2. Basic properties of the α-Farey map F_α, $0 \lt \alpha \lt 1$

First of all, note that there is a strong relation between the maps G_α from (1.3) and F_α from (1.4). For any $\alpha \in (0, 1)$, we get G_α as a induced transformation of F_α. This induced transformation is defined as follows.

For each $\alpha \in (0, 1)$ and $x\in \mathbb I_{\alpha} = [\alpha -1,\alpha)$, we put $j(0)=0$ if x = 0, and $j (x) = j_{\alpha}(x) = k$ if

1. (i) $x\ne 0$,

2. (ii) $F_{\alpha}^{\ell}(x) \notin (\frac{1}{1 + \alpha}, \frac{1}{\alpha}]$, $0 \le \ell \lt k$,

3. (iii) $F_{\alpha}^{k}(x) \in (\frac{1}{1 + \alpha}, \frac{1}{\alpha}]$.

Note that from definition (1.4) of F_α we then have that $F_{\alpha}^{k+1}(x)\in \mathbb I_{\alpha}$, which is the domain of G_α; see (1.3). In case $\tfrac{\sqrt{5}-1}{2} \lt \alpha \lt 1$ we further define $j(x)=0$ whenever $x\in [\tfrac{1}{\alpha +1} ,\alpha )$; see also Remark 1(i). As usual, we set that $F_{\alpha}^{0}(x) = x$. Now the induced transformation $F_{\alpha, J}$ is defined as:

\begin{equation*} F_{\alpha, J}(x) = F_{\alpha}^{j(x) + 1}(x)\quad \text{for}\ x \in \mathbb I_{\alpha}. \end{equation*}

The next proposition generalizes the result in [Reference Natsui30], where α was restricted to the interval $[\tfrac{1}{2}, 1]$.

Setting $A^{-} = \begin{pmatrix} -1 & 0 \\ 1 & 1 \end{pmatrix}$, $A^{+} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}$ and $A^{R} = \begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix}$, in view of (1.2) and (1.4) we can write F_α as:

\begin{equation*} F_{\alpha}(x) = \begin{cases} A^{-}x, & \text{if}\ x \in [\alpha - 1, 0), \\ A^{+}x, & \text{if}\ x \in [0, \frac{1}{1 + \alpha}], \\ A^{R}x, & \text{if}\ x\in (\frac{1}{1 + \alpha}, \frac{1}{\alpha}]. \end{cases} \end{equation*}

Define

\begin{equation*} A_{n}(x) = A(F_{\alpha}^{n-1}(x) )= \begin{cases} \begin{pmatrix} -1 & 0 \\ 1 & 1 \end{pmatrix} = (A^{-})^{-1}, & \text{if}\ F_{\alpha}^{n-1} (x) \in [\alpha - 1, 0), \\ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, & \text{if}\ F_{\alpha}^{n-1}(x) = 0,\\ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = (A^{+})^{-1}, & \text{if}\ F_{\alpha}^{n-1}(x) \in (0, \frac{1}{1 + \alpha}], \\ \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} = (A^{R})^{-1}, & \text{if}\ F_{\alpha}^{n-1}(x)\in (\frac{1}{1 + \alpha}, \frac{1}{\alpha}], \end{cases} \end{equation*}

for $x \in \mathbb I_{\alpha}$ and $n \ge 1$. We identify x with $(A_{1}(x), A_{2}(x), \ldots , A_{n}(x), \ldots)$. We will show that

\begin{equation*} \lim_{n \to \infty} \left(A_{1}(x) A_{2}(x) \cdot \cdots A_{n}(x)\right)(-\infty) = x. \end{equation*}

Put $\begin{pmatrix}s_{n} & u_{n} \\ t_{n} & v_{n} \end{pmatrix} = A_{1}(x) A_{2}(x) \cdots A_{n}(x)$ with $\begin{pmatrix}s_{0} & u_{0} \\ t_{0} & v_{0} \end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 &1 \end{pmatrix} $. Suppose that

\begin{equation*} x = \frac{\phantom{00}{\varepsilon_{\alpha, 1}(x)}\vert} {\vert{a_{\alpha, 1}(x)}} + \frac{\phantom{00}{\varepsilon_{\alpha, 2}(x)}\vert} {\vert{a_{\alpha, 2}(x)}} + \cdots + \frac{\phantom{00}{\varepsilon_{\alpha, n}(x)}\vert} {\vert{a_{\alpha, n}(x)}} + \cdots \end{equation*}

is the α-continued fraction expansion of $x \in \mathbb I_{\alpha}$. Recall that $\varepsilon_{\alpha, n}(x)= \text{sgn}(G_{\alpha}^{n-1}(x))$ and $a_{\alpha, n}(x) = \lfloor\frac{1}{|G_{\alpha}^{n-1}(x)|}+ 1 -\alpha \rfloor$ for $G_{\alpha}^{n-1}(x) \ne 0$. Then, from Proposition 1, it is easy to see that $(A_{1}(x), A_{2}(x), \ldots , A_{n}(x), \ldots )$ is of the form

(2.1)\begin{equation} (A^{\pm}, \underbrace{A^{+}, \ldots , A^{+}}_{a_{\alpha, 1}(x) -2}, A^{R}, A^{\pm}, \underbrace{A^{+}, \ldots , A^{+}}_{a_{\alpha, 2}(x) -2}, A^{R}, \ldots ) , \end{equation}

unless $A_{m} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ appears in (2.1) for some $m\geq 1$ (which happens when $x\in\mathbb Q$). Here $A^{\pm} = A^{-}$ or $A^{+}$ according to $\varepsilon_{\alpha, n}(x) = -1$ or +1, respectively. If $a_{\alpha, n}(x) = 1$, then we read $a_{\alpha, n} -2 = 0$ and delete $A^{\pm}$ before $A^{+}$. More precisely,

\begin{equation*} \begin{array}{lcl} A_{\sum_{j=1}^{n} a_{\alpha, j}(x)}(x) &= & A^{R}, \\ A_{\sum_{j =1}^{n} a_{\alpha, j}(x) + 1}(x) &= &\left\{\begin{array}{ccl} A^{-}, & \text{if} & \varepsilon_{\alpha, n}(x) = -1, \\ A^{+}, & \text{if} & \varepsilon_{\alpha, n}(x) = +1 \quad \text{and} \quad a_{\alpha, n+1}(x) \ge 2, \end{array} \right. \\ A_{\sum_{j=1}^{n} a_{\alpha, j}(x) + \ell}(x) & =& A^{+} \qquad \text{if} \,\,\,\, 2 \le \ell \lt a_{\alpha, n+1}, \end{array} \end{equation*}

with $\sum_{j = 1}^{0} a_{\alpha, j}(x) = 0$. As usual, we have for the G_α-convergents of x that:

\begin{equation*} \begin{pmatrix} 0 & \varepsilon_{\alpha, 1}(x) \\ 1 & a_{\alpha, 1}(x) \end{pmatrix} \begin{pmatrix} 0 & \varepsilon_{\alpha, 2}(x)\\ 1 & a_{\alpha, 2}(x) \end{pmatrix} \cdots \begin{pmatrix} 0 & \varepsilon_{\alpha, n}(x) \\ 1 & a_{\alpha, n}(x) \end{pmatrix} = \begin{pmatrix} p_{\alpha, n-1}(x) & p_{\alpha, n}(x) \\q_{\alpha, n-1}(x) & q_{\alpha, n}(x) \end{pmatrix}. \end{equation*}

We have the following result.

Lemma 1. For $k,n\in\mathbb N$, let $\Pi_k(x) := A_{1}(x) A_{2}(x) \cdots A_{k}(x)$, and $S_{n}(x) = \sum_{j=1}^{n} a_{\alpha, j}(x)$. Then, if $k = S_{n}(x)$,

\begin{equation*} \Pi_k(x) = \begin{pmatrix} p_{\alpha, n-1}(x) & p_{\alpha, n}(x) \\ q_{\alpha, n-1}(x) & q_{\alpha, n}(x) \end{pmatrix}. \end{equation*}

Furthermore, if $\ell \geq 1$ and $k = S_{n}(x) + \ell \lt S_{n+1}(x)$, we have

\begin{equation*} \Pi_k(x) = \begin{pmatrix} \ell p_{\alpha, n}(x)+\varepsilon_{\alpha, n+1}(x)p_{\alpha, n-1}(x) & p_{\alpha, n-1}(x) \\ \ell q_{\alpha, n}(x)+\varepsilon_{\alpha, n+1}(x)q_{\alpha, n-1}(x) & q_{\alpha, n-1}(x) \end{pmatrix}. \end{equation*}

With this result in mind, and analogously to the regular case (which is α = 1), we call

\begin{equation*} (A_{1}(x) A_{2}(x) \cdots A_{k}(x))(-\infty) = \frac{\ell p_{\alpha, n}(x) + \varepsilon_{\alpha,n+1}p_{\alpha, n-1}(x)}{\ell q_{\alpha, n}(x) + \varepsilon_{\alpha,n+1}q_{\alpha, n-1}(x)} \end{equation*}

the ( $(n+1, \ell)$th) α-mediant convergent of x for $\ell \gt 0$; see also (1.1) for the case α = 1.

From Lemma 1, the convergence of the mediant convergents follows:

Proposition 2. We have

\begin{equation*} \lim_{k \to \infty} (\Pi_{k}(x))(-\infty) = x. \end{equation*}

Proof. If x is rational, then the assertion follows easily. So we estimate $|x - (\Pi_{k}(x))(-\infty)|$ for an irrational x. We note that $(q_{\alpha, n}(x) :n \ge 1)$ is strictly increasing for any $x \in \mathbb I_{\alpha} \setminus \mathbb Q$, which follows from the fact that $a_{\alpha, n}(x) \ge 2$ if $\varepsilon_{\alpha, n}(x) = -1$ (for any $\alpha,\,\, 0 \lt \alpha \le 1$). This implies $\lim_{n \to \infty} q_{\alpha, n}(x) = \infty$ if x is irrational. As for the RCF, see e.g. [Reference Dajani and Kraaikamp10] or (1.1.14) in [Reference Iosifescu and Kraaikamp16], x can be written as:

(2.2)\begin{equation} \frac{(\ell p_{\alpha, n}(x) \pm p_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + p_{\alpha, n-1}(x)}{(\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + q_{\alpha, n-1}(x)} \text{or }\,\, \frac{p_{\alpha, n-1}(x)G_{\alpha}^{n}(x) + p_{\alpha, n}(x)} {q_{\alpha, n-1}(x)G_{\alpha}^{n}(x) + q_{\alpha, n}(x)}. \end{equation}

The estimate of the latter is easy since $(\Pi_{k}(x))(-\infty) = \frac{p_{\alpha, n-1}(x)}{q_{\alpha, n-1}(x)}$, $|G_{\alpha}^{n}(x)|$ $ \lt \max(\alpha, 1 - \alpha) \lt 1$, and $|p_{\alpha, n-1}(x)q_{\alpha, n}(x) - p_{\alpha, n}(x)q_{\alpha, n-1}(x)| = 1$. Anyway, it is the convergence estimate of the α-continued fraction expansion of x. In the former case, we see that $\left| x - (\Pi_{k}(x))(-\infty)\right|$ is equal to:

\begin{equation*} \left| \frac{(\ell p_{\alpha, n}(x) \pm p_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + p_{\alpha, n-1}(x)} {(\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + q_{\alpha, n-1}(x)} - \frac{\ell p_{\alpha, n}(x) \pm p_{\alpha, n-1}(x)} {\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x)} \right| , \end{equation*}

with $k = \sum_{j=1}^{n} a_{\alpha, j}(x) + \ell$. This can be estimated as

\begin{align*} {}&\left| \frac{\ell}{\left((\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + q_{\alpha, n-1}(x)\right) \left(\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x)\right)} \right|\\ =& \left| \frac{1}{\left(q_{\alpha, n}(x) \pm \frac{q_{\alpha, n-1}(x)}{\ell}\right) \left((\ell q_{\alpha, n}(x) \pm q_{\alpha, n-1}(x))F_{\alpha}^{k}(x) + q_{\alpha, n-1}(x)\right)} \right|\\ \lt & \,\, \frac{1}{q_{\alpha, n-1}(x)} \quad \to \quad 0 \end{align*}

Here we used the fact $F_{\alpha}^{k} (x)\ge 0$ for k not of the form $\sum_{j = 1}^{n} a_{\alpha, j}(x)$.

