

Now assume $\varepsilon (m-1)=1$ . It is clear that $\pi _{1}^{(m-1)}= \pi _{1}^{(m)}$ and $\pi _{0}^{(m)}(\beta _{1,m-1})= \pi _{0}^{(m)}(\beta _{0,m-1})-1.$ With $j = {\tilde \pi }^{(m-1)}(d)-1$ , it is straightforward from the definition of $\xi _j^{n,m}$ that

(3.15) $$ \begin{align} \xi_{{\tilde \pi}^{(m-1)}(d)-1}^{n,m} = e^{i\theta_{\beta_{0,m-1}}^{(m)}} ( \gamma_{0,\pi_{0}^{(m)}(\beta_{1,m-1})}^{n,m} - \gamma_{0,\pi_{0}^{(m)}(\beta_{0,m-1})}^{n,m} ) + \xi_{{\tilde \pi}^{(m-1)}(d)}^{n,m}. \end{align} $$

By equations (2.5) and (3.7), we have

(3.16) $$ \begin{align} \gamma_{0,j}^{n,m-1}= \begin{cases} \gamma_{0,j}^{n,m} , & 0\leq j < {\hat \pi}^{(m)}(d),\\ \gamma_{0,j+1}^{n,m} , & {\hat \pi}^{(m)}(d) \leq j < d, \end{cases} \end{align} $$

which in particular, by equation (3.8), gives $\gamma _{0,d-1}^{n,m-1}=\xi _{d}^{n,m}$ . Also, by equation (3.5), we have

(3.17) $$ \begin{align} \theta_{\alpha}^{(m)}= \begin{cases} \theta_{\alpha}^{(m-1)}, & \alpha \neq \beta_{0,m-1},\\ \theta_{\beta_{0,m-1}}^{(m-1)}+ \theta_{\beta_{1,m-1}}^{(m-1)}, & \alpha = \beta_{0,m-1}. \end{cases} \end{align} $$

The second statement in the lemma follows from combining this with equation (3.15) using the definition of ${\hat T}_{\alpha }^{n,m}$ .

Before proving our next lemma, note that we can write equation (2.3) as

(3.18) $$ \begin{align} {\hat \pi}^{(m-1)}(j)= \begin{cases} {\hat \pi}^{(m)}({ \tilde \pi }^{(m)}(d)+1), & j=d,\\ {\hat \pi}^{(m)}(j+1), & {\tilde \pi}^{(m-1)}(d)< j < d,\\ {\hat \pi}^{(m)}(j), & j \leq {\tilde \pi}^{(m)}(d). \end{cases} \end{align} $$

Our next two lemmas serve as technical tools for our next section. Most of their proofs consist of simple computations using our formulae and definitions. We highlight the main steps but do not present an exhaustive proof.

Lemma 3.4. Let $n\geq 1$ and $0< m \leq n$ . If $\varepsilon (m-1)=0$ and $\xi _{d-1}^{n,m-1}=\gamma _{1,d-1}^{n,m-1}$ , then

(3.19) $$ \begin{align} \hat{T}_{\alpha}^{(n,m-1)}(z)=\hat{T}_{\alpha}^{(n,m)}(z) \end{align} $$

for all $z \in \mathbb {C}$ and $\alpha \in \mathcal {A}\backslash \{ \beta _{1,m-1} \}$ .

Proof. For ${\tilde \pi }^{(m)}(d)< j < d$ , from the definition of $\hat {\pi }^{(m-1)}$ and since $\gamma _{0,j}^{n,m-1}= \gamma _{0,j}^{n,m}$ , we can write

$$ \begin{align*} \gamma_{0,{\hat \pi}^{(m-1)}(j)}^{n,m-1} - \gamma_{0,{\hat \pi}^{(m-1)}(j)}^{n,m-1} = \gamma_{0,{\hat \pi}^{(m)}(j+1)-1}^{n,m} - \gamma_{0,{\hat \pi}^{(m)}(j+1) }^{n,m}. \end{align*} $$

Since $\pi _{0}^{(m-1)}= \pi _{0}^{(m)}$ , $( \pi _{1}^{(m-1)})^{-1}(j) = ( \pi _{1}^{(m)})^{-1}(j+1)$ and as $j<d$ , using equation (3.14), we get

$$ \begin{align*} \theta_{( \pi_{1}^{(m-1)})^{-1}(j)}^{(m-1)} = \theta_{( \pi_{1}^{(m)})^{-1}(j+1)}^{(m)}. \end{align*} $$

As $\xi _{d-1}^{n,m-1}= \gamma _{1,d-1}^{n,m-1}$ and $\gamma _{1,d-1}^{n,m-1}=\gamma _{0,d}^{n,m}$ , the two expressions above give for ${\tilde \pi }^{(m)}(d)\leq {j < d}$ ,

(3.20) $$ \begin{align} \xi_{j}^{n,m-1}= \xi_{j+1}^{n,m}. \end{align} $$

Now assume $\alpha \in \mathcal {A}$ is such that $\pi _{1}^{(m)}(\alpha )>{\tilde \pi }^{(m)}(d)+1$ . By equation (3.20), we get $\xi _{\pi _{1}^{(m)}(\alpha )}^{n,m}=\xi _{\pi _{1}^{(m)}(\alpha )-1}^{n,m-1}$ , and since by equation (2.3) we have $\pi _{1}^{(m-1)}(\alpha )= \pi _{1}^{(m)}(\alpha )-1$ , this gives

$$ \begin{align*}\xi_{\pi_{1}^{(m)}(\alpha)}^{n,m}=\xi_{\pi_{1}^{(m-1)}(\alpha)}^{n,m-1}.\end{align*} $$

Since $\gamma _{0,j}^{n,m-1}= \gamma _{0,j}^{n,m}$ , the proof of the lemma in this case follows from the definition of $\xi _j^{n,m}$ and ${\hat T}_{\alpha }^{n,m}$ .

Note that $\pi _1^{(m-1)}(\beta _{0,m-1})=\pi _1^{(m)}(\beta _{1,m-1})$ and thus it follows from equation (3.20) that $ \xi _{\pi _{1}^{(m-1)}(\beta _{0,m-1})}^{n,m-1} = \xi _{\pi _{1}^{(m)}(\beta _{1,m-1})}^{n,m} $ . By equation (3.13), we get

(3.21) $$ \begin{align} \xi_{\pi_{1}^{(m-1)}(\beta_{0,m-1})}^{n,m-1} - \xi_{\pi_{1}^{(m)}(\beta_{0,m-1})}^{n,m} = e^{i\theta_{\beta_{1,m-1}}^{(m)}} ( \gamma_{0, {\hat \pi}^{(m-1)}(d)}^{n,m} - \gamma_{0, {\hat \pi}^{(m-1)}(d)-1}^{n,m} ). \end{align} $$

Since $\xi _{d-1}^{n,m-1}=\gamma _{1,d-1}^{n,m-1}$ , we have

(3.22) $$ \begin{align} \gamma_{0,d}^{n,m-1} - \gamma_{1,d-1}^{n,m-1} = e^{i\theta_{\beta_{1,m-1}}^{(m-1)}} ( \gamma_{0, {\hat \pi}^{(m-1)}(d)}^{n,m-1} - \gamma_{0, {\hat \pi}^{(m-1)}(d)-1}^{n,m-1}), \end{align} $$

which combined with equations (3.14) and (3.21), using the fact that $\gamma _{1,d-1}^{n,m-1} = \gamma _{0,d}^{n,m} $ , $\gamma _{0,j}^{n,m-1}=\gamma _{0,j}^{n,m}$ , when $j<d$ , and the definition of $\xi _j^{n,m}$ proves the lemma for $\alpha = \beta _{0,m-1}$ .

From equations (3.14), (3.18) and (3.22), as $\pi _{1}^{(m)}(\beta _{1,m-1})= \pi _{1}^{(m)}(\beta _{0,m})+1$ and $\gamma _{0,d}^{n,m}=\gamma _{1,d-1}^{n,m-1}$ , a trivial computation gives

$$ \begin{align*} \xi_{{ \tilde \pi }^{(m)}(d)+1}^{n,m} = e^{i\theta_{\beta_{0,m}}^{(m)}}( \gamma_{0, d}^{n,m-1} - \gamma_{0, d}^{n,m} ) + \xi_{{ \tilde \pi }^{(m)}(d)}^{n,m}. \end{align*} $$

By equations (3.18), (3.20) and from the definition of $\xi _{{ \tilde \pi }^{(m)}(d)-1}^{n,m-1}$ , we get

$$ \begin{align*} \xi_{{ \tilde \pi }^{(m)}(d)-1}^{n,m-1} = e^{i\theta_{\beta_{0,m}}^{(m-1)}} ( \gamma_{0, d-1}^{n,m-1} - \gamma_{0, d}^{n,m-1} ) + \xi_{{ \tilde \pi }^{(m)}(d)+1}^{n,m}. \end{align*} $$

Combining this with equations (3.14), (3.18) and noting that $\gamma _{0,d-1}^{n,m} = \gamma _{0,d-1}^{n,m-1} $ , the relation

$$ \begin{align*} \xi_{{ \tilde \pi }^{(m)}(d)-1}^{n,m-1} = \xi_{{ \tilde \pi }^{(m)}(d)-1}^{n,m} \end{align*} $$

simply follows from the definition of $\xi _j^{n,m}$ for $j=\tilde {\pi }^{(m)}(d)-1$ .

We now prove by induction on j that

(3.23) $$ \begin{align} \xi_{j}^{n,m-1} = \xi_{j}^{n,m} \end{align} $$

for $1 \leq j < \tilde {\pi }^{(m)}(d)$ .

Since $\pi _{0}^{(m-1)}= \pi _{0}^{(m)}$ , we get by equation (3.18) that $( \pi _{1}^{(m-1)})^{-1}(j) = ( \pi _{1}^{(m)})^{-1}(j)$ , and as $j<d$ , by equation (3.14), we have

(3.24) $$ \begin{align} \theta_{( \pi_{1}^{(m-1)})^{-1}(j)}^{(m-1)} = \theta_{( \pi_{1}^{(m)})^{-1}(j)}^{(m)}. \end{align} $$

Combined with equation (3.18), this gives

$$ \begin{align*}\xi_{j-1}^{n,m-1}= e^{i \theta^{(m)}_{ (\pi_1^{(m)})^{-1} (j) }} ( \gamma^{n,m}_{0, {\hat \pi}^{(m)}(j) -1 } - \gamma^{n,m}_{0, {\hat \pi}^{(m)}(j) } )+\xi_{j}^{n,m},\end{align*} $$

which, as $\xi _{j}^{n,m-1} = \xi _{j}^{n,m}$ , by equation (3.8), shows that $\xi _{j-1}^{n,m-1} = \xi _{j-1}^{n,m}$ , proving equation (3.23).

Now assume $\alpha \in \mathcal {A}$ is such that $\pi _{1}^{(m)}(\alpha )< \tilde {\pi }^{(m)}(d)$ . From equation (3.23), since $\pi _1^{(m-1)}(\alpha )= \pi _1^{(m)}(\alpha )$ , we get $\xi _{\pi _1^{(m-1)}(\alpha )}^{n,m-1} = \xi _{\pi _1^{(m)}(\alpha )}^{n,m}$ . This, combined with equation (3.24) and the definition of $\hat {T}_{\alpha }^{(n,m)}$ , proves our statement for $\pi _{1}^{(m)}(\alpha )< \tilde {\pi }^{(m)}(d)$ .

Since $\pi _1^{(m)}(\beta _{0,m-1})= \tilde {\pi }^{(m)}(d)$ and since we proved equation (3.19) for $\alpha = \beta _{0,m-1}$ and $\pi _{1}^{(m)}(\alpha )> \tilde {\pi }^{(m)}(d)+1$ , we get equation (3.19) for all $\pi _{1}^{(m)}(\alpha )\neq \tilde {\pi }^{(m)}(d)+1$ . As $\pi _{1}^{(m)}(\beta _{1,m-1})= \pi _{1}^{(m)}(\beta _{0,m})+1$ and $\pi _{1}^{(m)}(\beta _{1,m})=\tilde {\pi }^{(m)}(d)+1$ , we have equation (3.19) for all $\alpha \in \mathcal {A}\backslash \{ \beta _{1,m-1} \}$ .

