This section is devoted to study the space of mechanical systems \(\overline}}}\) as a metric space and different isometric representations. One is obtained by defining a new metric—called the Farey-metric \(d_F\)—on the interval [0, 1] inherited from the Farey number. Using results from [30]—linking Farey numbers and the dictionaries of Sturmian words—we show that the map

$$\begin [0,1] \rightarrow }}, \alpha \mapsto \Omega _\alpha \end$$

is isometric if [0, 1] is equipped with the Farey metric \(d_F\), see Theorem 3.8. Then, this map uniquely extends to an isometric surjective map \(\overline_F\rightarrow \overline}}}\). In addition, we give another representation of these spaces as the boundary of an infinite tree in Sect. 3.3.

3.1 The Farey Space

The Farey metric \(d_F:[0,1]\times [0,1]\rightarrow [0,\infty )\) on the interval [0, 1] is defined by

$$\begin d_F(\alpha , \beta ) := 0, & \alpha = \beta ,\\ 1, & \alpha \in \ \text \beta \in \,\\ \min \big \: \exists r \in F_m \text \alpha , \beta \in (r, r^*)\big \}, & \text \end\right. } \end$$

We will see that this defines an ultrametric on [0, 1], see Proposition 3.1. The set

$$\begin B_F(\alpha ,R) := \ \end$$

denotes the open ball around \(\alpha \in [0,1]\) of radius \(R>0\) in the Farey metric.

Proposition 3.1

The Farey metric \(d_F\) defines an ultrametric on [0, 1], i.e.,

$$\begin d_F(\alpha ,\beta )\le \max \big \, \qquad \alpha ,\beta ,\gamma \in [0,1]. \end$$

Moreover, the following holds for \(\alpha ,\beta \in [0,1]\) and \(m\in }\).

(a)

If \(d_F(\alpha ,\beta )=1\), then \(\alpha \in \\) or \(\beta \in \\).

(b)

If \(\alpha \ne \beta \) and \(\alpha \le r\le \beta \) holds for some \(r\in F_m\), then \(d_F(\alpha ,\beta )> \frac\).

(c)

If \(\alpha \in F_m\), then \(B_F\big (\alpha ,\frac\big )=\\).

(d)

If \(\alpha \not \in F_m\), then there exists an \(r\in F_m\) such that \(B_F\big (\alpha ,\frac\big )=(r,r^*)\).

(e)

We have \(|\alpha -\beta |\le 2d_F(\alpha ,\beta )\).

Proof

First note that for \(\alpha ,\beta \in [0,1]\),

$$\begin d_F(\alpha ,\beta ) \in \\cup \left\ \,:\, m\in }\right\} . \end$$

(3.1)

We first prove (a) to (e). Statement (a) follows immediately from the definition. For (b), suppose \(\alpha \ne \beta \) and \(\alpha \le r\le \beta \) holds for some \(r\in F_m\). Thus, there exists no \(s\in F_m\) such that \(\alpha ,\beta \in (s,s^*)\) proving \(d_F(\alpha ,\beta )> \frac\).

For (c), suppose \(\alpha \in F_m\). If \(\beta \in [0,1]\\), then (b) implies \(d_F(\alpha ,\beta )> \frac\) proving \(B_F\big (\alpha ,\frac\big )=\\).

For (d), suppose \(\alpha \not \in F_m\). Then, there is a unique \(r\in F_m\) such that \(\alpha \in (r,r^*)\). If \(\beta \in (r,r^*)\), then \(d_F(\alpha ,\beta )\le \frac<\frac\). If \(\beta \in [0,1](r,r^*)\), then (b) leads to \(d_F(\alpha ,\beta )> \frac\). Thus, \(B_F\big (\alpha ,\frac\big )=(r,r^*)\) follows using Equation (3.1).

For (e), note that the statement trivially holds if \(\alpha \in \\) or \(\beta \in \\). Let \(\alpha ,\beta \in (0,1)\) be different. By Equation (3.1) and (a), there is an \(m\in }\) such that \(d_F(\alpha ,\beta )=\frac\). Thus, there is an \(r\in F_m\) satisfying \(\alpha ,\beta \in (r,r^*)\). Then, \(|r-r^*|\le \frac\) holds by Lemma 2.4 implying

$$\begin |\alpha -\beta |\le \frac\le \frac = 2d_F(\alpha ,\beta ). \end$$

Finally, we show that \(d_F\) is an ultrametric. Clearly, \(d_F\) is symmetric and definite and it suffices to show \(d_F(\alpha ,\beta )\le \max \big \\) for all \(\alpha ,\beta ,\gamma \in [0,1]\). If \(d_F(\alpha ,\beta )=1\), then (a) yields \(\alpha \in \\) or \(\beta \in \\). Thus, \(\max \big \=1\) is concluded.