As mentioned in the introduction, the $(n+1, a_{\alpha, n+1}(x)-1)$th convergent is the same as the $(n+2, 1)$th convergent if $\varepsilon_{\alpha, n+2}(x) = -1$, i.e.,

\begin{equation*} \frac{(a_{\alpha, n+1}(x) -1)p_{\alpha, n}(x) +\varepsilon_{\alpha, n+1}(x) p_{\alpha, n-1}(x)}{(a_{\alpha, n+1}(x) -1)q_{\alpha, n}(x) +\varepsilon_{\alpha, n+1}(x) q_{\alpha, n-1}(x)} = \frac{p_{\alpha, n+1}(x) - p_{\alpha, n}(x)} {q_{\alpha, n+1}(x) - q_{\alpha, n}(x)}. \end{equation*}

In this sense, the map F_α makes a duplication if $\varepsilon_{\alpha, n}(x) = -1$. This duplication is k-fold if $(\varepsilon_{\alpha, n+ \ell}(x), a_{\alpha, n+\ell}(x)) = ( -1, 2)$ for $1\leq\ell\leq k$, i.e.,

\begin{eqnarray*} \frac{p_{\alpha, n}(x) - p_{\alpha, n-1}(x)}{q_{\alpha, n}(x) - q_{\alpha, n-1}(x)} &=& \frac{p_{\alpha, n+1}(x) - p_{\alpha, n}(x)}{q_{\alpha, n+1}(x) - q_{\alpha, n}(x)}\,\, =\,\,\cdots \\ &=& \frac{p_{\alpha, n+k}(x) - p_{\alpha, n+k-1}(x)}{q_{\alpha, n+k}(x) - q_{\alpha, n+k-1}(x)}. \end{eqnarray*}

We avoid this duplication using a suitable induced transformation. First, let us recall the definition of $F_{\alpha, \flat}$ for $\frac{1}{2}\leq\alpha \lt 1$; see (1.5). One can see that this map skips the $(n, a_{n+1}(x)-1)$th mediant convergent of $x \in \mathbb I_{\alpha}$ with $\varepsilon_{n+1}(x) = -1$. An important observation in [Reference Natsui30] is that for $\tfrac{1}{2}\leq\alpha \lt 1$ we have that $-\frac{x}{1+ x} \lt 1$ for any $\alpha -1\leq x \lt 0$. This does not apply anymore when $0 \lt \alpha \lt \frac{1}{2}$, as the definition of $F_{\alpha, \flat}$ should be on the interval $[\alpha -1, 1]$. To achieve this we “speed up” F_α and modify the definition of $F_{\alpha, \flat}$ as follows:

\begin{equation*} F_{\alpha, \flat}(x) = F_{\alpha}^{K(x)}(x), \end{equation*}

with $K(x) = \min \{k\geq 1 : F_{\alpha}^k(x) \in [\alpha - 1 , 1]$.

For $\alpha - 1\leq x \lt -\tfrac{1}{2}$, $F_{\alpha}(x) = - \frac{x}{1 + x} \gt 1$ (see also (1.4)), so $F_{\alpha}^{2}(x) \in \mathbb I_{\alpha}$. Thus $K(x) = 2$ and we have that $F_{\alpha}^{K(x)}(x) =\frac{1 + 2x}{x}$ in this case. For $x\in [-\frac{1}{2}, \frac{1}{2})$, one can easily see that $F_{\alpha}(x) \in [0, 1]$ and the same holds also for $x \in [\frac{1}{1 + \alpha}, 1]$. For $x \in [\frac{1}{2}, \frac{1}{1 + \alpha})$, we find that $K(x) = 2$ and $F_{\alpha, \flat}(x) = \frac{1 -2x}{x}$. Consequently, our new definition of $F_{\alpha, \flat}$ is the following:

(2.3)\begin{equation} F_{\alpha, \flat}(x) = \begin{cases} F_{\alpha}^2(x) = - \frac{1 + 2x}{x}, & \text{if}\ \alpha-1\leq x \lt - \frac{1}{2},\\ F_{\alpha}(x) = - \frac{x}{1+x}, & \text{if}\ -\frac{1}{2}\leq x \lt 0, \\ F_{\alpha}(x) = \frac{x}{1-x}, & \text{if}\ 0\leq x \lt \frac{1}{2}, \\ F_{\alpha}^2(x) = \frac{1-2x}{x}, & \text{if}\ \frac{1}{2}\leq x \lt \frac{1}{1 + \alpha}, \\ F_{\alpha}(x) = \frac{1-x}{x}, & \text{if}\ \frac{1}{1 + \alpha}\leq x\leq 1. \end{cases} \end{equation}

Clearly from (2.3) we have that for every $x\in [\alpha -1,1]$ the sequence $(F_{\alpha, \flat}^k(x))_{k\geq 0}$, which is the orbit of x under $F_{\alpha, \flat}$, is a subsequence of the sequence $(F_{\alpha}^n(x))_{n\geq 0}$ (the orbit of x under F_α). But then for every $x\in [\alpha -1,1]$ fixed there exists a (unique) monotonically increasing function $\hat{k}:\mathbb N\cup \{0\}\to\mathbb N\cup\{0\}$, such that $F_{\alpha, \flat}^{k}(x) = F_{\alpha}^{\hat{k}(k)}(x)$. Setting $\hat{k}=\hat{k}(k)$ for $k=0,1,\dots$, we put

\begin{equation*} \Pi_{\flat, k}(x) = \Pi_{\hat{k}}(x). \end{equation*}

From this definition, it is easy to derive the following.

Another possibility is by skipping $p_{n+1}(x) - p_{n}(x)$ and $q_{n+1}(x) - q_{n}(x)$ instead of $(a_{n+1}(x) -1))p_{n}(x) + \varepsilon_{n+1}(x) p_{n-1}(x)$ and $(a_{n+1}(x) -1))q_{n}(x) + \varepsilon_{n+1}(x)q_{n-1}(x)$ if $\varepsilon_{n+1}(x) = -1$. This can be done by the jump transformation $F_{\alpha, \sharp}$, defined as follows:

(2.4)\begin{equation} F_{\alpha, \sharp}(x) = \begin{cases} F_{\alpha}^{2}(x), & \text{if}\ x \lt 0,\\ F_{\alpha}(x), & \text{if}\ x\geq 0. \end{cases} \end{equation}

Note that the map $F_{\alpha, \sharp}$ from (2.4) is explicitly given by

(2.5)\begin{equation} F_{\alpha, \sharp}(x) = \begin{cases} - \frac{x}{1+2x}, & \text{if}\ \alpha-1\leq x \lt 0, \\ \frac{x}{1-x}, & \text{if}\ 0\le x \lt \frac{1}{1+\alpha}, \\ \frac{1 -x}{x}, & \text{if}\ \frac{1}{1 + \alpha} \le x \le \frac{1}{\alpha}. \end{cases} \end{equation}

This is well-defined for any $0 \lt \alpha \lt 1$. Indeed, the map $F_{\alpha, \sharp}$ from (2.5) skips $F_{\alpha}^{k+1}(x)$ if $F_{\alpha}^{k}(x) \lt 0$, which implies that there exists an $n\geq 1$ such that $G_{\alpha}^{n}(x) =F_{\alpha}^{k}(x)$ and $\varepsilon_{n}(x) = -1$. Thus we see that $\frac{p_{n}(x) - p_{n-1}(x)}{q_{n}(x) - q_{n-1}(x)}$ has been skipped in the sequence of the mediant convergents. In this note, we do not further consider this map $F_{\alpha, \sharp}$ since the discussion is almost the same as that of $F_{\alpha, \flat}$.

Now we consider $\frac{s_{k}(x)}{t_{k}(x)} = \Pi_{k}(x)(-\infty)$ with $s_{k}(x), t_{k}(x) \in \mathbb Z$, coprime, $t_{k} \gt 0$. From this and (2.2) we derive that

\begin{equation*} t_{k}^{2}(x) \left| x - \frac{s_{k}(x)}{t_{k}(x)}\right| = \left|F_{\alpha}^{k}(x) - \Pi_{k}^{-1}(-\infty)\right|^{-1} \end{equation*}

for $x \in [\alpha-1, \frac{1}{\alpha}]$ and $n\geq 1$. Note that $F_{\alpha}^{k}(x)$ can be interpreted as the future of x at time k, while $\Pi_{k}^{-1}(-\infty)$ is like the past of x at time k; see also Chapter 4 in [Reference Dajani and Kraaikamp10]. For this reason, it is interesting to find the closure of the set

\begin{equation*} \left\{\left(F_{\alpha}^{k}(x), \Pi_{k}^{-1}(x)(-\infty)\right) : \,\, x \in [\alpha-1, \frac{1}{\alpha}], \,\, k \gt 0 \right\}. \end{equation*}

This leads us to consider the following maps:

(2.6)\begin{equation} \hat{F}_{\alpha}(x, y) = \begin{cases} \left( - \frac{x}{1+x}, - \frac{y}{1+y}\right) , & \text{if}\ \alpha -1\leq x \lt 0, \\ \left( \frac{x}{1-x}, \frac{y}{1-y}\right) , & \text{if}\ 0\le x \lt \frac{1}{1 + \alpha}, \\ \left( \frac{1 -x}{x}, \frac{1 - y}{y}\right) , & \text{if}\ \frac{1}{1 + \alpha}\leq x \lt \frac{1}{\alpha}, \end{cases} \end{equation}

and

(2.7)\begin{equation} \hat{F}_{\alpha, \flat}(x, y) = \begin{cases} \left(- \frac{1+2x}{x}, - \frac{1+2y}{y} \right) , & \text{if}\ \alpha - 1\leq x \lt -\frac{1}{2}, \\ \left(- \frac{x}{1+x}, - \frac{y}{1+y}\right) , & \text{if}\ -\frac{1}{2}\leq x \lt 0,\\ \left(\frac{x}{1-x}, \frac{y}{1-y}\right) , & \text{if}\ 0\leq x \lt \frac{1}{2}, \\ \left(\frac{1 -2x}{x}, \frac{1 -2y}{y}\right) , & \text{if}\ \frac{1}{2}\leq x \leq \frac{1}{1 + \alpha}, \\ \left(\frac{1 -x}{x}, \frac{1 -y}{y}\right) , & \text{if}\ \frac{1}{1 + \alpha} \lt x \leq 1, \end{cases} \end{equation}

where (x, y) is in a ‘reasonable domain’ of the definition of each map, respectively. The question is to find this ‘reasonable domain’ for each case. This will be done in Theorem 1 for $\hat{F}_{\alpha}$ and in Theorem 2 for $\hat{F}_{\alpha, \flat}$. For example, for $\hat{F}_{\alpha}$ the domain will be the closure of

\begin{equation*} \left\{\left(F_{\alpha}^{k}(x), \left(A_{1}(x) \cdots A_{k}(x)\right)^{-1}(-\infty)\right): \,\, x \in \Big[ \alpha-1, \frac{1}{\alpha}\Big], \,\, k \gt 0 \right\} \end{equation*}

so that $\hat{F}_{\alpha}$ is bijective except for a set of Lebesgue measure 0. From this point of view, $\hat{F}_{\alpha}$ is the planar representation of the natural extension of F_α in the sense of Ergodic theory. Another point of view is that the characterization of quadratic surds by the periodicity of the map. Indeed, it is easy to see that $x \in (0, 1)$ is strictly periodic by the iteration of F if and only if it is a quadratic surd and its algebraic conjugate is negative. We can characterize the set of quadratic surds in a similar way with $\hat{G}^{*}_{\alpha}$, see the next section, and also $\hat{F}_{\alpha}$. We can apply the above to the construction of the natural extension of $F_{\alpha, \flat}$. Indeed, it is obtained as an induced transformation of $\hat{F}_{\alpha}$. In the next section, we give a direct construction of the natural extension of $\hat{F}_{\alpha}$ as a tower of the natural extension $\hat{G}_{\alpha}^{*}$ of G_α.