Note that by equation (2.4), we can write

(3.25) $$ \begin{align} {\hat \pi}^{(m-1)}(j)= \begin{cases} {\hat \pi}^{(m)}(j)-1, & \hat{\pi}^{(m)}(j)>\hat{\pi}^{(m)}(d)+1,\\ d, & \hat{\pi}^{(m)}(j)= \hat{\pi}^{(m)}(d)+1,\\ {\hat \pi}^{(m)}(j), & \hat{\pi}^{(m)}(j)<\hat{\pi}^{(m)}(d)+1. \end{cases} \end{align} $$

The following lemma provides a result similar to that of Lemma 3.4, but for the case $\varepsilon (m-1)=1$ . The main difference, compared with the previous case, comes from the fact that $\xi _{d-1}^{n,m-1}$ does not, beforehand, coincide with $\gamma _{1,d-1}^{n,m-1}$ , although we will later establish this equality.

Lemma 3.5. Let $n\geq 1$ and $0< m \leq n$ . If $\varepsilon (m-1)=1$ and $\xi _{d-1}^{n,m-1}=\xi _{d-1}^{n,m}$ , then for all $z \in \mathbb {C}$ and $\alpha \in \mathcal {A}\backslash \{ \beta _{0,m-1}, \beta _{1,m-1} \}$ , we have

(3.26) $$ \begin{align} \hat{T}_{\alpha}^{(n,m-1)}(z)=\hat{T}_{\alpha}^{(n,m)}(z) \end{align} $$

and

(3.27) $$ \begin{align} \hat{T}_{\beta_{0,m-1}}^{(n,m-1)}(z)=\hat{T}_{\beta_{0,m-1}}^{(n,m)} \circ (\hat{T}_{\beta_{1,m-1}}^{(n,m)})^{-1}(z). \end{align} $$

Proof. By equations (3.16) and (3.25), for all j such that $\hat {\pi }^{(m)}(j)\notin \{ \hat {\pi }^{(m)}(d), \hat {\pi }^{(m)}(d)+1 \}$ , we get

(3.28) $$ \begin{align} \gamma_{0,\hat{\pi}^{(m-1)}(j)}^{n,m-1}= \gamma_{0,\hat{\pi}^{(m)}(j)}^{n,m}, \end{align} $$

similarly, $\gamma _{0,\hat {\pi }^{(m-1)}(j)-1}^{n,m-1}= \gamma _{0,\hat {\pi }^{(m)}(j)-1}^{n,m}$ . In particular, for any $j \notin \{ \tilde {\pi }^{(m)}(\hat {\pi }^{(m)}(d)+1), d \}$ , we have

(3.29) $$ \begin{align} \gamma_{0,\hat{\pi}^{(m-1)}(j)-1}^{n,m-1}-\gamma_{0,\hat{\pi}^{(m-1)}(j)}^{n,m-1} = \gamma_{0,\hat{\pi}^{(m)}(j)-1}^{n,m}-\gamma_{0,\hat{\pi}^{(m)}(j)}^{n,m}. \end{align} $$

As $\pi _1^{(m-1)}=\pi _1^{(m)}$ and by equation (3.17), for all $j<d$ , we have

(3.30) $$ \begin{align} \theta_{(\pi_1^{(m-1)})^{-1}(j)}^{(m-1)} = \theta_{(\pi_1^{(m)})^{-1}(j)}^{(m)}. \end{align} $$

We now prove, by induction on j, that

(3.31) $$ \begin{align} \xi_{j}^{n,m-1}= \xi_{j}^{n,m} \end{align} $$

for $\tilde {\pi }^{(m-1)}(d)\leq j<d$ .

We have $\xi _{d-1}^{n,m-1}= \xi _{d-1}^{n,m}$ . Take $\tilde {\pi }^{(m-1)}(d)< j<d$ . As $\tilde {\pi }^{(m)}(\hat {\pi }^{(m)}(d)+1) = \tilde {\pi }^{(m-1)}(d)$ , we have that $ j \notin \{\tilde {\pi }^{(m)}(\hat {\pi }^{(m)}(d)+1), d \}$ , and hence by equations (3.29) and (3.30), we get $\xi _{j-1}^{n,m-1}= \xi _{j-1}^{n,m}$ . This shows that for any $\tilde {\pi }^{(m-1)}(d) \leq j<d$ , equation (3.31) holds.

By equations (3.16) and (3.25), we have $\gamma _{0,\hat {\pi }^{(m-1)}(d)}^{n,m-1}=\gamma _{0,\hat {\pi }^{(m)}(d)+1}^{n,m}$ and $\gamma _{0,\hat {\pi }^{(m-1)}(d)-1}^{n,m-1}=\gamma _{0,\hat {\pi }^{(m)}(d)-1}^{n,m}$ , and thus by equation (3.8), we get

$$ \begin{align*} \xi_{d-1}^{n,m-1}= e^{i \theta^{(m-1)}_{ \beta_{1,m-1} }} ( \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d) -1 } - \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d)+1 } )+\gamma_{0,d}^{n,m-1}. \end{align*} $$

Since $\xi _{d-1}^{n,m-1} = \xi _{d-1}^{n,m} $ and $\xi _{d}^{n,m}=\gamma _{0,d-1}^{n,m-1}$ , by combining this with the definition of $\xi _{d-1}^{n,m}$ and equation (3.30), we get

(3.32) $$ \begin{align} \gamma_{0,d-1}^{n,m-1}- \gamma_{0,d}^{n,m-1} = e^{i \theta_{\beta_{1,m-1}}^{(m-1)}} ( \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d) } - \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d)+1 }). \end{align} $$

By equation (3.8), with $j= \tilde {\pi }^{(m-1)}(d) -1$ , we have

$$ \begin{align*} \xi_{\tilde{\pi}^{(m-1)}(d)-1}^{n,m-1} = e^{i \theta_{\beta_{0,m-1}}^{(m-1)}} ( \gamma^{n,m-1}_{0, d-1 } - \gamma^{n,m-1}_{0, d } ) + \xi_{\tilde{\pi}^{(m-1)}(d)}^{n,m-1}, \end{align*} $$

which by equations (3.17), (3.31) and (3.32) gives

$$ \begin{align*} \xi_{\tilde{\pi}^{(m-1)}(d)-1}^{n,m-1} = e^{i \theta_{\beta_{0,m-1}}^{(m)}} ( \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d) } - \gamma^{n,m}_{0, {\hat \pi}^{(m)}(d)+1 } ) + \xi_{\tilde{\pi}^{(m-1)}(d)}^{n,m}, \end{align*} $$

and by equation (3.25), $\tilde {\pi }^{(m-1)}(d)= \tilde {\pi }^{(m)}(\hat {\pi }^{(m)}(d)+1)$ , combined with equation (3.8), this shows that $\xi _{\tilde {\pi }^{(m-1)}(d)-1}^{n,m-1} = \xi _{\tilde {\pi }^{(m-1)}(d)-1}^{n,m}$ .

Now assume, by induction in j, that for some $j<\tilde {\pi }^{(m-1)}(d)$ , we have $\xi _{j}^{n,m-1} = \xi _{j}^{n,m} $ . It is straightforward to see, by definition of $\xi _{j}^{n,m}$ , and equations (3.29) and (3.30) that $\xi _{j-1}^{n,m-1} = \xi _{j-1}^{n,m} $ . Since we had proved before that equation (3.31) holds for $\tilde {\pi }^{(m-1)}(d) \leq j < d$ , this shows that equation (3.31) is true for all $j<d$ .

Now, consider $\alpha \in \mathcal {A} \backslash \{ \beta _{0,m-1},\beta _{1,m-1} \}$ . By taking $j=\pi _{1}^{(m)}(\alpha )$ , we get $j \notin \{ \tilde {\pi }^{(m)}(\hat {\pi }^{(m)} (d)+1), d \}$ and by equation (3.28), we obtain $ \gamma _{0,\pi _{0}^{(m-1)}(\alpha )}^{n,m-1}= \gamma _{0,\pi _{0}^{(m)}(\alpha )}^{n,m}$ , and thus by equations (3.30), (3.31) and (3.9), we get equation (3.26).

By equation (3.9), for all $z \in \mathbb {C}$ , we get

$$ \begin{align*} \begin{array}{l} \hat{T}_{\beta_{0,m-1}}^{(n,m)} \circ \bigg(\hat{T}_{\beta_{1,m-1}}^{(n,m)}\bigg)^{-1}(z) = \xi_{\pi_1^{(m)}(\beta_{0,m-1})}^{n,m} \\\quad +\ e^{i \theta_{ \beta_{0,m-1} }^{ (m) }}\bigg[ e^{- i \theta_{\beta_{1,m-1}}^{(m)} } ( z - \xi_{\pi_1^{(m)}(\beta_{1,m-1})}^{n,m} ) + \gamma_{0,\pi_0^{(m)}(\beta_{1,m-1})}^{n,m} - \gamma_{0,\pi_0^{(m)}(\beta_{0,m-1})}^{n,m} \bigg] , \end{array} \end{align*} $$

which by Lemma 3.3 gives

$$ \begin{align*} \hat{T}_{\beta_{0,m-1}}^{(n,m)} \circ (\hat{T}_{\beta_{1,m-1}}^{(n,m)})^{-1}(z) = e^{i\theta_{\beta_{0,m-1}}^{(m-1)}}( z - \gamma_{0,d-1}^{n,m-1} ) + \xi_{0,{\tilde \pi}^{(m-1)}(d)-1}^{n,m} ,\end{align*} $$

and combined with equations (3.31) and (3.8) for $j= {\tilde \pi }^{(m-1)}(d)-1 $ , this gives

$$ \begin{align*} \hat{T}_{\beta_{0,m-1}}^{(n,m)} \circ (\hat{T}_{\beta_{1,m-1}}^{(n,m)})^{-1}(z) = e^{i\theta_{\beta_{0,m-1}}^{(m-1)}}( z - \gamma_{0,d}^{n,m-1} ) + \xi_{0,{\tilde \pi}^{(m-1)}(d)}^{n,m} .\end{align*} $$

By equation (3.9), this shows that equation (3.27) holds.

Consider now $J=\{ J_k \}_{0\leq k<r}$ , with $r \in \mathbb {N} $ , an ordered sequence of disjoint subintervals of I. Let $I'$ be a subinterval of I, we denote

$$ \begin{align*} J \cap I' = \{ J_k\cap I' : J_k \cap I' \neq \emptyset \}_{0\leq k<r}. \end{align*} $$

Recall we denote by $J^{(n+1)}$ the ordering of $\{ f_{\unicode{x3bb} ,\pi }^k(I^{(n)}\backslash I^{(n+1)}) \}_{0\leq k<r(n)}$ , where $r(n) = r_{\unicode{x3bb} ,\pi }^{n}(I_{\beta _{0,n}}^{(n)})$ .

Given $n \in \mathbb {N}$ , we define a sequence $\{k(m)\}_{0\leq m \leq n+1}$ of indices of $J^{(n+1)}$ as follows. Set $k(n+1)=0$ . For $0 \leq m < n+1$ , let $k(m)$ be equal to the number of disjoint subintervals in $J^{(n+1)}\cap I^{(m)}$ . It is clear, we have

$$ \begin{align*} J^{(n+1)} \cap (I^{(m-1)}\backslash I^{(m)}) = \{ J_{k} \}_{k(m)\leq k < k(m-1)} \end{align*} $$

for $0 < m \leq n+1$ .

Denote by $\beta (m) = \beta _{1-\varepsilon (m),m}$ , that is, the loser of $(\unicode{x3bb} ^{(m)},\pi ^{(m)})$ . The following two lemmas, which describe $J^{(n+1)}$ , will be needed for the following section.

