*(Underlying job: I am trying to construct an adic representation of a rotation.)*

The question involves an iterative construction. At step $n$, one constructs a partition $P_n$ of $(0,1)$ and a map $\beta_n \colon (0,1) \to (0,1)$ (it could be clearer to use some indices: $\beta_n \colon (0,1)_n \to (0,1)_{n+1}$). Roughly speaking this construction will allow to define a sequence of labels $P_n(x_n)$ of a point $x \in (0,1)$, and the problem is to retrieve $x$ from this sequence.

** (n=1)** Let $T_1$ be a rotation with irrational step $\theta \in (0,\frac12)$ and define the interval $J=(1-\theta,1)$. The integer $a= \lfloor \frac{1}{\theta} \rfloor -1 \geq 1$ is the smallest integer such that $(0,1)$ is covered by $J$, $T_1(J)$, $\ldots$, $T_1^{a+1}(J)$.

One has the periodic approximation of $T_1$ shown by the following tower decomposition with base $B=(a\theta, 1) = J \cup T_1^{a+1}(J)$:

Denote by $P_1$ the partition of $(0,1)$ whose blocks can be seen in the picture: $$ P_1 = \bigl\{J, T_1(J), \ldots, T_1^a(J), (a\theta, 1-\theta)\bigr\}. $$

The orbits of this periodic approximation define an equivalence relation: each point in $x\in J$ is glued with $T_1(x)$, $\ldots$, $T_1^a(x)$, and the class of a point $x \in (a\theta, 1-\theta)$ is the singleton $\{x\}$. We take the base $B$ as the set of representatives, and for every $x \in (0,1)$, we denote by $d_1(x) \in B$ the representative of the class of $x$.

Now, the transformation induced by $T_1$ on $B$ is isomorphic to a rotation $T_2$ with irrational step $$ \theta'=\frac{\theta}{1-a\theta}=\frac{1}{1+G(\theta)} > \frac{1}{2} $$ where $G$ is the Gauss function. The isomorphism $g_1\colon B \to (0,1)$ between the induced transformation and $T_2$ is the natural affine map. The map $\boxed{\beta_1=g_1 \circ d_1}$ sends the action space of $T_1$ to the action space of $T_2$.

** (n=2)** We proceed as before with $T_2$ and $J = (0, 1-\theta')$. This is the similar construction for the case of a step ($\theta'$) in $(\frac12, 1)$. This time, setting $\phi=1-\theta'$, one has a tower decomposition with base $B = (0, 1-a\phi)$, where $a = \lfloor \frac{1}{1-\theta} \rfloor -1$.

And we have the partition of $(0,1)$: $$ P_2 = \bigl\{J, T_2(J), \ldots, T_2^a(J), (\phi, 1-a\phi)\bigr\}, $$ and the representatives $d_2(x) \in B$.

Now, the transformation induced by $T_2$ on $B$ is isomorphic to the rotation $T_3$ with irrational step $$ \frac{1-(a+1)\phi}{1-a\phi}=\frac{G(\phi)}{1+G(\phi)} < \frac{1}{2}, $$ we denote by $g_2$ the isomorphism (affine map) and we set $\boxed{\beta_2=g_2 \circ d_2}$.

** (n)** We can continue so on. We get the partitions $P_n$ and the maps $\beta_n$.

My question is the following one. Denote $x_1=x$, $x_2= \beta_1(x_1)$, $\ldots$, $x_{n+1}=\beta_n(x_n)$, $\ldots$. Do we know (almost) every $x \in (0,1)$ if we know the block $P_n(x_n)$ of the partition $P_n$ to which $x_n$ belongs for every $n$ ?