3. The natural extension of F_α for $0 \lt \alpha \lt \frac{1}{2}$

As the case $\tfrac{1}{2}\leq\alpha\leq 1$ was discussed in [Reference Natsui30, Reference Natsui31], in the rest of this paper we will focus on the case $0 \lt \alpha \lt \tfrac{1}{2}$. We give some figures in the case of $\alpha = \sqrt{2} - 1$ for better understanding of the construction. We selected this value of α as an example as this is historically the first “more difficult” case; for $\alpha\in (\sqrt{2}-1,1]$ the natural extensions are simply connected regions which are the union of finitely many overlapping rectangles, while for $\alpha = \sqrt{2}-1$ the natural extension consists of two disjoint rectangles; see [Reference de Jong and Kraaikamp12, Reference Luzzi and Marmi24, Reference Moussa, Cassa and Marmi25]. In [Reference de Jong and Kraaikamp12] it is shown that for $\alpha\in\left( \frac{\sqrt{10}-3}{2},\sqrt{2}-1\right)$ there is a countably infinite number of disjoint connected regions. For $0 \lt \alpha \lt \sqrt{2}-1$ it is not so easy to describe $\Omega_{\alpha}$ explicitly; see the discussion at the end of $\S$ 2, and also [Reference de Jong and Kraaikamp12, Reference Kraaikamp, Schmidt and Steiner23–Reference Moussa, Cassa and Marmi25].

We start with the domain $\Omega_{\alpha}$ from [Reference Kraaikamp, Schmidt and Steiner23], given by the closure

\begin{equation*} \Omega_{\alpha} = \overline{\left\{\,\hat{G}^n_{\alpha}(x,-\infty )\, \Big| \,\, x\in [\alpha -1,\alpha)\right\}}, \end{equation*}

and the natural extension map $\hat{G}_{\alpha}:\Omega_{\alpha}\to\Omega_{\alpha}$, defined by

\begin{equation*} (x, y) \mapsto \begin{cases} \left(- \frac{1}{x} - b, \frac{1}{- y + b}\right) , & \text{if}\ x \lt 0, \\ \left( \frac{1}{x} - b, \frac{1}{y + b}\right) , & \text{if}\ x \gt 0, \end{cases} \end{equation*}

for $(x, y) \in \Omega_{\alpha}$.

Next we change y to $-\frac{1}{y}$, i.e., we consider

\begin{equation*} \Omega_{\alpha}^{*} = \left\{(x, y) : \Big( x, - \frac{1}{y}\Big) \in \Omega_{\alpha}\right\} , \end{equation*}

see Figure 3, and the map $\hat{G}_{\alpha}^{*}: \Omega_{\alpha}^{*}\to\Omega_{\alpha}^{*}$, defined by:

\begin{equation*} (x, y) \mapsto \begin{cases} \left(- \frac{1}{x} - b, - \frac{1}{y} -b\right) , & \text{if}\ x \lt 0, \\ \left( \frac{1}{x} - b, \frac{1}{y} - b\right), & \text{if}\ x \gt 0, \end{cases} \end{equation*}

where $b = \lfloor \left|\frac{1}{x}\right| + \alpha -1 \rfloor$; this gives another version of the natural extension, with which we work with for the rest of the paper. Recall from [Reference Kraaikamp, Schmidt and Steiner23] that $\hat{G}_{\alpha}:\Omega_{\alpha}\to\Omega_{\alpha}$ (and therefore $\hat{G}_{\alpha}^{*}$) is bijective except for a set of Lebesgue measure 0. The reason to move from $\Omega_{\alpha}$ to $\Omega_{\alpha}^{*}$ is that the first and second coordinate maps of G_α are similar. This allows for a more unified treatment. Also note that $\Omega_{\alpha}\subset [\alpha -1,\alpha]\times [0,1]$.

Figure 2.

$\hat{\Omega}_{\alpha,k,\pm}$ for $\alpha = \sqrt{2} - 1$

Figure 3.

$\Omega_{\alpha}^*$ for $\alpha = \sqrt{2} - 1$

Although formally $\alpha\not\in {\mathbb I}_{\alpha}$, we define k ₀ as the first digit of α in the G_α-expansion of α, i.e., $\frac{1}{k_{0} + \alpha}\leq\alpha \lt \frac{1}{k_{0} - 1 + \alpha}$. Furthermore, we define cylinders by

(3.1)\begin{equation} \begin{cases} \Omega_{\alpha, k_{0}, +}^{*} & =\,\,\, \{(x, y) \in \Omega_{\alpha}^{*} : \frac{1}{k_{0} + \alpha} \lt x \lt \alpha \}, \\[3pt] \Omega_{\alpha, k, +}^{*} & =\,\,\, \{(x, y) \in \Omega_{\alpha}^{*} : \frac{1}{k + \alpha} \lt x \leq\frac{1}{(k-1)+\alpha} \},\,\, \text{if}\ k \gt k_{0}, \\[3pt] \Omega_{\alpha, 2, -}^{*} & =\,\,\, \{(x, y) \in \Omega_{\alpha}^{*} : \alpha - 1 \lt x\leq - \frac{1}{2 + \alpha} \}, \\[3pt] \Omega_{\alpha, k, -}^{*} & =\,\,\, \{(x, y) \in \Omega_{\alpha}^{*} : - \frac{1}{(k-1) + \alpha} \lt x\leq - \frac{1}{k + \alpha} \},\,\, \text{if}\ k \geq 3. \end{cases} \end{equation}

Then we put

\begin{equation*} \left\{\begin{array}{ccl} \hat{\Omega}_{\alpha, k,-} & = & \left\{(x, y) : \left(-\frac{1}{x}, - \frac{1}{y}\right) \in \Omega_{\alpha, k, -}^{*}\right\} , \\ \hat{\Omega}_{\alpha, k,+} & = & \left\{(x, y) : \left(\frac{1}{x}, \frac{1}{y}\right) \in \Omega_{\alpha, k, +}^{*}\right\}. \end{array} \right. \end{equation*}

Note that if $(x,y)\in\hat{\Omega}_{\alpha,k,-}$ we have that x > 0 and $y\geq 0$; see Figure 2. For convenience we put $\Omega_{\alpha, k, +}^{*} = \emptyset$ for $2 \le k \lt k_{0}$. It is easy to see that

\begin{equation*} \Omega_{\alpha}^{*} = \left( \bigcup_{k = 2}^{\infty} (\hat{\Omega}_{\alpha, k, -} - (k, k)) \right) \cup \left( \bigcup_{k = k_{0}}^{\infty} (\hat{\Omega}_{\alpha, k, +} - (k, k))\right) \quad \text{(disj.~a.e.)}, \end{equation*}

where (disj. a.e.) means “disjoint except for a set of measure 0”. This disjointness follows from the next lemma.

Figure 4.

$\hat{\Omega}_{\alpha,k,\pm} - (1,1)$ for $\alpha = \sqrt{2} - 1$

Lemma 2. For every $k\in\mathbb N$, $k\geq 2$, we have:

\begin{equation*} \left( \hat{\Omega}_{\alpha, k+1, -} - (k+1, k+1) \right) \cap \left( \hat{\Omega}_{\alpha, k, +} - (k, k) \right) = \emptyset \quad \text{disj.~a.e.}, \end{equation*}

or equivalently,

\begin{equation*} \left( \hat{\Omega}_{\alpha, k+1, -} - (1, 1) \right) \cap \left( \hat{\Omega}_{\alpha, k, +} - (0, 0) \right) = \emptyset \quad \text{disj.~a.e.}; \end{equation*}

see Figure 4.

Proof. We see

\begin{equation*} \left( \hat{\Omega}_{\alpha, k, \pm} - (k, k) \right) = \hat{G}^{*}_{\alpha}\left(\Omega_{\alpha, k,\pm}^{*} \right). \end{equation*}

Then the assertion follows from the a.e.-bijectivity of $\hat{G}_{\alpha}^{*}$.

For $j \ge 1$, we define

\begin{equation*} \Upsilon_{\alpha, j} = \bigcup_{k=j+1}^{\infty}\left(\hat{\Omega}_{\alpha, k, -} - (k-j, k-j)\right) \cup \bigcup_{k=j+1}^{\infty}\left(\hat{\Omega}_{\alpha, k, +} - (k-j, k-j)\right) \end{equation*}

for $j \ge 2$, see Figure 5, and

\begin{equation*} \Upsilon_{\alpha} = \bigcup_{j=1}^{\infty} \Upsilon_{\alpha, j}. \end{equation*}

From Lemma 2, this is “disj. a.e.” We also see

\begin{equation*} \Upsilon_{\alpha, j}\cap \hat{\Omega}_{\alpha, j, +} = \emptyset \qquad \text{(disj.~a.e.)} , \end{equation*}

which implies

\begin{equation*} \Omega^{*}_{\alpha} \cap \left( \Upsilon_{\alpha} \right)^{-1} = \emptyset \qquad \text{(disj.~a.e.)}, \end{equation*}

where $\left( \Upsilon_{\alpha} \right)^{-1} = \left\{(x,y)\, :\, \left( \tfrac{1}{x},\tfrac{1}{y}\right) \in \Upsilon_{\alpha}\right\}$. Note that $(\Upsilon_{\alpha})^{-1} \subset \{(x, y) : x \gt 0\}$.

Now we will define the ‘reasonable domain’ V_α for the natural extension map $\hat{F}_{\alpha}$ from (2.6). We put $V_{\alpha} = \Omega_{\alpha}^{*}\cup \left(\Upsilon_{\alpha}\right)^{-1}$; see Figure 6. From the construction of V_α, it is not hard to see the following result.

Figure 5.

$\hat{\Omega}_{\alpha,k,\pm} - (\ell,\ell)$ and $\Upsilon_{\alpha, k}$ for $\alpha = \sqrt{2} - 1$

Figure 6.

$V_{\alpha}= \Omega_{\alpha}^{*}\cup (\Upsilon_{\alpha})^{-1}$ for $\alpha = \sqrt{2} - 1$

Proof. We show the following below. Then the assertion of the theorem is proved in exactly the same way as in [Reference Natsui31] in the case of $\tfrac{1}{2} \leq\alpha\leq 1$.

1. (i) The map $\hat{F}_{\alpha}$ defined on V_α is surjective.

2. (ii) The map $\hat{F}_{\alpha}$ is bijective except for a set of measure 0.

3. (iii) The measure $\frac{dx dy}{(x - y)^{2}}$ is the absolutely continuous ergodic invariant measure.

4. (iv) The Borel σ-algebra ${\mathcal B}(V_{\alpha})$ on V_α satisfies:

   \begin{equation*} {\mathcal B}(V_{\alpha}) = \sigma \left( \bigvee_{n=0}^{\infty} \hat{F}_{\alpha}^n\pi_1^{-1}{\mathcal B}([\alpha -1,\alpha))\right), \end{equation*}

   where ${\mathcal B}([\alpha -1,\alpha))$ is the Borel σ-algebra on $[\alpha -1,\alpha )$ and $\pi_1: V_{\alpha}\to [\alpha -1,\alpha)$ is the projection on the first coordinate.

For a.e.  $(x, y)\in\Omega_{\alpha}^{*}$, x ≠ 0, there exists a unique element $(x_{0}, y_{0}) \in\Omega_{\alpha}^{*}$ and a positive integer k such that

\begin{equation*} (x, y) = \hat{G}_{\alpha}^{*}(x_{0}, y_{0}) = \begin{cases} \left(-\frac{1}{x_{0}} - k, -\frac{1}{y_{0}} - k\right), & \text{if}\ x_{0} \lt 0, \\ \left(\frac{1}{x_{0}} - k, \frac{1}{y_{0}} - k\right), & \text{if}\ x_{0} \gt 0, \end{cases} \end{equation*}

since $(\Omega_{\alpha}^{*}, \hat{G}_{\alpha}^{*})$ is a natural extension of $(\mathbb I_{\alpha}, G_{\alpha})$; see [Reference Kraaikamp, Schmidt and Steiner23].