Lemma 3.7. For all $n \geq 0$ , we have

$$ \begin{align*} J^{(n+1)} \cap (I^{(n)} \backslash I^{(n+1)}) = \{ I^{(n)}\backslash I^{(n+1)} \}; \end{align*} $$

furthermore, for all $0<m\leq n$ , we have

(3.33) $$ \begin{align} J^{(n+1)} \cap (I^{(m-1)} \backslash I^{(m)}) = f_{\unicode{x3bb}^{(m-1)},\pi^{(m-1)}}( J^{(n+1)}\cap I_{\beta(m-1)}^{(m)} ). \end{align} $$

In particular, there exists a $k'(m)>0$ such that for all $k(m) \leq k < k(m-1) $ , we have

(3.34) $$ \begin{align} J_k= f_{\unicode{x3bb}^{(m-1)},\pi^{(m-1)}}( J_{k-k'(m)} ). \end{align} $$

4 Existence of a quasi-embedding

In this section, we introduce the notion of quasi-embedding and use it to relate the dynamics of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ with that of $T^{(n,m)}$ for any $n\geq 0$ and $0\leq m\leq n$ .

Definition 4.1. We say that $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ is quasi-embedded into $T^{(n,m)}$ , or that $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n,m)}$ for $x \in I' \subseteq I$ , if

$$ \begin{align*} T^{(n,m)}(\gamma^{(n)}(x))= \gamma^{(n)}(f_{\unicode{x3bb}^{(m)},\pi^{(m)}}(x)). \end{align*} $$

Intuitively, this means that $T^{(n,m)}$ and $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ are nearly topologically conjugate, the conjugacy failing only for points in $I \backslash I'$ .

The first lemma is a particular case of Theorem 4.3, where $n \geq 1$ and $m=n-1$ . We separate the cases $\varepsilon (m-1)=0,1$ as $T_{\alpha }^{(n,m)}$ is given by different expressions in each. For the case $\varepsilon (m-1)=0$ , we use Lemma 3.3 to obtain an expression for $T_{\beta _{1,n-1}}^{(n,n-1)}$ and distinguish between the cases $\alpha =\beta _{1,n-1}$ and $\alpha \neq \beta _{1,n-1}$ , as the first can be addressed directly while the second requires the use of Lemma 3.4. In the case $\varepsilon (m-1)=1$ , we also distinguish between $\alpha =\beta _{1,n-1}$ and $\alpha \neq \beta _{1,n-1}$ as, unlike the first, the latter case requires the use of Lemma 3.5.

Lemma 4.2. Let $n\geq 1$ and $\alpha \in \mathcal {A}$ . Then, $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}$ into $T^{(n,n-1)}$ for $x \in I^{(n-1)}\backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}( I^{(n)} ) $ . Furthermore, for all $z \in \mathbb {C}$ , we have

(4.1) $$ \begin{align} T_{\alpha}^{(n,n-1)}(z) = {\hat T}_{\alpha}^{(n,n-1)}(z). \end{align} $$

Proof. We distinguish the cases $\varepsilon (n-1)=0$ and $\varepsilon (n-1)=1$ .

Given $n\geq 1$ , assume $\varepsilon (n-1)=0$ . Lemma 3.3 for $m=n$ combined with equation (3.11) gives

(4.2) $$ \begin{align} T_{\beta_{1,n-1}}^{(n,n-1)} (z)= e^{i\theta_{\beta_{1,n-1}}^{(n-1)}}( z - \gamma_{0,{\hat \pi}^{(n-1)}(d)-1}^{n,n-1} ) + \gamma_{1,d-1}^{n,n-1} \end{align} $$

for all $z \in \mathbb {C}$ .

By Lemma 3.7, $J^{(n)} =\{ I^{(n-1)} \backslash I^{(n)} \}$ . Let $x \in I^{(n-1)}_{\beta _{1,n-1}}\backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}(I^{(n)})$ . Since we have $ f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}(x) \in I^{(n-1)} \backslash I^{(n)}$ , it follows from our definitions of breaking operator and breaking sequence that

(4.3) $$ \begin{align} \gamma^{(n)}(f_{\unicode{x3bb}^{(n-1)},\pi^{(n-1)}}(x)) = f_{\unicode{x3bb}^{(n-1)},\pi^{(n-1)}}(x) e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}} + |I^{(n)}|(1-e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}), \end{align} $$

since $x_{1,d-1}^{(n-1)}= |I^{(n)}|$ , $\gamma _{0, \hat {\pi }^{(n-1)}(d)-1}^{n,n-1} = x_{0, \hat {\pi }^{(n-1)}(d)-1}^{(n-1)}$ as $\gamma ^{(n)}(x)=x$ and $|I^{(n)}|=\gamma _{1,d-1}^{n,n-1}$ , this gives

(4.4) $$ \begin{align} \gamma^{(n)}(f_{\unicode{x3bb}^{(n-1)},\pi^{(n-1)}}(x))-e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}} \gamma^{(n)}(x) = \gamma_{1,d-1}^{n,n-1} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n-1)}(d)-1}^{n,n-1}. \end{align} $$

As $I_{\alpha }^{(n-1)}\backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}(I^{(n)}) = \emptyset $ for $\alpha \neq \beta _{1,n-1}$ , equation (4.2), together with equation (4.4), gives

$$ \begin{align*} T^{(n,n-1)}(\gamma^{(n)}(x))= \gamma^{(n)}(f_{\unicode{x3bb}^{(n-1)},\pi^{(n-1)}}(x)), \end{align*} $$

which proves that the map $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}$ into $T^{(n,n-1)}$ for $x \in I_{\alpha }^{(n-1)}\backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}( I^{(n)}) $ .

By continuity of $f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}$ in $I_{\beta _{1,n-1}}^{(n-1)}= [x_{0,\hat {\pi }^{(n-1)}(d)-1}^{(n-1)},x_{0,\hat {\pi }^{(n-1)}(d)}^{(n-1)})$ and from equation (4.4), we get

$$ \begin{align*} \gamma_{1,d-1}^{n,n-1} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n-1)}(d)-1}^{n,n-1} = \gamma_{1,d}^{n,n-1} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n-1)}(d)}^{n,n-1}, \end{align*} $$

which, combined with equations (4.2), (3.8) and (3.9), gives equation (4.1) for $\alpha = \beta _{1,n-1}$ .

Since $\xi _{d-1}^{n,n-1}=\gamma _{1,d-1}^{n,n-1}$ , by Lemma 3.4, we prove the second statement in our lemma for all $\alpha \in \mathcal {A}$ .

Now assume $\varepsilon (n-1)=1$ . It follows directly from our definitions of ${\hat T}_{\alpha }^{(n,m)} (z)$ , $T_{\alpha }^{(n,n)} (z)$ and $\xi _j^{n,m}$ using equation (3.12) that

(4.5) $$ \begin{align} {T}_{\beta_{1,n-1}}^{(n,n-1)} (z)= e^{ i \theta_{\beta_{1,n-1}}^{(n-1)} } ( z - \gamma_{0, \hat{\pi}^{(n)}(d) }^{n,n} ) + \gamma_{ 1,d }^{n,n} \end{align} $$

for all $z \in \mathbb {C}$ . Again, in this case, we also have $J^{(n)}=\{ I^{(n-1)} \backslash I^{(n)} \}$ and we can use equation (4.3) as before which, since $\theta _{\beta _{1,n-1}}^{(n-1)} = \theta _{\beta _{1,n}}^{(n)}$ , $\gamma _{1,d}^{n,n} = x_{1,d}^{(n)}$ and $\gamma _{0,\hat {\pi }^{(n)}(d)}^{n,n}= x_{0,\hat {\pi }^{(n)}(d)}^{(n)}$ , gives

(4.6) $$ \begin{align} \gamma^{(n)}(f_{\unicode{x3bb}^{(n-1)},\pi^{(n-1)}}(x))-e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}} \gamma^{(n)}(x) = \gamma_{1,d}^{n,n} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n)}(d)}^{n,n} \end{align} $$

for all $x \in I_{\beta _{1,n-1}}^{(n-1)} \backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}(I^{(n)})$ . As $I_{\alpha }^{(n-1)}\backslash f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}^{-1}(I^{(n)}) = \emptyset $ for $\alpha \neq \beta _{1,n-1}$ , combining equations (4.5) and (4.6), we prove the first statement in the lemma.

By continuity of $f_{\unicode{x3bb} ^{(n-1)},\pi ^{(n-1)}}$ in $I_{\beta _{1,n-1}}^{(n-1)}= [x_{0,\hat {\pi }^{(n-1)}(d)-1}^{(n-1)},x_{0,\hat {\pi }^{(n-1)}(d)}^{n-1})$ and from equation (4.6), we can relate the image by $\gamma ^{(n)}$ of the dth endpoint of the partitions associated to $f_{\unicode{x3bb} ^{(n-1)}, \pi ^{(n-1)}}$ and $f_{\unicode{x3bb} ^{(n)}, \pi ^{(n)}}$ as follows:

$$ \begin{align*} \gamma_{1,d}^{n,n-1} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n-1)}(d)}^{n,n-1} = \gamma_{1,d}^{n,n} - e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)}}\gamma_{0, \hat{\pi}^{(n)}(d)}^{n,n}. \end{align*} $$

As $\gamma _{1,d}^{n,n-1}= \xi _{d}^{n,n-1}$ , this, together with equations (4.5) and (3.9), proves equation (4.1) for $\alpha = \beta _{1,n-1}$ . Using the definition of $\xi _j^{n,m}$ , this can be rewritten as

$$ \begin{align*} \xi_{d-1}^{n,n-1}= e^{i {\theta}_{\beta_{1,n-1}}^{(n-1)} } ( \gamma^{n,n-1}_{0, {\hat \pi}^{(n-1)}(d) -1 } - \gamma^{n,n}_{0, {\hat \pi}^{(n)}(d) } )+\xi_{d}^{n,n}, \end{align*} $$

and since $ \gamma ^{n,n-1}_{0, {\hat \pi }^{(n-1)}(d) -1 } = \gamma ^{n,n}_{0, {\hat \pi }^{(n)}(d) -1 } $ , by equations (3.8) and (3.17), we get that $\xi _{d-1}^{n,n-1} = \xi _{d-1}^{n,n}$ . Hence, by Lemma 3.5 and equations (3.9), (3.10) and (3.12), we prove the second statement in the lemma for all $\alpha \in \mathcal {A}$ .

The following theorem establishes that $T_{\alpha }^{(n,m)} = {\hat T}_{\alpha }^{(n,m)}$ and that $\gamma ^{(n)}$ is a quasi- embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n,m)}$ except for points in a subinterval that decreases with n.

The proof is structured as follows. The theorem holds trivially in the case $n\geq 0$ and from Lemma 4.2, in the case $n \geq 1$ and $m=n-1$ . Next, we assume, by induction on m, that given a fixed $n\geq 1$ , the theorem is true for $T^{(n,m)}$ , with $0\leq m\leq n$ , and also for $T^{(n+1,m)}$ , with $0< m\leq n+1$ , and we prove it for $T^{(n+1,m-1)}$ .

We prove that $f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}$ is quasi-embedded into $T^{(n+1,m-1)}$ in $I_{\beta _{1,m-1}}^{(m-1)}\backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1} (I^{(n+1)})$ by induction in k, considering separate subintervals in $J^{(n+1)}$ . In particular, we achieve this by applying Lemmas A.1 and A.2 in an alternate way to extend the quasi-embedding throughout the interval. It follows that our theorem is true for $x \in I_{\beta _{1,m-1}}^{(m-1)}\backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1}(I^{(n+1)})$ .

To prove it is true for $I_{\alpha }^{(m-1)}\backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1}(I^{(n+1)})$ , with $\alpha \neq \beta _{1,m-1}$ , we separate cases $\varepsilon (m-1)=0$ and $\varepsilon (m-1)=1$ . In both cases, we distinguish between $\alpha \neq \beta _{0,m-1}$ , which requires the use of Lemmas 3.4 and 3.5, the proof of these two lemmas appear in the last section of the paper. The case $\alpha = \beta _{0,m-1}$ follows from a distinct straightforward argument.

Theorem 4.3. For all $n\geq 0$ and $0\leq m \leq n$ , $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n,m)}$ for $x \in I^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n)}) $ . Furthermore, for all $\alpha \in \mathcal {A}$ and $z \in \mathbb {C}$ , we have

(4.7) $$ \begin{align} T_{\alpha}^{(n,m)}(z) = {\hat T}_{\alpha}^{(n,m)}(z). \end{align} $$

Proof. Both statements in our theorem are trivial to prove for $n \geq 0$ and $m=n$ , as $I_{\alpha }^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n)} )= \emptyset $ . For $m=n-1$ , both statements follow directly from Lemma 4.2.