If $x_{0} \lt 0$, then we see

\begin{equation*} \left(-\frac{1}{x_{0}} - (k-1), -\frac{1}{y_{0}} - (k-1)\right) \in \Upsilon_{\alpha, 1}. \end{equation*}

This implies $\alpha \le -\frac{1}{x_{0}} - (k-1) \lt \alpha + 1$. We put

\begin{equation*} (x_{1}, y_{1}) = \left(\left(-\frac{1}{x_{0}} - (k-1)\right)^{-1}, \left(-\frac{1}{y_{0}} - (k-1)\right)^{-1} \right), \end{equation*}

and have $\frac{1}{1 + \alpha} \lt x_{1} \le \frac{1}{\alpha}$. From (2.6), we have that $(x, y) = \hat{F}_{\alpha}(x_{1}, y_{1})$.

If $x_{0} \gt 0$, then

\begin{equation*} \left(\frac{1}{x_{0}} - (k-1), \frac{1}{y_{0}} - (k-1)\right)\in \Upsilon_{\alpha, 1} \end{equation*}

and $(x, y) = \hat{F}_{\alpha}(x_{1}, y_{1})$ with

\begin{equation*} (x_{1}, y_{1}) = \left(\left(\frac{1}{x_{0}} - (k-1)\right)^{-1}, \left(\frac{1}{y_{0}} - (k-1)\right)^{-1} \right). \end{equation*}

We note that in both cases we have

(3.2)\begin{equation} (x_{1}, y_{1}) \in \Upsilon_{\alpha, 1}^{-1} \subset \left(\Upsilon_{\alpha}\right)^{-1}. \end{equation}

Next we consider the case $(x, y) \in \Upsilon_{\alpha}^{-1}$. This means $\left( \tfrac{1}{x}, \tfrac{1}{y}\right) \in \Upsilon_{\alpha, j}$ for some $j\geq 1$. We consider two cases.

Case (a): $\left(\tfrac{1}{x}, \tfrac{1}{y}\right) \in \hat{\Omega}_{\alpha, j+1, \pm} - (1, 1)$.

In this case, there exists $(x_{0}, y_{0}) \in \hat{\Omega}_{\alpha, j+1, \pm}$ such that $\left( \tfrac{1}{x}, \tfrac{1}{y}\right) = (x_{0} - 1, y_{0} - 1)$. Thus $(x_{1}, y_{1}) = \left(\pm \tfrac{1}{x_{0}}, \pm \tfrac{1}{y_{0}}\right) \in \Omega_{\alpha, j+1}^{*}$, which imlies

\begin{equation*} \frac{1}{j + 1 + \alpha} \lt \pm x_{1} \le \frac{1}{j + \alpha} \le \frac{1}{1 + \alpha}. \end{equation*}

Hence we find that $(x, y) = \hat F_{\alpha}(x_{1}, y_{1})$. Here we see

(3.3)\begin{equation} (x_{1}, y_{1}) \in \Omega^{*}_{\alpha}. \end{equation}

Case (b): $\left(\tfrac{1}{x}, \tfrac{1}{y}\right) \in \hat{\Omega}_{\alpha, k, \pm} - (k - j, k - j)$ for $k \gt j+1$.

In this case, we have

(3.4)\begin{equation} (x_{0}, y_{0}) := \left(\tfrac{1}{x} + 1, \tfrac{1}{y} + 1\right) \in \Omega_{\alpha, k \pm} - (k - j - 1, k - j - 1) \end{equation}

and then

(3.5)\begin{equation} (x_{1}, y_{1}) := (\tfrac{1}{x_{0}}, \tfrac{1}{y_{0}}) \in \Upsilon_{\alpha}^{-1}. \end{equation}

This shows

\begin{equation*} (x, y) = \hat{F}_{\alpha}(x_{1}, y_{1}). \end{equation*}

Consequently, we have the first statement. The second statement follows from (3.2), (3.3), (3.4) and (3.5). The third statement is also easy to obtain. Indeed, it is well-known that the measure given here is the absolutely continuous invariant measure for the direct product of the same linear fractional transformation. Because $\hat{G}_{\alpha}^{*}$ is an induced transformation of $\hat{F}_{\alpha}$ to $\Omega_{\alpha}^{*}$, the ergodicity of $\hat{F}_{\alpha}$ follows from that of $\hat{G}_{\alpha}^{*}$. The ergodicity of the latter is equivalent to that of G_α and it was proved by Luzzi and Marmi in [Reference Luzzi and Marmi24]. For the last statement, note that (2.1) allows one to identify a point x with a one-sided sequence of matrices with entries $A^{\pm}$, A^R. As a result, F_α, $\hat{F}_{\alpha}$ can be seen as a one-sided resp. two sided shifts, from which the fourth statement follows.

4. The natural extension of $F_{\alpha, \flat}$ for $0 \lt \alpha \lt \frac{1}{2}$

In the case of α = 1, F ₁ is the original Farey map F. We recall that

\begin{equation*} \hat{F}(x, y) =\begin{cases} \left(\frac{x}{1-x}, \frac{y}{1 - y}\right), & \text{if}\ 0\leq x \lt \frac{1}{2}, \\ \left(\frac{1 - x}{x}, \frac{1 - y}{y}\right), & \text{if}\ \frac{1}{2}\leq x \leq 1, \end{cases} \end{equation*}

defined on $V_{1}= \{(x, y) : 0 \le x \le 1, \,\, -\infty \le y \le 0 \}$ is the natural extension of F with the invariant measure $\hat{\mu}_{1}$ defined by $d\hat{\mu_{1}} = \frac{dx \, dy}{(x -y)^{2}}$. In particular, $\hat{F}$ is bijective on V ₁ except for a set of Lebesgue measure 0. It is easy to see that F ₁ and $F_{1, \flat}$ are the same. In the case of $\frac{1}{2} \le \alpha \lt 1$, the complete description was given in [Reference Natsui31]. Here we consider the case $0 \lt \alpha \lt \tfrac{1}{2}$ as a continuation of the previous section.

We put $V_{\alpha, \flat} = \{(x, y) \in V_{\alpha}, \,\, x \le 1\}$ and consider the induced transformation $\hat{F}_{\alpha, \flat}$ of $\hat{F}_{\alpha}$ to $V_{\alpha, \flat}$; see Figure 7.

Figure 7.

$V_{\alpha, \flat}= V_{\alpha} \cap \{(x, y) : x \le 1 \}$ for $\alpha = \sqrt{2}-1$

Recall the definition of the map $\hat{F}_{\alpha,\flat}$, as given in (2.7).

Put

\begin{equation*} \left\{\begin{array}{lcll} D_{1} & = & \left(V_{\alpha,-} + (1, 1)\right) & \left(\subset V_{1}\right) \\ D_{2} & = & V_{\alpha, +} \setminus D_{1} & \left(\subset V_{1}\right) \end{array} \right. \end{equation*}

with $V_{\alpha, -} = \{(x, y) \in V_{\alpha, \flat} : x \lt 0\}$ and $V_{\alpha, +} = \{(x, y) \in V_{\alpha, \flat} : x \ge 0\}$. We write $D = D_{1} \cup D_{2}$. By the definition we see that $D \subset V_{1}$. We define $\psi\, : \, D \to V_{\alpha}$ by

(4.1)\begin{equation} \psi (x, y) = \begin{cases} (x -1, y - 1), & \text{if}\ (x, y) \in D_{1}, \\ (x, y), & \text{if}\ (x, y) \in D_{2}, \end{cases} \end{equation}

see Figure 8. We note that

1. (i) D has positive Lebesgue measure since $V_{\alpha, -}$ has positive Lebesgue measure; see [Reference Kraaikamp, Schmidt and Steiner23],

2. (ii) $(\psi)^{-1} (\mu_{\alpha}) = \mu_{1}|_{D}$,

3. (iii) ψ is injective.

Theorem 3 We have $D = V_{1}$ and for a.e. $(x, y) \in V_{1}$,

(4.2)\begin{equation} \left((\psi)^{-1} \circ \hat{F}_{\alpha, \flat}\circ \psi\right) (x, y)= \hat{F}(x, y). \end{equation}

In other words, for any $0 \lt \alpha \lt \frac{1}{2}$, $\left( V_{\alpha, \flat}, \hat{\mu}_{\alpha, \flat}, \hat{F}_{\alpha, \flat} \right)$ is metrically isomorphic to $\left(V_{1}, \hat{\mu}_{1}, \hat{F} \right)$ by the isomorphism $\psi : V_{1} \to V_{\alpha, \flat}$.

Figure 8.

ψ ⁻¹ for $\alpha = \sqrt{2} -1$

5. Some applications

As stated in the Introduction, in $\S$5.1 we extend the result from [Reference Kraaikamp and Nakada22]. That is, we show that the set of normal numbers with respect to G_α is the same with that of $G_{\alpha'}$ for any α and $\alpha'$ in $(0, 1]$; see Theorem 4. To prove this result, we need the natural extensions of the α-Farey maps. In §5.2 we extend the result of [Reference Nakada and Natsui28], by proving that for a.e. α in $(0, 1)$, G_α is not ϕ-mixing; see §5.2 for the definition. To do so, we use the result of §5.1 together with a result from [Reference Carminati and Tiozzo8]. What we need are statements like “ $G_{\alpha}^{n}(\alpha - 1)$ is dense” and “there exist n ₀ and m ₀ such that $G_{\alpha}^{n_{0}}(\alpha - 1) = G_{\alpha}^{m_{0}}(\alpha)$.” The former follows from §5.1 and the latter from [Reference Carminati and Tiozzo8] for a.e. α.

5.1. Normal numbers

Given any finite sequence of non-zero integers $b_1,b_2,\dots, b_n$, we define the cylinder set $\langle b_{1}, \ldots, b_{n} \rangle_{\alpha}$ as

(5.1)\begin{equation} \langle b_{1}, \ldots, b_{n} \rangle_{\alpha} = \{x \in \mathbb I_{\alpha} : c_{\alpha, 1}(x) = b_{1}, \ldots, c_{\alpha, n}(x) = b_{n} \}, \end{equation}

where $c_{\alpha,j}(x)=\varepsilon_{\alpha,j}(x)a_{\alpha,j}(x)$ for $j=1,2,\dots,n$. An irrational number $x\in\mathbb I_{\alpha}$ is normal with respect to G_α if for any cylinder set $\langle b_{1}, \ldots, b_{n} \rangle_{\alpha}$,

\begin{equation*} \lim_{N \to \infty} \frac{\# \{0 \le m \le N-1 : G_{\alpha}^{m}(x) \in \langle b_{1}, \ldots, b_{n} \rangle_{\alpha} \}}{N} = \mu_{\alpha}(\langle b_{1}, \ldots, b_{n} \rangle_{\alpha}) \end{equation*}

holds, where µ_α is the absolutely continuous invariant probability measure for G_α. An irrational number $x\in (0, 1)$ is said to be α-normal if either $x\in [0,\alpha )$ and x is normal with respect to G_α, or $x\in [\alpha, 1)$ and x − 1 is normal with respect to G_α. In the sequel, we consider $\alpha = \lim_{\epsilon\downarrow 0} (\alpha -\epsilon )$ as an element of $\mathbb I_{\alpha}$.

Now we extend this notion to the 2-dimensional case. We set $\epsilon_{\alpha}(x,y)$ to be equal to $\varepsilon_{\alpha}(x)$. Since $\hat{G}_{\alpha}^{*}$ is bijective (a.e.), we can define $\epsilon_{\alpha,n}$ and $a_{\alpha, n}$ for $n \le 0$ by $\epsilon_{\alpha,n}(x, y) = \epsilon_{\alpha}({{{\hat {G^{*}_{\alpha}}}}}^{n-1}(x, y))$ and $a_{\alpha,n}(x, y) = a_{\alpha} ({{{\hat {G^{*}_{\alpha}}}}}^{n-1}(x, y))$.