Given $n \geq 0$ , we now assume the following.

(H1). For all $0\leq m' \leq n$ and $\alpha \in \mathcal {A}$ , $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m')},\pi ^{(m')}}$ into $T^{(n,m')}$ for $x \in I^{(m')}\backslash f_{\unicode{x3bb} ^{(m')},\pi ^{(m')}}^{-1}( I^{(n)} )$ , and that for all $z \in \mathbb {C}$ ,

$$ \begin{align*} T_{\alpha}^{(n,m')} (z)={ \hat T}_{\alpha}^{(n,m')} (z). \end{align*} $$

(H2). Given $0 < m \leq n+1$ , we also assume that for all $\alpha \in \mathcal {A}$ , $\gamma ^{(n+1)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n+1,m)}$ for $x \in I^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n+1)} )$ , and that for $z \in \mathbb {C}$ ,

(4.8) $$ \begin{align} T_{\alpha}^{(n+1,m)} (z)={ \hat T}_{\alpha}^{(n+1,m)} (z). \end{align} $$

We need to relate the breaking sequence at the $(m-1)$ -step of the Rauzy induction with our map $ T_{\alpha }^{(n+1,m-1)}$ . We do this in a separate appendix at the end of the paper.

Case 1. Fix $\alpha =\beta _{1,m-1}$ . The Rauzy induction is either of type 1 or type 0 and we have $\beta _{\epsilon ,m}= (\pi _{\epsilon }^{(m)})^{-1}(d)$ . We prove now that $\gamma ^{(n+1)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}$ into $T^{(n+1,m-1)}$ for all $x \in I_{\alpha }^{(m-1)}\backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1}(I^{(n+1)})$ , that is,

(4.9) $$ \begin{align} \gamma^{(n+1)}(f_{\unicode{x3bb}^{(m-1)},\pi^{(m-1)}}(x) ) = T^{(n+1,m-1)}(\gamma^{(n+1)}(x)). \end{align} $$

Step 4. Combining equations (A.9) and (A.6), for any $x \in I_{\alpha }^{(m-1)} \backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1}( I^{(n+1)} ) $ and $z \in \mathbb {C} $ replacing $x= x_{0,\hat {\pi }^{(m-1)(d)}}^{(m-1)} - \delta $ and taking $\delta \rightarrow 0^+$ , we get

(4.10) $$ \begin{align} T_{\alpha}^{(n+1,m-1)}(z)= e^{i \theta_{\alpha}^{(m-1)}}( z - \gamma_{0,\hat{\pi}^{(m-1)}(d)}^{n+1,m-1} ) + \gamma_{0,d}^{n+1,m-1}, \end{align} $$

and this can be written as

$$ \begin{align*} T_{\beta_{1,m-1}}^{(n+1,m-1)}(z) = {\hat T}_{\beta_{1,m-1}}^{(n+1,m-1)}(z). \end{align*} $$

In the next cases, we establish a relation between $T_{\alpha }^{(n+1,m-1)}$ , when $\alpha \neq \beta _{1,m-1}$ , and the breaking sequence at the step $n+1$ .

Note first that since we are assuming that $\gamma ^{(n+1)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n+1,m)}$ for $x \in I_{\alpha }^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n+1)}) $ , it follows that for these values of x, we have

(4.11) $$ \begin{align} T^{(n+1,m)}(\gamma^{(n+1)}(x))= \gamma^{(n+1)}(f_{\unicode{x3bb}^{(m)},\pi^{(m)}}(x)), \end{align} $$

with $\alpha \neq \beta _{1,m-1}$ , we get

$$ \begin{align*} T_{\beta_{1,m-1}}^{(n+1,m-1)}(z)= e^{i \theta_{\beta_{1,m-1}}^{(m-1)}}( z - \gamma_{0,\hat{\pi}^{(m-1)}(d)-1}^{n+1,m-1} ) + \gamma_{0,d-1}^{n+1,m-1}, \end{align*} $$

which, by equations (A.12) and (3.8), shows that $\xi _{d-1}^{n+1,m-1} = \gamma _{1,d-1}^{n+1,m-1} $ . Hence, by Lemma 3.4, we get that $ T_{\alpha }^{(n+1,m-1)} = {\hat T}_{\alpha }^{(n+1,m-1)} $ .

By equations (3.11) and (4.8), we get that $ T_{\alpha }^{(n+1,m-1)} = T_{\alpha }^{(n+1,m)} $ , and by equation (A.13), we get

$$ \begin{align*} T_{\alpha}^{(n+1,m-1)}(\gamma^{(n+1)}(x))= \gamma^{(n+1)}(f_{\unicode{x3bb}^{(m)},\pi^{(m)}}(x)) \end{align*} $$

for $x \in I_{\alpha }^{(m)} \backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n+1)} ) $ . Since $f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}(x) = f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}(x) $ , for $x \in I_{\alpha }^{(m)}$ , we get

$$ \begin{align*}T_{\alpha}^{(n+1,m-1)}(\gamma^{(n+1)}(x))= \gamma^{(n+1)}(f_{\unicode{x3bb}^{(m-1)},\pi^{(m-1)}}(x)).\end{align*} $$

We proved that for all $z \in \mathbb {C}$ ,

$$ \begin{align*} T_{\alpha}^{(n+1,m-1)} (z)={ \hat T}_{\alpha}^{(n+1,m-1)} (z), \end{align*} $$

and from equation (A.14), we get for all $\alpha \in \mathcal {A}$ , $\gamma ^{(n+1)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}$ into $T^{(n+1,m-1)}$ for $x \in I^{(m-1)}\backslash f_{\unicode{x3bb} ^{(m-1)},\pi ^{(m-1)}}^{-1}( I^{(n+1)})$ . Thus, for all $0\leq m \leq n+1$ and $\alpha \in \mathcal {A}$ , we have that equations (4.8) and (A.13) hold, and therefore $\gamma ^{(n+1)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n+1,m)}$ for $x \in I^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n+1)} )$ .

This shows that for all $n\geq 0$ , $0\leq m \leq n$ and $\alpha \in \mathcal {A}$ that $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}$ into $T^{(n,m)}$ for $x \in I^{(m)}\backslash f_{\unicode{x3bb} ^{(m)},\pi ^{(m)}}^{-1}( I^{(n)})$ and for all $z \in \mathbb {C}$ , we have equation (4.7). This finishes our proof.

5 Existence of embeddings of IETs into PWIs

In this section, we prove the existence of non-trivial embeddings of IETs into PWIs. We introduce the family $\mathcal {F}_{\theta }$ of PWIs which are $\theta $ -adapted to an IET $(\unicode{x3bb} ,\pi )$ and show, in Theorem 5.1, that when $\gamma _{\theta }^{(n)}$ converges to a topological embedding $\gamma _{\theta }$ , then the latter is an isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any $\theta $ -adapted PWI. We recall some classical notions of the theory of IETs, in particular, the Zorich cocycle and the characterization of its Oseledets flags and associated Lyapunov spectrum, as well as the translation surface of genus $g(\mathfrak {R})$ associated to an IET. We use these results to prove an important bound on a quantity determined by the Rauzy cocycle in Lemma 5.7. We introduce a submanifold $W_{[\unicode{x3bb} ],\pi }^{\delta }$ of the torus $\mathbb {T}^{\mathcal {A}}$ related to the Oseledets flags of the Zorich cocycle for the underlying IET and use Lemma 5.7 to determine a bound for the sequence $\theta ^{(n)}$ when $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ . This result together with Theorem 5.1 are the key ingredients in the proof of Theorem 5.9, which states that for a full measure set of IETs, if $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , then $\gamma _{\theta }^{(n)}$ converges to a Lipschitz map $\gamma _{\theta }$ , which is an isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any $\theta $ -adapted PWI. The embedding resulting from Theorem 5.9 may, however, be trivial. Thus, we define a submanifold $\mathcal {W}_{[\unicode{x3bb} ],\pi }^{\delta } \subset W_{[\unicode{x3bb} ],\pi }^{\delta }$ which we show, in Theorem 5.11, has full measure when $g(\mathfrak {R})\geq 2$ , for which the embedding $\gamma _{\theta }$ is guaranteed to be non-trivial.

Given $(\unicode{x3bb} ,\pi )\in \mathbb {R}_+^{\mathcal {A}}\times \mathfrak {S}({\mathcal {A}})$ , recall we denote by $\Theta _{\unicode{x3bb} ,\pi }$ the set of all $\theta \in \mathbb {T}^{\mathcal {A}}$ such that for all $n \geq 0$ , $\gamma _{\theta }^{(n)}:I \rightarrow \mathbb {C}$ is an injective map. Let $\Theta _{\unicode{x3bb} ,\pi }'$ denote the set of all $\theta \in \Theta _{\unicode{x3bb} ,\pi }$ for which there exists a topological embedding $\gamma _{\theta }:I \rightarrow \mathbb {C}$ such that for all $x \in I$ ,

$$ \begin{align*} \gamma_{\theta}(x)= \lim_{n \rightarrow +\infty}\gamma_{\theta}^{(n)}(x). \end{align*} $$

Furthermore, given $\theta \in \Theta _{\unicode{x3bb} ,\pi }'$ , we say that a PWI $T:X \rightarrow X$ together with a partition $\{X_{\alpha }\}_{\alpha \in \mathcal {A}}$ is $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ if for all $\alpha \in \mathcal {A}$ :

1. (i) $X_{\alpha }\supseteq \gamma _{\theta }(I_{\alpha })$ ;

2. (ii) with $x_j= x_{0,j}^{(0)}$ and

   (5.1) $$ \begin{align} T_{\alpha}(z)=e^{i \theta_{\alpha}}( z - \gamma_{\theta}(x_{\pi_0(\alpha)-1}) ) + \gamma_{\theta}(f_{\unicode{x3bb},\pi}(x_{\pi_0(\alpha)-1})), \end{align} $$
   for all $z \in \mathbb {C}$ , we have $T(z)=T_{\alpha }(z)$ for all $z \in X_{\alpha }$ .

We denote the family of PWIs which are $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ by $\mathcal {F}_{\theta }$ .

Recall that we say there is an embedding of an IET $(I,f_{\unicode{x3bb} ,\pi })$ into a PWI $(X,T)$ if there exists a topological embedding $\gamma :I \rightarrow \mathbb {C}$ such that for all $x\in I$ ,

$$ \begin{align*} \gamma\circ f_{\unicode{x3bb},\pi}(x) = T\circ \gamma (x). \end{align*} $$

Given $x \in I$ , consider the family $\Pi (x)$ of points $0=t_0<t_1<\cdots <t_N=x$ . Given $\theta \in \Theta _{\unicode{x3bb} ,\pi }'$ , define a map $\mathcal {L}_{\theta }: I \rightarrow \mathbb {R}_+ $ by

$$ \begin{align*} \mathcal{L}_{\theta}(x)= \sup_{(t_0,\ldots,t_N) \in \Pi(x)} \sum_{j=0}^{N-1}| \gamma_{\theta}(t_{j+1}) - \gamma_{\theta}(t_{j}) |. \end{align*} $$

We say a map $\gamma _{\theta }$ is an isometric embedding of an IET $(I,f_{\unicode{x3bb} ,\pi })$ into a PWI $(X,T)$ if it is an embedding and $\mathcal {L}_{\theta }(x)=x$ for all $x \in I$ .

The following theorem states that when $\gamma _{\theta }^{(n)}$ converges to a topological embedding $\gamma _{\theta }$ , it is also an isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any PWI which is $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ . The proof follows from estimates related to the facts that the restriction of any PWI in $\mathcal {F}_{\theta }$ to $\gamma _{\theta }(I)$ can be approximated by the map $T^{(n,0)}$ with increasing precision as $n \rightarrow +\infty $ , and that Theorem 4.3 guarantees that $\gamma ^{n}$ is a quasi-embedding of $f_{\unicode{x3bb} ,\pi }$ into $T^{(n,0)}$ for points in $I \backslash f_{\unicode{x3bb} ,\pi }^{-1}(I^{(n)})$ , which implies that the conjugacy between these two maps only fails to hold for points in an interval which is decreasing with n.

Proof. For any map $g:I \rightarrow \mathbb {C}$ , denote $\| g \|_{\infty } = \sup _{x \in I}|g(x)|$ .

As $f_{\unicode{x3bb} ,\pi }$ is a bijective map, we have

$$ \begin{align*} \| \gamma_{\theta} \circ f_{\unicode{x3bb},\pi} - \gamma_{\theta}^{(n)} \circ f_{\unicode{x3bb},\pi} \|_{\infty} = \| \gamma_{\theta} - \gamma_{\theta}^{(n)} \|_{\infty}, \end{align*} $$

which as $\theta \in \Theta _{\unicode{x3bb} ,\pi }'$ , shows that

(5.2) $$ \begin{align} \lim_{n \rightarrow +\infty}\| \gamma_{\theta} \circ f_{\unicode{x3bb},\pi} - \gamma_{\theta}^{(n)} \circ f_{\unicode{x3bb},\pi} \|_{\infty}=0. \end{align} $$

From equation (3.9) and Theorem 4.3, for any $\alpha \in \mathcal {A}$ and $x \in I$ , we have

$$ \begin{align*} T_{\alpha}^{(n,0)}(\gamma_{\theta}^{(n)}(x)) = e^{i \theta_{\alpha}}( \gamma_{\theta}^{(n)}(x) - \gamma_{0,\pi_0(\alpha)-1}^{n,0} ) + \gamma_{1,\pi_1(\alpha)-1}^{n,0}, \end{align*} $$

and by equation (5.1) applying the triangle inequality, we get

$$ \begin{align*} \begin{array}{lr} \| T_{\alpha}^{(n,0)} \circ \gamma_{\theta}^{(n)} - T_{\alpha} \circ \gamma_{\theta} \|_{\infty} \leq \\ \| \gamma_{\theta}^{(n)} - \gamma_{\theta} \|_{\infty} + | \gamma_{\theta}(x_{\pi_0(\alpha)-1}) - \gamma_{0,\pi_0(\alpha)-1}^{n,0}| + |\gamma_{1,\pi_1(\alpha)-1}^{n,0} - \gamma_{\theta}(f_{\unicode{x3bb},\pi}(x_{\pi_0(\alpha)-1})) |, \end{array} \end{align*} $$

which, as $\theta \in \Theta _{\unicode{x3bb} ,\pi }'$ , shows that

(5.3) $$ \begin{align} \lim_{n \rightarrow +\infty} \| T^{(n,0)} \circ \gamma_{\theta}^{(n)} - T \circ \gamma_{\theta} \|_{\infty}=0. \end{align} $$

By Theorem 4.3, $\gamma ^{(n)}$ is a quasi-embedding of $f_{\unicode{x3bb} ,\pi }$ into $T^{(n,0)}$ for all $x \in I \backslash f_{\unicode{x3bb} ,\pi }^{-1}(I^{(n)})$ and thus, we have

$$ \begin{align*} \gamma^{(n)}( f_{\unicode{x3bb},\pi}(x) ) = T^{(n,0)}( \gamma^{(n)}(x) ),\end{align*} $$

in particular, this gives

$$ \begin{align*} \| \gamma^{(n)} \circ f_{\unicode{x3bb},\pi} - T^{(n,0)} \circ \gamma^{(n)} \|_{\infty} \leq \sup_{x \in f_{\unicode{x3bb},\pi}^{-1}(I^{(n)}) } | \gamma^{(n)}( f_{\unicode{x3bb},\pi}(x) ) - T^{(n,0)}( \gamma^{(n)}(x) ) |.\end{align*} $$

For a sufficiently large $N>0$ , we have $f_{\unicode{x3bb} ,\pi }^{-1}(I^{(n)}) \subseteq I_{\pi _1^{-1}(1)} $ , whenever $n>N$ .

As $T^{(n,0)}(\gamma _{\theta }^{(n)}(x_{\hat {\pi }^{(n)}(1)}^{(n)}))= x_1^{(n)} \in \overline {I^{(n)}} $ and since $T_{\pi _1^{-1}(1)}^{(n,0)}$ is an isometry, we get that

$$ \begin{align*} \sup_{x \in f_{\unicode{x3bb},\pi}^{-1}(I^{(n)}) } | T^{(n,0)}( \gamma^{(n)}(x) ) | \leq 2 |I^{(n)}|.\end{align*} $$

Since $\sup _{x \in f_{\unicode{x3bb} ,\pi }^{-1}(I^{(n)}) } | \gamma ^{(n)}( f_{\unicode{x3bb} ,\pi }(x) ) | \leq |I^{(n)}|$ and $|I^{(n)}| \rightarrow 0$ as $n \rightarrow + \infty $ , this shows that

(5.4) $$ \begin{align} \lim_{n \rightarrow +\infty} \| \gamma^{(n)} \circ f_{\unicode{x3bb},\pi} - T^{(n,0)} \circ \gamma^{(n)} \|_{\infty}=0. \end{align} $$

By the triangle inequality, we have

$$ \begin{align*} \| \gamma_{\theta} \circ f_{\unicode{x3bb},\pi} - T \circ \gamma_{\theta} \|_{\infty} &\leq \| \gamma_{\theta} \circ f_{\unicode{x3bb},\pi} - \gamma_{\theta}^{(n)} \circ f_{\unicode{x3bb},\pi} \|_{\infty} + \| \gamma^{(n)} \circ f_{\unicode{x3bb},\pi} - T^{(n,0)} \circ \gamma^{(n)} \|_{\infty}\\ &\quad+ \| T^{(n,0)} \circ \gamma_{\theta}^{(n)} - T \circ \gamma_{\theta} \|_{\infty}. \end{align*} $$

Taking the limit as $n \rightarrow + \infty $ and by equations (5.2), (5.3) and (5.4), we get

$$ \begin{align*} \gamma_{\theta}\circ f_{\unicode{x3bb},\pi}(x) = T \circ \gamma_{\theta} (x) \end{align*} $$

for all $x \in I$ , which proves that $\gamma _{\theta }$ is an embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into $(X,T)$ .

Finally, given $x {\kern-1pt}\in{\kern-1pt} I$ , consider $0{\kern-1pt}={\kern-1pt}t_0{\kern-1pt}<{\kern-1pt}t_1{\kern-1pt}<\cdots {\kern-1pt}<{\kern-1pt}t_N=x$ . For all $n{\kern-1pt}\geq{\kern-1pt} 0$ , $\gamma _{\theta }^{(n)} \in \mathcal {PL}(|I|)$ from which follows that $| \gamma _{\theta }^{(n)}(t_{j+1}) - \gamma _{\theta }^{(n)}(t_{j}) |= | t_{j+1}-t_{j} |$ for any $j=0,\ldots ,N-1$ . Hence, as $\theta \in \Theta _{\unicode{x3bb} ,\pi }'$ , we get

$$ \begin{align*} x = \sum_{j=0}^{N-1}| \gamma_{\theta}^{(n)}(t_{j+1}) - \gamma_{\theta}^{(n)}(t_{j}) | \rightarrow \sum_{j=0}^{N-1}| \gamma_{\theta}(t_{j+1}) - \gamma_{\theta}(t_{j}) |, \quad \textrm{as } n \rightarrow + \infty, \end{align*} $$

which shows that $\mathcal {L}_{\theta }(x)=x$ , finishing our proof.

Following [Reference Avila and Forni7, Reference Avila and Leguile8], let $\mathbb {P}_+^{\mathcal {A}} = \mathbb {P}(\mathbb {R}_+^{\mathcal {A}}) \simeq \mathbb {P}_+^{\mathcal {A}}$ denote the projectivization of $\mathbb {R}_{+}^{\mathcal {A}}$ . Let $\mathfrak {R} \subseteq \mathfrak {S}({\mathcal {A}})$ be a Rauzy class. Since $\mathcal {R}$ commutes with dilations on $\mathbb {R}_{+}^{\mathcal {A}}$ , it projectivizes to a map $\mathcal {R}_R : \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R}$ called the Rauzy renormalization map which is defined in the complement of countably many hyperplanes. Moreover, we have that if $[\unicode{x3bb} ] = [\unicode{x3bb} ']$ , then $B_R(\unicode{x3bb} ',\pi )=B_R(\unicode{x3bb} ,\pi )$ for any $\pi \in \mathfrak {R}$ , and hence the application $([\unicode{x3bb} ],\pi ) \rightarrow B_R([\unicode{x3bb} ],\pi )$ is well defined. We refer to this cocycle as the Rauzy cocycle as well.

An induction scheme $\mathcal {S}: \mathbb {R}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow \mathbb {R}_{+}^{\mathcal {A}} \times \mathfrak {R}$ is an acceleration of Rauzy induction if there exists an integral application $m: \mathbb {R}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow \mathbb {Z}_+$ such that for every $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_{+}^{\mathcal {A}} \times \mathfrak {R}$ , we have $m(a\unicode{x3bb} ,\pi )= m(\unicode{x3bb} ,\pi )$ for all $a>0$ and

$$ \begin{align*}\mathcal{S}(\unicode{x3bb},\pi) = \mathcal{R}^{m(\unicode{x3bb},\pi)}(\unicode{x3bb},\pi).\end{align*} $$

It is direct to see that $\mathcal {S}$ also commutes with dilations on $\mathbb {R}_{+}^{\mathcal {A}}$ and hence it projectivizes to a map $\mathcal {S}_R : \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R}$ which we call an acceleration of Rauzy renormalization. Moreover, we have that if $A:\mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow SL({\mathcal {A}},\mathbb {Z})$ defines a cocycle over $\mathcal {S}$ , then its projectivization $([\unicode{x3bb} ],\pi ) \rightarrow A([\unicode{x3bb} ],\pi )$ is well defined.

A flag, on an N-dimensional vector space F, is a decreasing family of vector subspaces $\{F^j\}_{j=1,\ldots ,k+1}$ , with $k\leq N$ ,

$$ \begin{align*} F = F^1 \supsetneq F^2 \supsetneq \cdots \supsetneq F^k \supsetneq \{0\}= F^{k+1}. \end{align*} $$

The flag is said to be complete if $k=N$ and $\dim F^j = N+1-j$ for all $j=1,\ldots ,N$ .

The following well-known result follows from Oseledets theorem [Reference Oseledec18].

Theorem 5.2. Let $\mathfrak {R} \subseteq \mathfrak {S}({\mathcal {A}})$ be a Rauzy class, $\mathcal {S}_R: \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R}$ be an acceleration of Rauzy renormalization which is measurable with respect to an ergoric measure $m_{\mathfrak {R}}$ and let $A:\mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R} \rightarrow SL({\mathcal {A}},\mathbb {Z})$ be a $m_{\mathfrak {R}}$ -measurable cocycle over $\mathcal {S}_R$ .

There exist $\kappa (\mathfrak {R}) \in \mathbb {N}$ , real numbers $\nu _1(\mathfrak {R})>\cdots >\nu _{\kappa (\mathfrak {R})}(\mathfrak {R})$ and for $m_{\mathfrak {R}}$ -almost every (a.e.) $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R}$ , there exists a flag $ \mathbb {R}^{\mathcal {A}} = V_{[\unicode{x3bb} ],\pi }^1 \supsetneq \cdots \supsetneq V_{[\unicode{x3bb} ],\pi }^{\kappa (\mathfrak {R})} \supsetneq \{0\} = V_{[\unicode{x3bb} ],\pi }^{\kappa (\mathfrak {R})+1} $ such that $A([\unicode{x3bb} ],\pi ) \cdot V_{[\unicode{x3bb} ],\pi }^{j}=V_{\mathcal {S}_R([\unicode{x3bb} ],\pi )}^{j}$ and

$$ \begin{align*} \lim_{n\rightarrow +\infty}\frac{1}{n}\log \| A^{(n)}([\unicode{x3bb}],\pi)\cdot v \| = \nu_j(\mathfrak{R}) \end{align*} $$

for all $v \in V_{[\unicode{x3bb} ],\pi }^{j} \backslash V_{[\unicode{x3bb} ],\pi }^{j+1}$ , $j=1,\ldots ,\kappa (\mathfrak {R})$ .