We can define also $c_{\alpha, j}(x, y) = \epsilon_{\alpha, j}(x, y) a_{\alpha, j}(x, y)$. With these definitions, we extends the notion of a $(k,\ell)$-cylinder set for $-\infty \lt k \lt \ell \lt \infty$ by:

\begin{equation*} \langle b_{k}, b_{k+1}, \ldots, b_{\ell} \rangle_{\alpha, (k, \ell)} = \{(x, y) \in \Omega_{\alpha}^{*} : c_{\alpha, k}(x) = b_{k}, \ldots, c_{\alpha, \ell}(x) = b_{\ell} \}. \end{equation*}

Then we can define normality of an element $(x, y) \in \Omega_{\alpha}^{*}$: (x, y) is said to be normal with respect to $\hat{G}^{*}_{\alpha}$ if for any sequence of integers $(b_{k}, b_{k+1}, \ldots , b_{\ell})$, $-\infty \lt k \lt \ell \lt \infty$

\begin{align*} {} &\lim_{N \to \infty} \frac{1}{N}\sharp\{1 \le n \le N : c_{k+n-1}(x,y) = b_{k}, \ldots,c_{\ell + n-1}(x, y) = b_{\ell}\} \\ = & \,\, \hat{\mu}_{\alpha}( \langle b_{k}, b_{k+1}, \ldots , b_{\ell} \rangle_{\alpha, (k, \ell)}), \end{align*}

where $\hat{\mu}_{\alpha}$ is the absolutely continuous invariant probability measure for $\hat{G}^{*}_{\alpha}$, satisfying $d\hat{\mu}_{\alpha} = C_{\alpha} \tfrac{dx dy}{(x - y)^{2}}$ with the normalizing constant C_α. According to this definition, it is easy to see that (x, y) is normal with respect to $\hat{G}^{*}_{\alpha}$ if and only if x is normal with respect to G_α (independent of the choice of y). For example, one may choose $y = -\infty$. We will show the following result.

Theorem 4 The set of α-normal numbers is the same with that of 1-normal numbers with respect to $G = G_{1}$.

The proof for $\sqrt{2} - 1\leq \alpha \lt \tfrac{1}{2}$ is basically the same as the case $\tfrac{1}{2} \le \alpha \le 1$. In what follows, we give the proof of this theorem mainly keeping in mind the case $0 \lt \alpha \lt \sqrt{2} - 1$. In particular, for $0 \lt \alpha \lt \tfrac{3 - \sqrt{5}}{2}$. (By [Reference Kraaikamp, Schmidt and Steiner23] and [Reference Nakada27], the size of $\Omega_{\alpha}^{*}$ with respect to the measure $\tfrac{dx dy}{(x - y)^{2}}$ is equal to that of ${\hat{G}}_{\frac{1}{2}}^{*}$ for $\tfrac{3 - \sqrt{5}}{2}\leq \alpha \lt \tfrac{1}{2}$ and is larger than it for $0 \lt \alpha \lt \tfrac{3 - \sqrt{5}}{2}$.)

We define an induced map $\hat{F}_{\alpha, \flat, 2}$ of $\hat{F}_{\alpha, \flat}$. To do so, first we define an induced map $\hat{F}_{\alpha, \flat, 1}$. We put

(5.2)\begin{equation} V_{\alpha, \flat, 1} = \Omega_{\alpha}^{*} \cup \left\{\left( - \frac{x}{1+ x}, - \frac{y}{1 + y}\right) : (x, y) \in \Omega_{\alpha}^{*}, -\tfrac{1}{2} \le x \lt 0 \right\}. \end{equation}

We note that the second part of the right side is

\begin{equation*} \left\{(x, y) : x \le 1, \,\,\left(\tfrac{1}{x}, \tfrac{1}{y}\right) \in \cup_{k=2}^{\infty}\hat{\Omega}_{\alpha, k, -} - (1, 1) \right\}. \end{equation*}

Hence we see $V_{\alpha, \flat, 1} = V_{\alpha, \flat} \cap \{(x, y): y \le -1\}$. Then $\hat{F}_{\alpha, \flat, 1}$ is the induced map of $\hat{F}_{\alpha, \flat}$ to $\hat{V}_{\alpha, \flat, 1}$. We will write it explicitly. Recall the definition of $\hat{\Omega}_{\alpha, k, \pm}$, (3.1).

1. (i) If $\alpha -1 \leq x \lt -\tfrac{1}{2}$, then $\hat{F}_{\alpha, \flat}(x, y) \in \Omega_{\alpha}^{*} \subset V_{\alpha, \flat, 1}$, see (5.2), and $\hat{F}_{\alpha, \flat, 1}(x, y) = \hat{F}_{\alpha, \flat}(x, y) = \hat{G}_{\alpha}^{*}(x, y)$.

2. (ii) If $-\tfrac{1}{2} \le x \lt 0$, then $\hat{F}_{\alpha}(x, y) = \left(-\tfrac{x}{1+x}, -\tfrac{y}{1+y}\right)$ which implies $ 0 \lt -\tfrac{x}{1+x} \le 1$ and $ -\tfrac{y}{1+y} \le -1$. Thus we have $\hat{F}_{\alpha, \flat, 1}(x, y) = \hat{F}_{\alpha, \flat}(x, y)$.

3. (iii) If $(x, y) \in \Omega_{\alpha, k, +}^{*}$ for some $k \ge k_{0}$, then $\hat{F}_{\alpha}(x, y) = \left(\tfrac{x}{1 -x}, \tfrac{y}{1 - y}\right) \in \hat{\Omega}_{\alpha, k, +}$. This shows $\tfrac{x}{1 - x} \lt 1$ but $\tfrac{y}{1 -y} \gt 1$. Hence $\hat{F}_{\alpha}(x, y) \notin V_{\alpha, \flat, 1}$. The same hold for $\hat{F}_{\alpha}^{\ell}(x, y)$ for $2 \le \ell \le k-1$ and then $\hat{F}_{\alpha}^{k}(x, y) (= \hat{G}_{\alpha}^{*}(x, y)) \in \Omega_{\alpha}^{*} \subset V_{\alpha, \flat, 1}$.

4. (iv) If $0 \le x \le 1$ and $(\tfrac{1}{x}, \tfrac{1}{y}) \in \hat{\Omega}_{\alpha, k , -} - (1, 1)$, then $(\tfrac{1}{x}, \tfrac{1}{y}) - (\ell, \ell) \in \hat{\Omega}_{\alpha, k, -} - (\ell - 1, \ell -1) =: (u_{\ell}, v_{\ell})$ for $2 \le \ell \le k-1$. This implies $0 \gt -\tfrac{1}{v_{\ell}} \gt -1$ and so $\left( \tfrac{1}{u_{\ell}}, \tfrac{1}{v_{\ell}}\right) \in V_{\alpha, \flat, 1}$. Moreover, $u_{k-1}^{-1} \in \mathbb I_{\alpha}$, where $(\tfrac{1}{x}, \tfrac{1}{y}) - (k-1, k-1) = (u_{k}, v_{k})$. In other words, there exists $(u', v') \in \Omega_{\alpha}^{*}$ such that $\left(\tfrac{1}{u'}, \tfrac{1}{v'}\right) - (k, k) = \left(\tfrac{1}{x}, \tfrac{1}{y}\right) - (k-1, k-1)$. Hence we have

   \begin{equation*} {\phantom{XXX}}\hat{F}_{\alpha, \flat, 1}(x, y) = \left(\tfrac{1}{x}- (k-1), \tfrac{1}{y} - (k-1) \right) \left( = \hat{G}_{\alpha}^{*}(u', v') \right). \end{equation*}

Consequently, we see that $\hat{F}_{\alpha, \flat, 1}(x, y)$ satisfies:

(5.3)\begin{equation} \left\{\begin{array}{ll} \hat{F}_{\alpha, \flat}(x, y) = G_{\alpha}^{*}(x,y), & \text{if } \alpha - 1 \le x \lt -\tfrac{1}{2} \\ \hat{F}_{\alpha, \flat}(x, y), & \text{if } -\tfrac{1}{2} \le x \lt 0 \\ \hat{G}_{\alpha}^{*}(x, y), & \text{if } 0 \le x \,\, \text{and} \,\, (x, y)\in \Omega_{\alpha}^{*} \\ \left(\tfrac{1}{x} - (k - 1), \tfrac{1}{y} - (k - 1)\right) , & \text{if } 0 \leq x \leq 1, \text{and}\\ & (\tfrac{1}{x}, \tfrac{1}{y}) \in \hat{\Omega}_{\alpha, k , -}- (1, 1). \end{array} \right. \end{equation}

Next we consider $V_{\alpha, \flat, 2} \subset V_{\alpha, \flat, 1}$, which is defined as follows: $V_{\alpha, \flat, 2} = V_{\alpha, \flat, -} \cup V_{\alpha, \flat, +}$, with

\begin{equation*} \left\{\begin{array}{lll} V_{\alpha, \flat, -} &= & V_{\alpha, \flat, 1} \cap \{(x, y) : x \lt 0, y\le -2\}\\ & = & \Omega_{\alpha}^{*} \cap \{(x, y) : x \lt 0, y\le -2\} , \\ && \\ V_{\alpha, \flat, +} & = & V_{\alpha, \flat, 1} \cap \{(x, y) : x \ge 0\}. \end{array} \right. \end{equation*}

We let $\hat{F}_{\alpha, \flat, 2}$ be the induced transformation on $V_{\alpha, \flat, 2}$. We see that $V_{1, \flat, 2} = V_{1, \flat, +} = W$, where W is defined as:

(5.4)\begin{equation} W = [0,1]\times [-\infty,-1] \end{equation}

and $V_{\alpha, \flat, -} = \emptyset$. Recall the map ψ as defined in (4.1), and notice that when ψ ⁻¹ is restricted to $V_{\alpha,\flat,2}$, one finds:

\begin{equation*} \psi^{-1}(x, y) = \left\{\begin{array}{lll} (x+1, y+1), & \text{if} & (x, y) \in V_{\alpha, \flat, -}, \\ (x, y), &\text{if} & (x, y) \in V_{\alpha, \flat, +}. \end{array} \right. \end{equation*}

Then from Theorem 3, we have $\psi^{-1}(V_{\alpha, \flat, 2}) = W$, with W from (5.4).

Theorem 5 The induced map $\hat{F}_{\alpha, \flat, 2}$ of $\hat{F}_{\alpha, \flat, 1}$ to $V_{\alpha, \flat, 2}$ is metrically isomorphic to the natural extension of the Gauss map G ₁, where the absolutely continuous invariant probability measure for $\hat{F}_{\alpha, \flat, 2}$ is given by $C_{\alpha, \flat, 2} \frac{dx dy}{(x - y)^{2}}$, with the normalizing constant $C_{\alpha, \flat, 2}$, see Figures 7, 9, and Figure 8.

Figure 9.

$V_{\alpha, \flat, 1}$ and $V_{\alpha, \flat, 2} $ for $\alpha = \sqrt{2}-1$

Proof. It is easy to see that the induced map of $\hat{F}_{1}$ on W is the natural extension of the Gauss map G. Indeed we see, with W from (5.4), that $\hat{G}^{*} = \hat{F}_{1}|_W$:

\begin{equation*} (x, y) \mapsto \left(\tfrac{1}{x} -\left\lfloor \frac{1}{x} \right\rfloor, \tfrac{1}{y} - \left\lfloor \frac{1}{x} \right\rfloor \right) \end{equation*}

is bijective on W on (a.e.). Since $\psi^{-1}V_{\alpha, \flat, 2} = W$, the conjugacy $\psi^{-1}\circ \hat{F}_{\alpha, \flat, 2} \circ \psi = \hat{G}^{*}_{1}$ follows from Theorem 3 and basic fact on induced transformations from Ergodic theory.

The next step is the definition of normal numbers associated both with $\hat{F}_{\alpha, \flat, 1}$ and $\hat{F}_{\alpha, \flat,2}$.

We define a digit function $\delta(x, y)$ and get a sequence $(\delta_{n}(x, y):n \ge 1)$ in the following way:

(5.5)\begin{equation} \delta(x, y) = \left\{\begin{array}{ll} \delta_{-, k} & \text{if } (x, y)\in \Omega_{\alpha, k, -}^{*}, \,\, k \ge 2, \\ \delta_{+, k} & \text{if } (x, y)\in \Omega_{\alpha, k, +}^{*}, \,\, k \ge k_{0}, \\ \delta_{0, 2} & \text{if } (x, y)\in \left\{(x, y): \left(\tfrac{1}{x}, \tfrac{1}{y}\right) \in \hat{\Omega}_{\alpha, 2, -} - (1, 1), x \le 1 \right\} ,\\ \delta_{0, k} & \text{if } (x, y)\in \left\{(x, y) : \left( \tfrac{1}{x}, \tfrac{1}{y}\right) \in \hat{\Omega}_{\alpha, k, -} - (1, 1)\right\}, \,\,k \gt 2, \end{array}\right. \end{equation}

and $\delta_{n}(x, y) = \delta(\hat{F}_{\alpha, \flat, 1}^{n-1}(x, y))$, $n \ge 1$, for $(x, y) \in V_{\alpha, \flat, 1}$. It is easy to see that the set of sequences $(\delta_{n}(x, y))$ separates points of $V_{\alpha, \flat, 1}$.