The spaces $V_{[\unicode{x3bb} ],\pi }^j$ are called Oseledets subspaces and the numbers $\nu _j(\mathfrak {R})$ are called the Lyapunov exponents of the cocycle. The integer $\dim V_{[\unicode{x3bb} ],\pi }^j -\dim V_{[\unicode{x3bb} ],\pi }^{j+1}$ is called the multiplicity of the Lyapunov exponent $\nu _j(\mathfrak {R})$ and it is constant in a full measure set. The Lyapunov spectrum of the cocycle is the set of its Lyapunov exponents counted with multiplicity.

In [Reference Veech22], Veech proved that Rauzy renormalization admits an absolutely continuous ergodic measure. This measure, however, is not finite and thus the Rauzy cocycle is not measurable with respect to it.

In [Reference Zorich26], Zorich defined an acceleration of Rauzy induction as follows. Given $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_{+}^{\mathcal {A}} \times \mathfrak {S}({\mathcal {A}})$ , let $n(\unicode{x3bb} ,\pi )$ denote the smallest $n \in \mathbb {N}$ such that $\varepsilon (n) \neq \varepsilon (0)$ and set

$$ \begin{align*} \mathcal{Z}(\unicode{x3bb},\pi)= \mathcal{R}^{n(\unicode{x3bb},\pi)}(\unicode{x3bb},\pi). \end{align*} $$

The map $\mathcal {Z}$ is called Zorich induction and it projectivizes to a map $\mathcal {Z}_R {\kern-1pt}: {\kern-1pt}\mathbb {P}_{+}^{\mathcal {A}} {\kern-1pt}\times{\kern-1pt} \mathfrak {R}{\kern-1pt} \rightarrow {\kern-1pt}\mathbb {P}_{+}^{\mathcal {A}} {\kern-1pt}\times{\kern-1pt} \mathfrak {R}$ called Zorich renormalization.

Define the matrix function $B_{Z} : \mathbb {R}^{\mathcal {A}} \times \mathfrak {R} \rightarrow SL({\mathcal {A}},\mathbb {Z})$ by

$$ \begin{align*} B_{Z}(\unicode{x3bb},\pi) = B(\unicode{x3bb}^{( n(\unicode{x3bb},\pi) -1 )},\pi^{( n(\unicode{x3bb},\pi) -1 )}) \cdot \cdots \cdot B(\unicode{x3bb}^{(1)},\pi^{(1)}) \cdot B(\unicode{x3bb},\pi).\end{align*} $$

The Zorich cocycle is the linear cocycle over the Zorich induction $(\mathcal {Z},B_{Z})$ on $\mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R}\times \mathbb {R}^{\mathcal {A}}$ . Its projectivization $(\mathcal {Z}_R,B_{Z})$ is well defined and also called Zorich cocycle.

Let $\| \cdot \|$ denote a matrix norm on $SL({\mathcal {A}},\mathbb {Z})$ and let $\| A\|_{0}= \max \{ \| A \|, \| A \|^{-1} \}$ for any $A \in SL({\mathcal {A}},\mathbb {Z})$ . Recall we denote $\log ^+y= \max \{\log (y) ,0 \}$ for any $y>0$ .

Theorem 5.4. [Reference Zorich26]

Let $\mathfrak {R}\subset \mathfrak {S}({\mathcal {A}})$ be a Rauzy class. Then,

$$ \begin{align*} \int_{\mathbb{P}_{+}^{\mathcal{A}}\times \mathfrak{R} } \log^+ \| B_Z \|_0 \,d\mu_{\mathfrak{R}} < + \infty. \end{align*} $$

In particular, $B_Z$ is a measurable cocycle with respect to $\mu _{\mathfrak {R}} $ .

Recall the linear map $\Omega _{\pi }$ in equation (2.1). Let $H_{\pi }$ be the image subspace of $\Omega _{\pi }$ , that is, $H_{\pi }=\Omega _{\pi }(\mathbb {R}^{\mathcal {A}})$ . From [Reference Avila and Forni7, Reference Veech23], it follows that

(5.5) $$ \begin{align} B(\unicode{x3bb},\pi)\cdot H_{\pi} = H_{\pi^{(1)}}, \end{align} $$

from which follows that $\dim H_{\pi }$ only depends on the Rauzy class $\mathfrak {R} \subset \mathfrak {S}({\mathcal {A}})$ of $\pi $ .

A translation surface (as defined in [Reference Avila and Forni7]) is a surface with a finite number of conical singularities endowed with an atlas such that coordinate changes are given by translations in $\mathbb {R}^2$ . Given $(\unicode{x3bb} ,\pi ) \in \mathbb {R}_+^{\mathcal {A}} \times \mathfrak {R}$ , it is possible (see for instance [Reference Veech22]) to associate, via a suspension construction, a translation surface, with genus $g(\mathfrak {R})\geq 1$ and $\kappa $ singularities depending only on $\mathfrak {R}$ . Moreover, $\dim H_{\pi } = 2 g(\mathfrak {R})$ .

By equation (5.5), it is direct to see that $H_{\pi }$ is an invariant subspace for both Rauzy and Zorich cocycles. Hence, we can consider restrictions $B_R([\unicode{x3bb} ],\pi )|_{H_\pi }$ and $B_Z([\unicode{x3bb} ],\pi )|_{H_\pi }$ as integral cocycles over $\mathcal {R}_R$ and $\mathcal {Z}_R$ , respectively, which we call restricted Rauzy and Zorich cocycles. To simplify the notation, we, at times, write $B_R([\unicode{x3bb} ],\pi )$ and $B_Z([\unicode{x3bb} ],\pi )$ instead of $B_R([\unicode{x3bb} ],\pi )|_{H_\pi }$ and $B_Z([\unicode{x3bb} ],\pi )|_{H_\pi }$ .

As a consequence of Theorems 5.2 and 5.4, for any Rauzy class $\mathfrak {R}\subset \mathfrak {S}({\mathcal {A}})$ , there exist $k(\mathfrak {R}) \in \mathbb {N}$ such that for $\mu _{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_{+}^{\mathcal {A}} \times \mathfrak {R}$ , there exists a flag of Oseledets subspaces $ H_{\pi } = F_{[\unicode{x3bb} ],\pi }^1 \supsetneq \cdots \supsetneq F_{[\unicode{x3bb} ],\pi }^{k(\mathfrak {R})} \supsetneq \{0\} = F_{[\unicode{x3bb} ],\pi }^{k(\mathfrak {R})+1} $ with an associated Lyapunov spectrum

$$ \begin{align*}\vartheta_{1}(\mathfrak{R})>\cdots> \vartheta_{k(\mathfrak{R})}(\mathfrak{R}). \end{align*} $$

In [Reference Zorich26], it is shown that $k(\mathfrak {R}) \leq 2 g(\mathfrak {R})$ and that $\vartheta _j(\mathfrak {R})= - \vartheta _{k(\mathfrak {R})+1-j}(\mathfrak {R})$ for all $j =1,\ldots ,k(\mathfrak {R})$ . In [Reference Avila and Viana9], the authors proved that the Lyapunov spectrum of the restricted Zorich cocycle is simple on every Rauzy class, that is, all Lyapunov exponents have multiplicity 1. Consequently, the spectral properties of the restricted Zorich cocycle can be summarized as follows.

Theorem 5.5. Let $\mathfrak {R}\subset \mathfrak {S}({\mathcal {A}})$ be a Rauzy class. There exist Lyapunov exponents,

$$ \begin{align*} \vartheta_1(\mathfrak{R})> \cdots>\vartheta_{g(\mathfrak{R})}(\mathfrak{R})>0>\vartheta_{g(\mathfrak{R})+1}(\mathfrak{R})\!=\!-\vartheta_{g(\mathfrak{R})}(\mathfrak{R})>\cdots>\vartheta_{2g(\mathfrak{R})}(\mathfrak{R})=-\vartheta_{1}(\mathfrak{R}), \end{align*} $$

and, for $\mu _{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_{+}^{\mathcal {A}}\times \mathfrak {R}$ , there exists a complete flag

$$ \begin{align*} H_{\pi} = F_{[\unicode{x3bb}],\pi}^1 \supsetneq \cdots \supsetneq F_{[\unicode{x3bb}],\pi}^{2g(\mathfrak{R})} \supsetneq \{0\} = F_{[\unicode{x3bb}],\pi}^{2g(\mathfrak{R})+1} \end{align*} $$

such that $B_{Z}([\unicode{x3bb} ],\pi )|_{H_\pi } {\kern-1pt}\cdot{\kern-1pt} F_{[\unicode{x3bb} ],\pi }^{j}{\kern-1pt}={\kern-1pt}F_{\mathcal {Z}_R([\unicode{x3bb} ],\pi )}^{j}$ . For all $v \in F_{[\unicode{x3bb} ],\pi }^{j} \backslash F_{[\unicode{x3bb} ],\pi }^{j+1}$ , $j=1,\ldots , 2g(\mathfrak {R})$ ,

$$ \begin{align*} \lim_{n\rightarrow +\infty}\frac{1}{n}\log \| B_{Z}([\unicode{x3bb}],\pi)|_{H_\pi}\cdot v \| = \vartheta_j(\mathfrak{R}). \end{align*} $$

We say $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_{+}^{\mathcal {A}}\times \mathfrak {R}$ is generic if $([\unicode{x3bb} ],\pi )$ is in the full measure set of $\mathbb {P}_{+}^{\mathcal {A}}\times \mathfrak {R}$ from Theorem 5.5.

Let $\|\cdot \|_1:SL({\mathcal {A}},\mathbb {Z})\rightarrow \mathbb {R}_+$ be the norm,

$$ \begin{align*}\|A\|_1= \sum_{\alpha \in {\mathcal{A}}}\sum_{\beta \in {\mathcal{A}}} |A_{\alpha \beta}|.\end{align*} $$

Denote by $\textrm {Leb}$ the Lebesgue measure in $\mathbb {P}_{+}^{\mathcal {A}}$ and by $c_{\mathfrak {R}}$ the counting measure in a Rauzy class $\mathfrak {R}$ . The following theorem is a restatement of a result by Marmi, Moussa and Yoccoz [Reference Marmi, Moussa and Yoccoz16], and gives a bound for the growth of the Zorich cocycle for a full measure set of $([\unicode{x3bb} ],\pi )$ . The proof can be found in [Reference Marmi, Moussa and Yoccoz16, §4.7].

Theorem 5.6. [Reference Marmi, Moussa and Yoccoz16]

For $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ and $\varepsilon '>0$ , there exists $C_{\varepsilon '}>0$ such that for any $m\geq 0$ ,

$$ \begin{align*} \| B_{Z}( \mathcal{Z}_R^{m}([\unicode{x3bb}],\pi) ) \|_1 < C_{\varepsilon'} \| B_{Z}^{(m)}( [\unicode{x3bb}],\pi ) \|_1^{\varepsilon'}. \end{align*} $$

Given $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_{+}^{\mathcal {A}}\times \mathfrak {R}$ and $m \geq 0$ , denote the sum of the m first Zorich acceleration times by

$$ \begin{align*} s^m([\unicode{x3bb}],\pi) = \sum_{k<m} n (\mathcal{Z}_R^{k}([\unicode{x3bb}],\pi)).\end{align*} $$

So far, the choice of vector norm $\| \cdot \|$ has not been relevant as Theorem 5.2 does not depend on any particular choice. However, in what follows, we consider $\| \cdot \|$ to be the euclidean norm.

In the following lemma, we combine estimates from Theorems 5.5 and 5.6 to obtain an important bound for the growth of the Rauzy cocycle, restricted to $F_{[\unicode{x3bb} ],\pi }^{g(\mathfrak {R})+1} \backslash \{0\}$ , for a full measure set of parameters.