Let $(e_{n}: 1 \le n \le \ell)$ be a block of $\delta_{j, k}$’s. Then we define a cylinder set of length $\ell \ge 1$ by

\begin{equation*} \langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha, \flat, 1} = \{(x, y) \in V_{\alpha, \flat, 1} : \delta_{n}(x, y) = e_{n}, 1 \le n \le \ell\}. \end{equation*}

We denote by $\mu_{\alpha, \flat, 1}$ the absolutely continuous invariant probability measure for $\hat{F}_{\alpha, \flat, 1}$. An element $(x, y) \in V_{\alpha, \flat, 1}$ is said to be α-1-Farey normal if

(5.6)\begin{eqnarray} &&\lim_{N \to \infty} \frac{1}{N} \sharp\{n : 1 \le n \le N, \,\, \hat{F}_{\alpha, \flat, 1}^{n-1}(x, y) \in \langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 1}\} \nonumber \\ && \qquad \,\, = \mu_{\alpha, \flat, 1}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,1}) \end{eqnarray}

holds for every cylinder set of length $\ell \ge 1$. We can define the notion of the α-2-Farey normality in a similar way, compare (5.6) and (5.7). We may use the same notation $\delta(x, y)$ restricted on $V_{\alpha, \flat, 2}$; c.f. (5.5). However, we use $\eta(x, y)$ to describe the difference between $\hat{F}_{\alpha, \flat, 1}$ and $\hat{F}_{\alpha, \flat, 2}$: from a digit function $\eta(x, y)$ we get a sequence $(\eta_{n}(x, y):n \ge 1)$ as follows.

\begin{equation*} \eta(x, y) = \left\{\begin{array}{lll} \delta_{-, k} & \text{if} & (x, y) \in \Omega_{\alpha, k, -}^{*}, \,\, k \ge 2 ,\,\, y\le -2,\\ \delta_{+, k} & \text{if} & (x, y) \in \Omega_{\alpha, k +}^{*}, \,\,\, k \ge k_{0}, \\ \delta_{0, 2} & \text{if} & (x, y) \in \left\{\left(\tfrac{1}{x} , \tfrac{1}{y} \right) \in \Omega_{\alpha, 2, -}^{*}-(1,1), x \le 1 \right\} ,\\ \delta_{0, k} &\text{if}& (x, y) \in \left\{\left(\tfrac{1}{x}, \tfrac{1}{y} \right) \in \Omega_{\alpha, k, -}^{*} -(1,1)\right\}, \,\,k \gt 2, \end{array}\right. \end{equation*}

and $\eta_{n}(x, y) = \eta(\hat{F}_{\alpha, \flat, 2}^{n-1}(x, y))$, $n \ge 1$, for $(x, y) \in V_{\alpha, \flat, 2}$. It is easy to see that the set of sequences $(\eta_{n}(x, y))$ separates points of $V_{\alpha, \flat, 2}$.

Let $(e_{n} : 1 \le n \le \ell)$ be a block $\delta_{j, k}$’s, then we define a cylinder set of length $\ell \ge 1$ by

\begin{equation*} \langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha, \flat, 2} = \{(x, y) \in V_{\alpha, \flat, 2} : \eta_{n}(x, y) = e_{n}, 1 \le n \le \ell\}. \end{equation*}

We denote by $\mu_{\alpha, \flat, 2}$ the absolutely continuous invariant probability measure for $\hat{F}_{\alpha, \flat, 2}$. An element $(x, y) \in V_{\alpha, \flat, 2}$ is said to be α-2-Farey normal if

(5.7)\begin{eqnarray} &&\lim_{N \to \infty} \frac{1}{N} \sharp\{n : 1 \le n \le N, \,\, \hat{F}_{\alpha, \flat, 2}^{n-1}(x, y) \in \langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2}\} \nonumber \\ && \qquad \,\, = \mu_{\alpha, \flat, 2}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}) \end{eqnarray}

holds for every cylinder set of length $\ell \ge 1$. Here we have to be careful with the measures $\mu_{\alpha, \flat, 1}$ and $\mu_{\alpha, \flat, 2}$, which take different values only by the normalizing constants for any measurable set $A \subset V_{\alpha, \flat, 2}$.

The proof of Theorem 4 is done in steps. We first prove that under the induced transformation $\hat{F}_{\alpha,\flat,1}$, α-1-Farey normality is equivalent to α-normality. After that we proceed to the induced system $\hat{F}_{\alpha,\flat,2}$, that is isomorphic to the Gauss map $\hat{G}_1$, and show that α-2-Farey normality is equivalent to normality w.r.t. $\hat{G}_1$. On the other hand, one can prove that for points in the domain of the $\hat{F}_{\alpha,\flat,2}$ map, a point is α-1-Farey normal if and only if it is α-2-Farey normal. From the above equivalences, one then concludes that α-normality is equivalent to 1-normality.

Define $r_{1} := 1$ and set for $j \ge 2$, $r_{j} = r_{j}(x, y):= n $ whenever $\hat{F}_{\alpha, \flat, 1}^{n-1}(x, y) \in \Omega_{\alpha}^{*}$ and $\hat{F}_{\alpha, \flat, 1}^{m}(x, y) \notin \Omega_{\alpha}^{*}$, $r_{j-1}\leq m \lt n$.

Lemma 3. Suppose that

\begin{equation*} (x, y) \in \left\{(x, y) : x \le 1, \,\,\left(\tfrac{1}{x}, \tfrac{1}{y}\right) \in \bigcup_{k=2}^{\infty} \hat{\Omega}_{\alpha, k, -} - (1, 1) \right\}. \end{equation*}

Then $(x, y)$ is α-1-Farey normal if and only if $\hat{F}_{\alpha, \flat, 1}(x, y) \in \Omega_{\alpha}^{*}$ is α-1-Farey normal.

Proof. From the 4th line of the right side of (5.3), we have $\hat{F}_{\alpha, \flat, 1}(x, y) \in \Omega_{\alpha}^{*}$. Then the equivalence of the normality is easy to follow.

From this lemma, it is enough to restrict the α-1-Farey normality only for $(x, y) \in \Omega_{\alpha}^{*}$.

Lemma 4. An element $(x_{0}, y_{0}) \in \Omega_{\alpha}^{*}$ is α-1-Farey normal if and only if x ₀ is α-normal.

Proof. Suppose that $r_{i} = r_{i}(x_0,y_0)$, and decompose $\mathbb N$ as $\mathbb N_{1}\cup \mathbb N_{2}$ with

\begin{equation*} \mathbb N_{1} = \{r_i\, :\, i\geq 1\} (= \{n\in\mathbb N\, :\, \delta_{n}(x_{0}, y_{0}) = \delta_{\pm, k}\,\, \text{for some}\,\,k \}), \end{equation*}

and

\begin{equation*} \mathbb N_{2} = \mathbb N\setminus\mathbb N_1 = \{n\in\mathbb N : \delta_{n}(x_{0}, y_{0}) = \delta_{0, k}\,\, \text{for some}\,\,k \}, \end{equation*}

which corresponds to

\begin{equation*} \hat{F}_{\alpha, \flat, 1}^{n-1}(x_{0}, y_{0}) \in \Omega_{\alpha}^{*}\,\, \text{if}\,\, n\in\mathbb N_{1}, \end{equation*}

and

\begin{equation*} \hat{F}_{\alpha, \flat, 1}^{n-1}(x_{0}, y_{0}) \in \left\{(x, y) : \left(\tfrac{1}{x}, \tfrac{1}{y}\right)\in \cup_{k=2}^{\infty} \hat{\Omega}_{\alpha, k, -} - (1, 1)\right\}\,\, \text{if}\,\, n\in\mathbb N_{2}. \end{equation*}

Remark 5.1. We easily find that the following properties hold:

1. (i) $\hat{F}_{\alpha, \flat, 1}(\langle \delta_{-, 2}\rangle_{\alpha, \flat, 1} \cap \{(x, y) : -\tfrac{1}{2} \le x \lt 0 \} ) = \langle \delta_{0, 2} \rangle_{\alpha, \flat, 1}$,

2. (ii) $\hat{F}_{\alpha, \flat, 1}(\langle \delta_{-, 2}\rangle_{\alpha, \flat, 1} \cap \{(x, y) : \alpha -1 \le x \lt -\tfrac{1}{2} \}) = \langle \delta_{-, k} \rangle_{\alpha, \flat, 1}$ for some $k\geq 2$,

3. (iii) $\hat{F}_{\alpha, \flat, 1}(\langle \delta_{-, k}\rangle_{\alpha, \flat, 1} ) = \langle \delta_{0, k} \rangle_{\alpha, \flat, 1} \,\,\text{for}\,\,k \ge 3$,

4. (iv) $\hat{F}_{\alpha, \flat, 1}(\langle \delta_{+, k}\rangle_{\alpha, \flat, 1} ) \subset \Omega_{\alpha}^{*}$.

Furthermore, if $n \in \mathbb N_{1}$ and either $\delta_{n}(x_{0}, y_{0}) = \delta_{-, k}$, $k \ge 3$, or $\delta_{n}(x_{0}, y_{0}) = \delta_{-,2}$ and the first coordinate of $\hat{F}_{\alpha, \flat, 1}^{n-1}(x_{0},y_{0})$ is in $\left(-\tfrac{1}{2}, 0 \right)$, then $n+1\in \mathbb N_{2}$ and $n+2 \in \mathbb N_{1}$. Otherwise $n+1 \in\mathbb N_{1}$.

We note that $r_{k+1} - r_{k} = 1$ or 2 for any $k\ge1$. Moreover, if $\delta_{n+1}(x_{0}, y_{0}) = \delta_{0,k}$ then $\delta_{n}(x_{0}, y_{0}) = \delta_{-, k}$.

The properties from Remark 5.1 show that we can reproduce $(\delta_{n}(x_{0}, y_{0}): j \ge 1)$ if $(c_{j} : j \ge 1)$ is given as a sequence of integers, or equivalently, $(\delta_{r_{j}}(x_{0}, y_{0}): j \ge 1)$; c.f. (5.1). Indeed, for $(x, y) \in \Omega_{\alpha}^{*}$ we see the following:

(5.8)\begin{equation} \begin{array}{lcl} \delta_{n}(x,y) = \delta_{-, k} & \iff & \delta_{n+1}(x, y) = \delta_{0, k} \,\, \text{if} \,\, k \ge 3 \\ \delta_{n}(x,y) = \delta_{-, 2} & \Rightarrow & \left\{\begin{array}{l} \delta_{n+1}(x, y) = \delta_{0, 2},\,\, \\ \text{and}\,\, \delta_{n+2}(x, y) = \delta_{+,k},\,\, k\geq k_0 \\ \qquad\text{or} \\ \delta_{n+ 1}(x, y) = \delta_{-,k},\,\, k\geq 2 \end{array}\right. \end{array} \end{equation}

and for $\ell\geq k_0$,

(5.9)\begin{equation} \delta_{n}(x,y) = \delta_{+, \ell} \,\, \Rightarrow \,\, \left\{\begin{array}{l} \delta_{n+1}(x, y) = \delta_{+,k} \,\, k\geq k_0\\\qquad\text{or} \\ \delta_{n+ 1}(x,y) = \delta_{-,k} \,\, k\geq 2. \end{array} \right. \end{equation}

So for any $(x, y) \in \Omega_{\alpha}^{*}$, from (5.8) and (5.9), we have

\begin{equation*} \begin{array}{lll} \delta_{n}(x,y) = \delta_{0, k} & \Rightarrow & \delta_{n-1}(x, y) = \delta(\hat{F}_{\alpha, \flat, 1}^{-1}(x, y) )= \delta_{-, k} \,\, \text{if}\,\,k\ge 3, \\ \delta_{n}(x, y) = \delta_{0, 2} & \Rightarrow & \delta_{n-1}(x, y) = \delta_{-, 2} \,\, \text{and} \,\, \delta_{n+1}(x, y) = \delta_{+, \ell}, \,\, \ell \geq k_{0}. \end{array} \end{equation*}

Let $(b_{i} : 1 \leq i \leq n)$ (see (5.1)) be a sequence of non-zero integers such that $\langle b_{1}, b_{2}, \ldots, b_{n} \rangle_{\alpha} \ne \emptyset$. From the above discussion, if $(x_0,y_0)\in \langle b_{1}, b_{2}, \ldots, b_{n} \rangle_{\alpha}$, we can construct a sequence $(\hat{\delta}_{1}, \hat{\delta}_{2}, \ldots, \hat{\delta}_{m})$ satisfying $\hat{\delta}_{r_{i}} = b_{i}$, where r_i is the ith occurrence of the form $\delta_{\pm, k}$ with $r_{1} = 1$ (since we start in $\hat{\Omega}_{\alpha}^{*}$) and $r_{n} = m$ or m − 1, and such that $(x_0,y_0)\in \langle \hat{\delta}_1, \ldots, \hat{\delta}_m \rangle_{\alpha,\flat,1}$. The latter happens when $\hat{\delta}_{m} = \delta_{0, 2}$. Similarly, we can construct $(b_{1}, b_{2}, \ldots, b_{n})$ from $(\hat{\delta}_{1}, \hat{\delta}_{2}, \ldots ,\hat{\delta}_{m})$, where $b_{n}= \hat{\delta}_{m}$ or $b_{n} = \hat{\delta}_{m-1}$.