Lemma 5.7. For $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , there exists $K\geq 1$ such that for all $v \in F_{[\unicode{x3bb} ],\pi }^{g(\mathfrak {R})+1} \backslash \{0\}$ , we have

(5.6) $$ \begin{align} \sum_{n=0}^{+\infty} \|B_{R}^{(n)}([\unicode{x3bb}],\pi)\cdot v \| < K \| v \|. \end{align} $$

Proof. By Theorem 5.5, for $\mu _{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ and any $0<\eta <1$ , there exists $K_{\eta }>0$ such that for every $m \geq 0$ ,

$$ \begin{align*} \| B_{Z}^{(m)}( [\unicode{x3bb}],\pi ) \|_1< K_{\eta} e^{\eta^{-1}\vartheta_1(\mathfrak{R}) m}. \end{align*} $$

As, by Theorem 5.4, $\mu _{\mathfrak {R}}$ has positive density, this also holds for $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ . Combined with Theorem 5.6, for $\varepsilon '=\tfrac 14\eta ^2 \vartheta _{g(\mathfrak {R})}(\mathfrak {R}) / \vartheta _1(\mathfrak {R}) $ , this gives

(5.7) $$ \begin{align} \| B_{Z}( \mathcal{Z}_R^{m}([\unicode{x3bb}],\pi) ) \|_1< K_{\eta} C_{\varepsilon'} e^{({1}/{4})\eta\vartheta_{g(\mathfrak{R})}(\mathfrak{R}) m} \end{align} $$

for $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ .

By Theorem 5.5, we also get that for $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , there exists $K_{\eta }'>0$ , such that, for any $v \in F_{[\unicode{x3bb} ],\pi }^{g(\mathfrak {R})+1} \backslash \{0\}$ , we have

(5.8) $$ \begin{align} \| B_{Z}^{(m)}( [\unicode{x3bb}],\pi ) \cdot v \|< K_{\eta}' e^{-\eta\vartheta_{g(\mathfrak{R})}(\mathfrak{R}) m} \| v \|. \end{align} $$

Let $\mathcal {E}_{\eta }$ denote the set of $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ for which there exists $K_{\eta }">0$ such that

(5.9) $$ \begin{align} \| B_{Z}( \mathcal{Z}_R^{m}([\unicode{x3bb}],\pi) ) \|_1^{2} \cdot \| B_{Z}^{(m)}( [\unicode{x3bb}],\pi ) \cdot v \| < K_{\eta}" e^{-({1}/{2}) \eta\vartheta_{g(\mathfrak{R})}(\mathfrak{R}) m} \| v \| \end{align} $$

for all $v \in F_{[\unicode{x3bb} ],\pi }^{g(\mathfrak {R})+1} \backslash \{0\}$ and $m\geq 0$ . By combining equations (5.7) and (5.8), we get that $\mathcal {E}_{\eta } $ is a set of full $\textrm {Leb} \times c_{\mathfrak {R}}$ measure.

Now, fix $0<\eta <1$ and $ ([\unicode{x3bb} ],\pi ) \in \mathcal {E}_{\eta } $ . For $n \geq 0$ , let $ M(n) = \max \{ m \geq 0 : s^m([\unicode{x3bb} ],\pi ) \leq n \}$ . Also, given positive integers $k_1<k_2$ , we denote

$$ \begin{align*}B^{(k_1,k_2)}([\unicode{x3bb}],\pi) = B([\unicode{x3bb}^{(k_2)}],\pi^{(k_2)}) \cdot B([\unicode{x3bb}^{(k_2-1)}],\pi^{(k_2-1)}) \cdot \cdots \cdot B([\unicode{x3bb}^{(k_1)}],\pi^{(k_1)}). \end{align*} $$

We have

(5.10) $$ \begin{align} \| B_{R}^{(n)}([\unicode{x3bb}],\pi) \cdot v \| \leq \max_{s^{M(n)}([\unicode{x3bb}],\pi) \leq k < n} \| B^{(s^{M(n)}([\unicode{x3bb}],\pi),k)}([\unicode{x3bb}],\pi) \cdot B_Z^{( M(n) )} ([\unicode{x3bb}],\pi) \cdot v \|. \end{align} $$

It is clear that we have

$$ \begin{align*} \max_{s^{M(n)}([\unicode{x3bb}],\pi) \leq k < n} \| B^{(s^{M(n)}([\unicode{x3bb}],\pi),k)}([\unicode{x3bb}],\pi) \|_1 \leq \| B_Z ( \mathcal{Z}_R^{ M(n) }([\unicode{x3bb}],\pi) ) \|_1 , \end{align*} $$

hence, from equation (5.10), for all $n \geq 0$ , we get

$$ \begin{align*}\| B_{R}^{(n)}([\unicode{x3bb}],\pi) \cdot v \| \leq \| B_{Z}( \mathcal{Z}_R^{M(n)}([\unicode{x3bb}],\pi) ) \|_1 \cdot \| B_{Z}^{(M(n))}( [\unicode{x3bb}],\pi ) \cdot v \|,\end{align*} $$

which, combined with the fact that for all $m \geq 0$ , we have $n(\mathcal {Z}_R^m([\unicode{x3bb} ],\pi )) \leq \| B_{Z}( \mathcal {Z}_R^{m}([\unicode{x3bb} ], \pi ) ) \|_1$ , gives

$$ \begin{align*} \sum_{n=0}^{+\infty} \|B_{R}^{(n)}([\unicode{x3bb}],\pi)\cdot v \| \leq \sum_{m=0}^{+\infty} \| B_{Z}( \mathcal{Z}_R^{m}([\unicode{x3bb}],\pi) ) \|_1^{2} \cdot \| B_{Z}^{(m)}( [\unicode{x3bb}],\pi ) \cdot v \|. \end{align*} $$

This, combined with equation (5.9), which holds since $([\unicode{x3bb} ],\pi ) \in \mathcal {E}_{\eta }$ , shows that by taking $K=\max \{K_{\eta }" (1-e^{-1/2\eta \vartheta _{g(\mathfrak {R})}(\mathfrak {R})})^{-1},1\}$ , we get equation (5.6) as intended.

Recalling equation (2.8), note that for any $\unicode{x3bb} , \unicode{x3bb} ' \in \mathbb {R}_{+}^{\mathcal {A}}$ such that $[\unicode{x3bb} ] = [\unicode{x3bb} ']$ , we have $B_{\mathbb {T}^{\mathcal {A}}}(\unicode{x3bb} ,\pi )= B_{\mathbb {T}^{\mathcal {A}}}(\unicode{x3bb} ',\pi )$ and thus $B_{\mathbb {T}^{\mathcal {A}}}(\unicode{x3bb} ,\pi )$ admits a projectivization which we denote $B_{\mathbb {T}^{\mathcal {A}}}([\unicode{x3bb} ],\pi )$ and also call projection of the Rauzy cocycle on $\mathbb {T}^{\mathcal {A}}$ .

Recall that $p:\mathbb {R}^{\mathcal {A}} \rightarrow \mathbb {T}^{\mathcal {A}}$ is the natural projection,

$$ \begin{align*} p(v) = ( ( v )_{\alpha}\,\mod\! 2 \pi )_{\alpha \in \mathcal{A}} \quad \textrm{for all } v \in \mathbb{R}^{\mathcal{A}}.\end{align*} $$

The flat torus is the torus $\mathbb {T}^{\mathcal {A}}$ viewed as a Riemannian manifold equipped with the flat Riemannian metric, which is the pushforward under p of the euclidean metric in $\mathbb {R}^{\mathcal {A}}$ . The flat Riemannian metric induces a distance on the torus $d_{\mathbb {T}^{\mathcal {A}}}: \mathbb {T}^{\mathcal {A}} \times \mathbb {T}^{\mathcal {A}} \rightarrow \mathbb {R}_+$ such that

$$ \begin{align*}d_{\mathbb{T}^{\mathcal{A}}}(\theta,\theta') = \inf \{ \| v-v' \| : v \in p^{-1}(\theta), \ v' \in p^{-1}(\theta') \} \end{align*} $$

for any $\theta , \theta ' \in \mathbb {T}^{\mathcal {A}}$ .

Given $\delta>0$ and a generic $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , let

$$ \begin{align*} E_{[\unicode{x3bb}],\pi}^{\delta} = \{ v \in F_{[\unicode{x3bb}],\pi}^{g(\mathfrak{R})+1} \backslash \{0\} : \| v \| <\delta \} ,\end{align*} $$

and let $ W_{[\unicode{x3bb} ],\pi }^{\delta } = p ( E_{[\unicode{x3bb} ],\pi }^{\delta } ) $ .

Recall equation (3.5), which, given $\theta \in \mathbb {T}^{\mathcal {A}}$ , defines a sequence $\{\theta ^{(n)}\}_{n\geq 0}$ on $\mathbb {T}^{\mathcal {A}}$ , which is used to construct the breaking sequence $\{\gamma _{\theta }^{(n)}\}_{n\geq 0}$ . The following lemma states that for a full measure set of $([\unicode{x3bb} ],\pi )$ , and for sufficiently small $\delta>0$ and all $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , the sum of all $d_{\mathbb {T}^{\mathcal {A}}}(\theta ^{(n)},0)$ is bounded.

Lemma 5.8. For $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , there exists $K\geq 1$ and $\delta>0$ such that for all $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , we have

(5.11) $$ \begin{align} \sum_{n=0}^{+\infty} d_{\mathbb{T}^{\mathcal{A}}}(\theta^{(n)},0) < K d_{\mathbb{T}^{\mathcal{A}}}(\theta,0). \end{align} $$

Proof. By Lemma 5.7, for $\textrm {Leb} \times c_{\mathfrak {R}}$ -a.e. $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , there exists $K>1$ such that for all $v \in E_{[\unicode{x3bb} ],\pi }^{\delta }$ , with $\delta = \pi \cdot K^{-1}$ for all $n\geq 0$ , we have

$$ \begin{align*} \| B_R^{(n)}([\unicode{x3bb}],\pi)\cdot v \| <\pi. \end{align*} $$

Moreover, it is clear that if $\| v\| < \pi $ , we have $d_{\mathbb {T}^{\mathcal {A}}}(p(v),0) = \| v \|$ , and thus, for all $n\geq 0$ , we have

(5.12) $$ \begin{align} d_{\mathbb{T}^{\mathcal{A}}}(p ( B_R^{(n)}([\unicode{x3bb}],\pi)\cdot v ),0) = \| B_R^{(n)}([\unicode{x3bb}],\pi)\cdot v \|. \end{align} $$

Also note that as $\delta \leq \pi $ , the restriction $p|_{E_{[\unicode{x3bb} ],\pi }^{\delta }} : E_{[\unicode{x3bb} ],\pi }^{\delta } \rightarrow W_{[\unicode{x3bb} ],\pi }^{\delta }$ is a bijection and thus $p^{-1}(\theta ) \cap E_{[\unicode{x3bb} ],\pi }^{\delta }$ contains a single point which we denote by $p_{\delta }^{-1}(\theta )$ . Take $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ . It is clear by equation (2.8) that we have

$$ \begin{align*} B_{\mathbb{T}^{\mathcal{A}}}^{(n)}([\unicode{x3bb}],\pi) \cdot \theta = p ( B_{R}^{(n)}([\unicode{x3bb}],\pi) \cdot p_{\delta}^{-1}(\theta) ) , \end{align*} $$

which, combined with equation (5.12), yields $ d_{\mathbb {T}^{\mathcal {A}}}(B_{\mathbb {T}^{\mathcal {A}}}^{(n)}([\unicode{x3bb} ],\pi ) \cdot \theta ,0) = \| B_{R}^{(n)}([\unicode{x3bb} ],\pi ) \cdot p_{\delta }^{-1}(\theta ) \|$ for all $n \geq 0$ . By equation (3.5) and Lemma 5.7, this gives equation (5.11), finishing our proof.

We say a map $\gamma : I \rightarrow \mathbb {C}$ is Lipschitz if $\{(\mathrm {Re}(\gamma (x)), \mathrm {Im}(\gamma (x))) : x \in I \}$ is the graph of a Lipschitz map. The following theorem shows that for a generic $([\unicode{x3bb} ],\pi )$ and sufficiently small $\delta>0$ , when $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , the sequence $\gamma _{\theta }^{(n)}$ converges to a Lipschitz map $\gamma _{\theta }$ , which is an isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any PWI that is $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ .

Proof. Consider the space $\mathcal {C}(I,\mathbb {C})$ of continuous maps from the interval I to $\mathbb {C}$ . Note that this is a Banach space for the supremum norm $\|.\|_{\infty }$ . We also have that $\gamma _{\theta }^{(n)} \in \mathcal {C}(I,\mathbb {C})$ for all $n \geq 0$ , since $\gamma _{\theta }^{(0)}$ is continuous and, by Lemma 3.1, $\mathfrak {Br}( \theta _{\beta _{1,n-1}}^{(n-1)} , J^{(n)}) \cdot \mathcal {C}(I,\mathbb {C})\subseteq \mathcal {C}(I,\mathbb {C})$ .

Take any $\varphi \in (0, \pi /2 )$ . By Lemma 5.8, there exists a set $\mathcal {E}\subseteq \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ of full $\textrm {Leb} \times c_{\mathfrak {R}}$ measure such that for every $([\unicode{x3bb} ],\pi ) \in \mathcal {E}$ , there exists $K\geq 1$ and $0<\delta < \varphi K^{-1}$ such that for all $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , we have equation (5.11).

Take $([\unicode{x3bb} ],\pi ) \in \mathcal {E}$ and $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ . For all $x \in I$ , we have

$$ \begin{align*} | \gamma_{\theta}^{(n+1)}(x)-\gamma_{\theta}^{(n)}(x)|= | \mathfrak{Br}( \theta_{\beta_{1,n}}^{(n)}, J^{(n+1)}) \cdot \gamma_{\theta}^{(n)}(x)-\gamma_{\theta}^{(n)}(x) |. \end{align*} $$

Denoting, as in equation (3.4), by $r(n)$ the number of intervals of $J^{(n+1)}$ , by equation (3.1), this gives

$$ \begin{align*} | \gamma_{\theta}^{(n+1)}(x)-\gamma_{\theta}^{(n)}(x)|\leq \max_{k < r(n)}\{ |\underline{\epsilon}_k| , |\overline{\epsilon}_k| \} +\sup_{x \in I}|\gamma_{\theta}^{(n)}(x)(1-e^{i \theta_{\beta_{1,n}}^{(n)} })|. \end{align*} $$

Since

$$ \begin{align*} \sup_{x \in I}| \gamma_{\theta}^{(n)}(x)(1-e^{i \theta_{\beta_{1,n}}^{(n)} }) | \leq 2 |\unicode{x3bb}|\sin( \theta_{\beta_{1,n}}^{(n)} /2), \end{align*} $$

by Lemma 3.2, we get

$$ \begin{align*} | \gamma_{\theta}^{(n+1)}(x)-\gamma_{\theta}^{(n)}(x)|\leq 4|\unicode{x3bb}|\sin( \theta_{\beta_{1,n}}^{(n)}/2 ). \end{align*} $$

Therefore, as $\theta _{\beta _{1,n}}^{(n)} \leq d_{\mathbb {T}^{\mathcal {A}}}(\theta ^{(n)},0)$ , there exists $C>0$ such that for all $n\geq 0$ ,

$$ \begin{align*} | \gamma_{\theta}^{(n+1)}(x)-\gamma_{\theta}^{(n)}(x)|\leq C |\unicode{x3bb}| d_{\mathbb{T}^{\mathcal{A}}}(\theta^{(n)},0). \end{align*} $$

Now take m, $n \in \mathbb {N}$ such that $m>n$ . Note that we have

$$ \begin{align*} \| \gamma_{\theta}^{(m)}- \gamma_{\theta}^{(n)} \|_{\infty}\leq \sum_{k=0}^{m-n-1}\| \gamma_{\theta}^{(m-k)}- \gamma_{\theta}^{(m-k-1)} \|_{\infty}, \end{align*} $$

and therefore,

(5.13) $$ \begin{align} \| \gamma_{\theta}^{(m)}- \gamma_{\theta}^{(n)} \|_{\infty}\leq C |\unicode{x3bb}| \sum_{k=n}^{m-1} d_{\mathbb{T}^{\mathcal{A}}}(\theta^{(k)},0). \end{align} $$

From equation (5.11), by taking a sufficiently large $N>0$ and considering $N<n<m$ , the right-hand side of equation (5.13) can be made arbitrarily small. Thus, $\{ \gamma _{\theta }^{(n)} \}_{n \geq 0}$ is a Cauchy sequence in $\mathcal {C}(I,\mathbb {C})$ and, therefore, it must converge to a unique limit $\gamma _{\theta } \in \mathcal {C}(I,\mathbb {C})$ .

As for all $n \geq 0$ , $\gamma _{\theta }^{(n)} \in \mathcal {C}(I,\mathbb {C})$ , by equation (3.6), it is simple to see that for any $x, y \in I$ , $x \neq y$ , we have

$$ \begin{align*}\frac{ | \mathrm{Im}(\gamma_{\theta}^{(n)}(x)) - \mathrm{Im}(\gamma_{\theta}^{(n)}(y)) | }{ | \mathrm{Re}(\gamma_{\theta}^{(n)}(x)) - \mathrm{Re}(\gamma_{\theta}^{(n)}(y)) |} \leq \tan \bigg( \sum_{n=1}^{+ \infty} \theta_{\beta_{1,n-1}}^{(n-1)} \bigg). \end{align*} $$

For any map $\gamma :I\rightarrow \mathbb {C}$ , its Lipschitz constant $L(\gamma )$ is given by

$$ \begin{align*} L(\gamma) = \sup_{x, y \in I, \ x\neq y} \frac{ | \mathrm{Im}(\gamma(x)) - \mathrm{Im}(\gamma(y)) | }{ | \mathrm{Re}(\gamma(x)) - \mathrm{Re}(\gamma(y)) |}. \end{align*} $$

Hence, in particular, we get

$$ \begin{align*} \arctan(L(\gamma_{\theta}^{(n)}))\leq \sum_{n=0}^{+ \infty} d_{\mathbb{T}^{\mathcal{A}}}(\theta^{(n)},0), \end{align*} $$

which, as $\delta < \varphi K^{-1}$ , by equation (5.11), gives $ \arctan (L(\gamma _{\theta }^{(n)}))\leq \varphi $ . Clearly, $L(\gamma _{\theta })\!\leq \!\sup _{n\geq 0}L(\gamma _{\theta }^{(n)})$ , and as $\varphi <\pi /2$ , this shows that $L(\gamma _{\theta } )< + \infty $ and thus $\gamma _{\theta }$ is a Lipschitz map. In particular, it is continuous and injective, and thus a topological embedding.

This proves that $W_{[\unicode{x3bb} ],\pi }^{\delta }\subseteq \Theta _{\unicode{x3bb} ,\pi }' $ and therefore by Theorem 5.1, for any $\theta \in W_{[\unicode{x3bb} ],\pi }^{\delta }$ , $\gamma _{\theta }$ is an isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any PWI that is $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ .

As described in [Reference Ashwin, Goetz, Peres and Rodrigues6], we can extend Rauzy–Veech induction to PWIs which admit embeddings of IETs as follows. Assume $(I,f_{\unicode{x3bb} ,\pi })$ has an embedding by $\gamma _{\theta }$ into $(X,T)$ . Define the map $\mathcal {S}(T)$ as the first return map under T to $X^*$ , where

$$ \begin{align*} X^*=\begin{cases} \bigcup_{\alpha \neq \beta_0} X_{\alpha} \cup (X_{\beta_{0}} \cap T(X_{\beta_{1}})) & \textrm{if} \ (\unicode{x3bb},\pi) \ \textrm{has type} \ 0,\\ \bigcup_{\alpha \neq \beta_0} X_{\alpha} & \textrm{if} \ (\unicode{x3bb},\pi) \ \textrm{has type} \ 1. \end{cases} \end{align*} $$

It is clear that $(X^*,\mathcal {S}(T))$ is again a $d'$ -PWI, with possibly $d' \neq d$ . Denote by $\mathcal {A}'$ an alphabet with $d'$ symbols and denote by $\{X_{\alpha '}^*\}_{\alpha ' \in \mathcal {A}'}$ the partition of $X^*$ . It is simple to see that there is a collection of d symbols $\mathcal {A} \subseteq \mathcal {A}'$ , possibly after relabelling, such that $X_{\alpha '}^* \cap \gamma _{\theta }(I^{(1)}) \neq \emptyset $ if and only if $\alpha ' \in \mathcal {A}$ . Define $X'= \bigcup _{\alpha \in \mathcal {A}}X_{\alpha }^*$ .

Now, $(X',\mathcal {S}(T))$ is $\theta ^{(1)}$ -adapted to $(\unicode{x3bb} ^{(1)},\pi ^{(1)})$ and, by [Reference Ashwin, Goetz, Peres and Rodrigues6, Theorem 2.3], the restriction of $\gamma _{\theta }$ to $I^{(1)}$ is an embedding of $(I^{(1)},f_{\unicode{x3bb} ^{(1)},\pi ^{(1)}})$ into $(X',\mathcal {S}(T))$ .

It is thus possible to iterate this procedure by setting $(X^{(0)}, \mathcal {S}^{(0)}(T)) = (X,T)$ and $( X^{(n)}, \mathcal {S}^{(n)}(T) ) = ( (X^{(n-1)})', \mathcal {S}(\mathcal {S}^{(n-1)}(T)) )$ for $n \geq 1$ . The following lemma easily follows from Theorem 5.1.

Given a generic $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ and $\delta>0$ , note that $W_{[\unicode{x3bb} ],\pi }^{\delta }$ defines a $g(\mathfrak {R})$ -dimensional submanifold embedded in the torus $\mathbb {T}^{\mathcal {A}}$ . Pulling back the flat metric by the embedding map, it is possible to construct a $g(\mathfrak {R})$ -volume form and thus define a positive measure $m_{g(\mathfrak {R})}$ on $W_{[\unicode{x3bb} ],\pi }^{\delta }$ .

Denote the projection on $\mathbb {T}^{\mathcal {A}}$ of the Oseledets subspace $F_{[\unicode{x3bb} ],\pi }^{2g(\mathfrak {R})}$ by $W_{[\unicode{x3bb} ],\pi }^{SS} = p( F_{[\unicode{x3bb} ],\pi }^{2g(\mathfrak {R})}) $ . Note that $W_{[\unicode{x3bb} ],\pi }^{SS}$ is a one-dimensional submanifold embedded in $\mathbb {T}^{\mathcal {A}}$ .

For any $n\geq 0$ and $\theta \in \mathbb {T}^{\mathcal {A}}$ , let $B_{\mathbb {T}^{\mathcal {A}}}^{(-n)}([\unicode{x3bb} ],\pi )\cdot \theta = \{ \theta ' \in \mathbb {T}^{\mathcal {A}} : B_{\mathbb {T}^{\mathcal {A}}}^{(n)}([\unicode{x3bb} ],\pi )\cdot \theta ' = \theta \}.$ Consider

$$ \begin{align*} \mathcal{W}_{[\unicode{x3bb}],\pi}^{\delta} = W_{[\unicode{x3bb}],\pi}^{\delta} \backslash \bigg( W_{[\unicode{x3bb}],\pi}^{SS} \cup \bigcup_{n=0}^{+\infty} B_{\mathbb{T}^{\mathcal{A}}}^{(-n)}([\unicode{x3bb}],\pi)\cdot 0 \bigg). \end{align*} $$

Recall the definitions of arc, linear and non-trivial embeddings in §1. The following theorem establishes that for any Rauzy class $\mathfrak {R}$ such that $g(\mathfrak {R})\geq 2$ and for a full measure set of $([\unicode{x3bb} ],\pi ) \in \mathbb {P}_+^{\mathcal {A}} \times \mathfrak {R}$ , when $\theta \in \mathcal {W}_{[\unicode{x3bb} ],\pi }^{\delta } $ for sufficiently small $\delta>0$ , $\gamma _{\theta }$ is a non-trivial isometric embedding of $(I,f_{\unicode{x3bb} ,\pi })$ into any PWI $(X,T)$ that is $\theta $ -adapted to $(\unicode{x3bb} ,\pi )$ . Since $(I,f_{\unicode{x3bb} ,\pi })$ is topologically conjugated to the restriction of $(X,T)$ to the image of the embedding $\gamma _{\theta }(I)$ , we have that the latter map is one-to-one and therefore $\gamma _{\theta }(I)$ is an invariant set for $(X,T)$ . Moreover, $\gamma _{\theta }(I)$ is a curve which is not a union of line segments or circle arcs. Thus, Theorem A follows directly from Theorem 5.11, which we now state and prove.