To show the statement of the lemma, first we assume that $(x_{0}, y_{0})$ is α-1-Farey normal. It is easy to see that:

(5.10)\begin{eqnarray} &&\lim_{N \to \infty}\frac{1}{N} \sharp\{n : 1 \le n \le N, n \in \mathbb N_{2}\} = \lim_{K \to \infty} \mu_{\alpha, \flat, 1}\left( \bigcup_{k=2}^{K} \langle \delta_{0,k} \rangle_{\alpha, \flat,1}\right) \notag \\ && \qquad \,\, = \mu_{\alpha, \flat, 1}\left( \bigcup_{k=2}^{\infty} \langle \delta_{0,k} \rangle_{\alpha, \flat,1}\right). \end{eqnarray}

The equality (5.10) also shows that

(5.11)\begin{equation} \lim_{N \to \infty}\frac{1}{N} \sharp\{n : 1 \le n \le N, n \in \mathbb N_{1}\} = \mu_{\alpha, \flat, 1}(\hat{\Omega}_{\alpha}^{*} ) , \end{equation}

By the same argument, we can show that for any sequence $(\hat{\delta}_{1},\ldots ,\hat{\delta}_{m})$,

\begin{equation*} \mu_{\alpha, \flat,1}(\langle \hat{\delta}_{1}, \hat{\delta}_{2}, \ldots , \hat{\delta}_{m} \rangle_{\alpha, \flat, 1}) = \mu_{\alpha, \flat, 1}(\langle b_{1},b_{2}, \ldots , b_{n} \rangle_{\alpha}) \end{equation*}

if $\hat{\delta}_{m}$ is of the form $\delta_{\pm, k}$, otherwise

\begin{equation*} \mu_{\alpha, \flat,1}(\langle \hat{\delta}_{1}, \hat{\delta}_{2}, \ldots , \hat{\delta}_{m} \rangle_{\alpha, \flat, 1}) = \sum_{\ell_{0}}^{\infty} \mu_{\alpha, \flat, 1}(\langle b_{1}, b_{2}, \ldots , b_{n}, \ell \rangle_{\alpha}). \end{equation*}

We have

(5.12)\begin{align} {} & \frac{1}{N} \sharp\{j : 1 \le k\le N, \delta_{j} = b_{1}, \delta_{j+1} = b_{2}, \ldots , \delta_{j+ n -1} = b_{n}\} \notag \\ = & \frac{\hat{N}}{N} \frac{1}{\hat{N}} \sharp\{j : 1 \le j\le \hat{N}, c_{j} = b_{1}, c_{j+1} = b_{2}, \ldots, c_{j+ n -1} = b_{n}\} \end{align}

where $\hat{N}= \max \{j : r_{j} \le N \}$ and $c_{j} = \varepsilon_{j}(x_{0} \cdot a_{j}(x_{0})$.

With this notation, the left side converges to $\mu_{\alpha, \flat, 1}(\langle \hat{\delta}_{1}, \ldots, \hat{\delta}_{m}\rangle_{\alpha, \flat, 1} )$ and the first term of the right side goes to $\hat{\mu}_{\alpha, \flat, 1} (\Omega_{\alpha}^{*})$ (see (5.11)). Thus the second term of the right side goes to

\begin{equation*} \frac{\mu_{\alpha, \flat, 1}(\langle \hat{\delta}_{1}, \hat{\delta}_{2}, \ldots, \hat{\delta}_{m}\rangle_{\alpha, \flat, 1} )} {\mu_{\alpha, \flat, 1}(\Omega_{\alpha}^{*})}. \end{equation*}

as $N \to \infty$ and its numerator is

\begin{equation*} \mu_{\alpha, \flat, 1}(\{(x, y) : c_{r_{1}}(c, y) = b_{1}, c_{r_{2}}(x, y) = b_{2}, \ldots, c_{r_{n}}(x, y) = b_{n}\}. \end{equation*}

The definition of µ_α and $\mu_{\alpha, \flat, 1}$ implies that $\tfrac{1}{\mu_{\alpha, \flat,1}(\hat{\Omega}_{\alpha}^{*})}$ changes $\mu_{\alpha, \flat, 1}$ to µ_α. Consequently, we get the limit of the second term as $\mu_{\alpha}(\langle b_{1}, b_{2}, \ldots, b_{n} \rangle_{\alpha})$. This shows that $(x_{0}, y_{0})$ is normal with respect to $\hat{G}^{*}_{\alpha}$, which implies the α-normality of x ₀.

Next we suppose that $(x_{0}, y_{0})$ is not α-1-Farey normal. We want to show that $(x_{0}, y_{0})$ is also not α-normal. We check the equality (5.11) again. If $\tfrac{\hat{N}}{N}$ does not conveges to $\mu_{\alpha, \flat,1}(\Omega_{\alpha}^{*})$, then it is easy to see that x ₀ is not α-normal. On the other hand if $\lim_{N \to \infty} \tfrac{\hat{N}}{N} = \mu_{\alpha, \flat,1} (\Omega_{\alpha}^{*})$, then, by the same argument, we see that the second term of the right side of (5.12) does not converge to $\mu_{\alpha}(\langle b_{1}, b_{2}, \ldots, b_{n} \rangle_{\alpha})$, which shows that x ₀ is not α-normal. Indeed, from Remark 5.1 we can construct a sequence $(\hat{\delta}_{1}, \hat{\delta}_{2}, \ldots , \hat{\delta}_{m})$ such that $\hat{\delta}_{r_{j}} = \delta_{{\rm sgn}(b_{j}), \, |b_{j}|}$ and $m = r_{J }= \hat{N}$, i.e., $\hat{\delta}_{r_{j}} = \delta_{{\rm sgn}, |b_{j}| }$, $1 \le j \le J$ and for other $\ell$ ( $\ell \ne r_{j}$, $1 \le j \le J$) are of the form $\delta_{0, k}$, and it is determined uniquely by $\hat{\delta}_{\ell - 1}$ since $\ell -1$ is r_j for some $1 \le j \le J$. Indeed, $\hat{\delta}_{\ell} = \delta_{0, k}$ implies $\hat{\delta}_{\ell -1} = \delta_{-, k}$. Note that $\hat{\delta}_{\ell -1} = \delta_{-, 2}$ does not mean $\hat{\delta}_{\ell} = \delta_{0, 2}$ since $\ell -1 = r_{j}$, $\ell = r_{j+1}$ can happen. But $\hat{\delta}_{\ell -1} = \delta_{-, k}$, $k \ge 3$, implies $r_{j}+1 \ne r_{j+1}$. Then we can show the estimate in the above.

The same idea shows that the α-1-Farey normality is equivalent to the α-2-Farey normality. Here we note that $\hat{F}_{\alpha, \flat,2}$ is an induced map of $\hat{F}_{\alpha, \flat, 1}$ and for $(x,y)\in V_{\alpha, \flat, 2}$, $\eta_n(x,y)\eta_{n+1}(x,y)\neq \delta_{- , 2}\delta_{0,2}$. So in the η-code of a point (x, y) the digit $\delta_{0,2}$ serves as a marker for the missing preceding digit $\delta_{-,2}$ in the corresponding δ-code of (x, y).

Lemma 5. An element $(x, y) \in \Omega_{\alpha}^{*}$ is α-2-Farey normal if and only if it is α-1-Farey normal.

Proof. Sketch of the proof

From the sequence $\delta_{n}(x, y)$, we can construct $\eta_{m}(x, y) \in V_{\alpha, \flat, 2}$ by deleting the digit $\delta_{-,2}$ that is followed by a digit $\delta_{0,2}$. More precisely,

\begin{eqnarray*} &&\delta_{n-1}(x, y), \delta_{n}(x, y) = \delta_{-,2},\,\, \delta_{n+1}((x, y) = \delta_{0,-2} \\ &\Rightarrow& \eta_{m} = \delta_{n-1}(x, y), \eta_{m+1}(x, y) = \delta_{n+1}, \end{eqnarray*}

where m is the cardinality of n such that $\delta_{n}\delta_{n+1} = \delta_{-, 2}\delta_{0,2}$. On the other hand, given the sequence $(\eta_{m}(x, y) : - \infty \lt m \lt \infty)$, we can construct the sequence $(\delta_{n}(x, y) : -\infty \lt n \lt \infty)$ by inserting $\delta_{-,2}$ before every occurrence of every $\delta_{0,2}$. Following the proof of Lemma 4, we get the result.

Lemma 6. An element $(x_{0}, y_{0}) \in \Omega_{\alpha}^{*}$ is α-2-normal if and only if $\psi^{-1}(x_{0}, y_{0}) \in W$ is normal with respect to $\hat{G}^{*}$.

Proof. Following the above proofs, cylinder sets associated with $\hat{F}_{\alpha, \flat,2}$ are approximated by each other (using ψ and ψ ⁻¹); see Theorem 5. Suppose that $(x_{0}, y_{0})$ is α-2-Farey normal. Every cylinder set associated with $\hat{G}^{*}(= \hat{G}_{1}^{*})$ is a rectangle. To be more precise, a cylinder set $\langle a_{1}, a_{2}, \ldots, a_{n} \rangle_{1}$ is of the form $\left[\frac{p_{n}}{q_{n}}, \frac{p_{n} + p_{n-1}}{q_{n} + q_{n-1}} \right) \times [-\infty, -1]$, or $\left(\frac{p_{n} + p_{n-1}}{q_{n} + q_{n-1}}, \frac{p_{n}}{q_{n}} \right] \times [-\infty, -1]$. We divide it into three parts such that $\eta_{1}(x, y) = \delta_{\sharp, k}$, $\sharp = +, 0$, and −. Then ψ ⁻¹-image of each part is a countable union of $\hat{F}_{\alpha, \flat,2}$ cylinder sets, just as discussed in the above. Hence we can prove that $(x_{0}, y_{0})$ (or $(x_{0}+1, y_{0}+1)$ is normal with respect to $\hat{G}^{*}$ in the same way.

Now suppose that $(x_{0}, y_{0})$ is not α-2-normal and $\psi^{-1}(x_{0}, y_{0})$ is normal with respect to $\hat{G}^{*}$. Then there exist ϵ > 0 and a cylinder set with respect to $\hat{F}_{\alpha, \flat, 2}$ such that either

(5.13)\begin{eqnarray} &&\lim_{N \to \infty} \frac{1}{N} \# \{n : 0 \le m \le N-1, \,\, \hat{F}_{\alpha, \flat, 2}^{m}(x_0, y_0) \in \langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2}\} \nonumber \\ && \qquad \,\, \gt \mu_{\alpha, \flat, 2}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}) + \epsilon , \end{eqnarray}

or

(5.14)\begin{eqnarray} &&\lim_{N \to \infty} \frac{1}{N} \# \{n : 0 \le m \le N-1, \,\, \hat{F}_{\alpha, \flat, 2}^{m}(x_0, y_0) \in \langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2}\} \nonumber \\ && \qquad \,\, \lt \mu_{\alpha, \flat, 2}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}) - \epsilon , \end{eqnarray}

and

(5.15)\begin{eqnarray} &&\lim_{N\to\infty} \frac{1}{N} \# \{0\leq m\leq N-1:\hat{G^{*}}^{m}(\psi^{-1}(x_{0}, y_{0})) \in \langle b_{1}, \ldots, b_{n} \rangle_{1,(1,n)} \} \nonumber \\ && \qquad \,\, = \hat{\mu}(\langle b_{1}, \ldots, b_{n} \rangle_{1, (1, n)}) \end{eqnarray}

for any sequence of positive integers $(b_{1}, b_{2}, \ldots, b_{n})$, where $\hat{\mu}$ is the measure defined by $\tfrac{1}{\log 2} \tfrac{dx dy}{(x - y)^{2}}$. Note that $\langle \cdots \rangle_{1, (1,n)}$ means a cylinder set with respect to $\hat{G}^{*} = \hat{G}_{\alpha}^{*}$ with α = 1.

We start by assuming (5.13) and (5.15) hold, and show it will lead to a contradiction. Since the set of cylinder sets associated with $\hat{G}^{*}$ generates the Borel σ-algebra, there exist a finite number of pairwise disjoint cylinder sets

\begin{equation*} \langle b_{j,1}, b_{j, 2}, \ldots, b_{j, k_{j}}\rangle_{1,(1, k_{j})}, \qquad 1 \le j \le M \lt \infty\ \text{and}\ 1 \le k_{j} \lt \infty, \end{equation*}

such that

\begin{equation*} \psi^{-1}(\langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2}) \subset \bigcup_{j = 1}^{M} \langle b_{j,1}, b_{j, 2}, \ldots, b_{j, k_{j}}\rangle_{1,(1, k_{j})} \end{equation*}

and

\begin{equation*} \hat{\mu} \left( \bigcup_{j = 1}^{M} \langle b_{j,1}, b_{j, 2}, \ldots, b_{j, k_{j}}\rangle_{1}\right) \lt \hat{\mu}(\psi^{-1}(\langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2})) + \tfrac{1}{2}\epsilon. \end{equation*}

Since ψ is an isomorphism (see Theorem 5), we have

\begin{equation*} \hat{F}_{\alpha,\flat,2}(x_0,y_0) = \psi \hat{G}^{*}\psi^{-1}(x_0,y_0) \end{equation*}

and

\begin{equation*} \mu_{\alpha, \flat, 2}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}) = \hat{\mu}(\psi^{-1}(\langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2})). \end{equation*}

Thus,

\begin{eqnarray*} && \lim_{N \to \infty} \frac{1}{N} \# \Big\{0 \leq m \leq N-1:\,\, \hat{F}_{\alpha, \flat, 2}^{m}(x_0,y_0)\in \langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2}\Big\} \\ &\leq & \lim_{N \to \infty} \frac{1}{N} \# \Big\{0 \leq m \leq N-1:\,\, \hat{G^{*}}^{m}(\psi^{-1}(x_{0},y_{0})) \in \bigcup_{j = 1}^{M} \langle b_{j,1}, \ldots, b_{j, k_{j}}\rangle_{1} \Big\}\\ &=& \hat{\mu}\left( \bigcup_{j = 1}^{M} \langle b_{j,1}, b_{j, 2}, \ldots, b_{j, k_{j}}\rangle_{1}\right)\\ & \lt & \hat{\mu}(\psi^{-1}(\langle e_{1}, e_{2}, \ldots, e_{\ell} \rangle_{\alpha, \flat, 2})) + \tfrac{1}{2}\epsilon \\ &=& \mu_{\alpha, \flat, 2}(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}) + \tfrac{1}{2}\epsilon. \end{eqnarray*}

Combining this with (5.13) yields $\epsilon \lt \tfrac{1}{2}\epsilon$, which is a contradiction.

On the other hand, if (5.14) and (5.15) hold, a proof similar to the one above but now approximating the cylinder $\psi^{-1}\left(\langle e_{1}, e_{2}, \ldots, e_{\ell}\rangle_{\alpha,\flat,2}\right)$ from “inside” by a union of cylinders leads to the same contradiction.

Proof. Proof of Theorem 4

This is a direct consequence of Lemmas 3, 4, 5 and 6.

5.2. Non-ϕ-mixing property

We start with some definitions of mixing properties. Let $(\Omega, \mathfrak{B}, P)$ be a probability space. For sub σ-algebras $\mathcal{A}$ and $\mathcal{B}\subset\mathfrak{B}$, we put

\begin{equation*} \phi (\mathcal A, \mathcal{B}) = \sup \left\{ \left|\frac{P(A \cap B)}{P(A)} - P(B) \right| : A \in \mathcal A, \,\, B \in \mathcal{B}, \,\, P(A) \gt 0\right\}. \end{equation*}

Suppose that $(X_{n} : n \ge 1)$ is a stationary sequence of random variables. We denote by $\mathcal F_{m}^{n}$ the sub-σ algebra of $\mathfrak B$ generated by $X_{m}, X_{m+1}, X_{m+2}, \ldots , X_{n}$. We define $\phi(n) = \sup_{m \ge 1} \phi(\mathcal F_{1}^{m}, \mathcal F_{m+n}^{\infty})$ and $\phi^{*}(n) = \sup_{m \ge 1} \phi(\mathcal F_{m+n}^{\infty}, \mathcal F_{1}^{m})$.

The process $(X_{n}:n \ge 1)$ is said to be ϕ-mixing if $\lim_{n \to \infty} \phi(n) = 0$ and reverse ϕ-mixing if $\lim_{n \to \infty} \phi^{*}(n) = 0$, respectively.

G_α is said to be ϕ-mixing (or reverse ϕ-mixing) if $(a_{\alpha, n}, \epsilon_{\alpha, n})$ is ϕ-mixing (or reverse ϕ-mixing), respectively. In [Reference Nakada and Natsui28], it is shown that G_α is not ϕ-mixing for a.e. α, $\tfrac{1}{2} \le \alpha \le 1$. On the other hand, G_α is reverse ϕ-mixing for every α, $0 \lt \alpha \le 1$, which follows from [Reference Aaronson and Nakada1].

In [Reference Nakada and Natsui28], it is shown that G_α is weak Bernoulli for any $\tfrac{1}{2}\leq \alpha \le 1$ but is not ϕ-mixing. It is not hard to show that G_α is weak Bernoulli for any $0 \lt \alpha \lt \tfrac{1}{2}$ following the proof given in [Reference Nakada and Natsui28]. On the other hand, it follows that G_α is reverse ϕ-mixing for any $0 \lt \alpha \le 1$; see [Reference Aaronson and Nakada1]. In the proof of the next Theorem we outline how one can extend the proofs of [Reference Nakada and Natsui28] to the case $0 \lt \alpha \lt \tfrac{1}{2}$.

Theorem 6 For almost every α, $0 \lt \alpha \lt 1$, G_α is not ϕ-mixing.

Remark 5.2. One has ϕ-mixing whenever the orbit of $\alpha -1$ and the left-orbit of α are ultimately periodic. This is the case when α is rational or quadratic irrational.

Proof. Sketch of the proof

The proof of the non-ϕ-property in [Reference Nakada and Natsui28] is based on the following two properties:

1. (i) For almost every α, $\tfrac{1}{2} \leq \alpha \leq 1$, $(G_{\alpha}^{n}(\alpha) : n \ge 0)$ is dense in $\mathbb I_{\alpha}$.

2. (ii) For every α, $\tfrac{1}{2} \lt \alpha \lt \tfrac{\sqrt{5} - 1}{2}$, $G_{\alpha}^2(\alpha) = G_{\alpha}^{2}(\alpha -1)$ and for every α, $\tfrac{\sqrt{5}-1}{2} \le \alpha \lt 1$, $G_{\alpha}^{2}(\alpha) = G_{\alpha}(\alpha - 1)$, respectively.

The first statement follows from the fact that the set of normal numbers w.r.t. α is independent of α ([Reference Kraaikamp and Nakada22]). Because of Theorem 4 above, we can extend (i) to almost every $0 \lt \alpha \le 1$.

The second statement is generalized in [Reference Carminati and Tiozzo8]: for almost every α, there exists $n, m$ such that $G_{\alpha}^{n}(\alpha) = G_{\alpha}^{m}(\alpha - 1)$. From this, we can show that thin cylinders exist for almost every α and for any δ > 0. To be more precise, for any δ > 0, a cylinder set $\mathcal C = \langle c_{\alpha, 1}, c_{\alpha, 2}, \ldots , c_{\alpha, \ell} \rangle_{\alpha}$ is said to be a δ-thin-cylinder if

1. a) $G_{\alpha}^{\ell}(\mathcal C)$ is an interval,

2. b) $G_{\alpha}^{\ell}:\, \mathcal C \to G_{\alpha}^{\ell}(\mathcal C)$ is bijective,

and

1. c) $|G_{\alpha}^{\ell}(\mathcal C)| \lt \delta$.

Once we have a sequence of δ_n-thin cylinders with $\delta_{n} \searrow 0$, the proof is completely the same as one given in [Reference Nakada and Natsui28] if we choose α so that the matching property holds and $\alpha - 1$ is α-normal. For in this case, there exist n ₀, m ₀ such that $G_{\alpha}(\alpha - 1)^{n_{0}} = G_{\alpha}^{m_{0}}(\alpha)$ (matching property). Moreover, there exists $n_{\delta} \gt \max (n_{0}, m_{0})$ such that $\min (|\alpha - G_{\alpha}^{n_{\delta}}(\alpha -1)|, |(\alpha - 1) - G_{\alpha}^{n_{\delta}}(\alpha -1)| \lt \delta)$, which follows from the normality of $\alpha -1$.

We suppose that $|(\alpha - 1) - G_{\alpha}(\alpha -1)| \lt \delta$. Because of the matching property, see [Reference Carminati and Tiozzo8], either $\langle c_{1}(\alpha - 1), c_{2}(\alpha - 1), \ldots, c_{n_{\delta}}(\alpha -1) \rangle_{\alpha}$ or $\langle c_{1}(\alpha), c_{2}(\alpha), \ldots, c_{n_{\delta}}(\alpha) \rangle_{\alpha}$ is an δ-thin-cylinder set. This is because of the following: If α is normal, then it means α is not rational nor quadratic. The iteration $G_{\alpha}^{n}$ associated with α and $G_{\alpha}^{m}$ associated with $\alpha - 1$ are linear fractional transformations. Hence $\alpha \mapsto G_{\alpha}^{n}(\alpha)$ and $(\alpha \mapsto G_{\alpha}^{m}\circ " - 1")(\alpha)$ define the same linear fractional transformation, otherwise α is a fixed point of a linear fractional transformation which means α is either rational or quadratic. We denote by L_r, $L_{\ell}$ and S the linear fractional transformations which induce $G_{\alpha}^{n}(\alpha)$, $G_{\alpha}^{m}(\alpha -1)$ and $x \mapsto x - 1$, respectively. Then $L_{r}(\langle c_{1}(\alpha),\ldots, c_{n_{\delta}}(\alpha) \rangle_{\alpha}) = (L_{\ell}\circ S) (\langle c_{1}(\alpha), \ldots, c_{n_{\delta}}(\alpha) \rangle_{\alpha}) $. This shows that $G_{\alpha}^{n}(\langle c_{1}(\alpha), \ldots, c_{n_{\delta}}(\alpha) \rangle_{\alpha}) $ and $G_{\alpha}^{m}(\langle c_{1}(\alpha - 1), \ldots, c_{n_{\delta}}(\alpha -1) \rangle_{\alpha})$ have one common end point $G_{\alpha}^{n}(\alpha)$ and no common inner point. In the case of $|\alpha - G_{\alpha}(\alpha -1)| \lt \delta$, the same holds exactly. In this way, we can choose a sequence of δ_n-thin cylinders and Theorem 6 follows in exactly the same way as in [Reference Nakada and Natsui28].