Now, suppose \(d_F(\alpha ,\beta )<1\). Then, there is an \(m\in }\) and an \(r\in F_m\) such that \(d_F(\alpha ,\beta )=\frac\) and \(\alpha ,\beta \in (r,r^*)\). If \(\gamma \not \in (r,r^*)\), then \(d_F(\alpha ,\gamma )> \frac=d_F(\alpha ,\beta )\) follows from (b). Next suppose \(\gamma \in (r,r^*)\). Since \(d_F(\alpha ,\beta )=\frac\), there is an \(s\in F_\) such that \(\alpha \le s \le \beta \). Thus, either \(\gamma <s\) or \(\gamma \ge s\). If \(\gamma <s\), then (b) yields \(d_F(\gamma ,\beta )>\frac\). Hence, Equation (3.1) leads to \(d_F(\gamma ,\beta )\ge \frac=d_F(\alpha ,\beta )\) proving the claim. If \(\gamma \ge s\), one concludes similarly \(d_F(\alpha ,\gamma )\ge \frac=d_F(\alpha ,\beta )\) finishing the proof. \(\square \)

It is not difficult to check that [0, 1] equipped with \(d_F\) is not a complete metric space. For instance, \((\frac)_}}\) is a Cauchy sequence but it admits no limit with respect to \(d_F\). Denote by \([0,1]_F\) the interval [0, 1] equipped with the topology inherited from \(d_F\). Furthermore, \(\overline_F\) denotes the corresponding completion with respect to \(d_F\).

Definition 3.2

The topology on \(\overline_F\) induced by \(d_F\) is called Farey topology, and the metric space \((\overline_F,d_F)\) is called Farey space.

Remark

A basis for the Farey topology on [0, 1] is given by

$$\begin }}=\big \}\cap [0,1] \big \}. \end$$

Note that Proposition 3.1 (e) asserts that the Farey topology is finer than the Euclidean topology, which can be deduced also from the bases of the topologies. In particular, convergence in \([0,1]_F\) implies convergence in [0, 1]. In addition, we point out that while \(\overline_F\) is a totally disconnected space, it has plenty isolated points, which are all the rational numbers in [0, 1].

Since we are dealing with different topologies on [0, 1], we denote the Euclidean limit of \((x_n)_}} \subseteq [0,1]\) by \(\lim _E x_n\) and the limit in the Farey topology by \(\lim _F x_n\) (provided they exist).

Proposition 3.3

Let \((\alpha _n)_}}\subseteq [0,1]_F\) be a Cauchy sequence with respect to \(d_F\). Then, exactly one of the following assertions hold:

(a)

The limit \(\alpha :=\lim _E \alpha _n\in [0,1]\) is irrational and \(\alpha =\lim _F \alpha _n\).

(b)

The limit \(\alpha :=\lim _E \alpha _n\in [0,1]\) is rational and

(b1)

\(\alpha _n = \alpha \) for n large enough;

(b2)

\(\alpha _n > \alpha \) for n large enough;

(b3)

\(\alpha _n < \alpha \) for n large enough.

Proof

According to Proposition 3.1 (e), the limit \(\alpha :=\lim _E \alpha _n\in [0,1]\) exists as \((\alpha _n)_}}\) is Cauchy with respect to \(d_F\). If \(\alpha \) is irrational, then for each \(m\in }\), there is a unique \(r\in F_m\) satisfying \(\alpha \in (r,r^*)\). Thus, Proposition 3.1 (d) implies that for each \(m\in }\), there is an \(n_m\in }\) such that \(\alpha _n\in B_F(\alpha ,\frac)=(r,r^*)\) for \(n\ge n_m\). Hence, \(\alpha =\lim _F \alpha _n\) follows.

Next suppose \(\alpha =\frac\) with p and q coprime. Assume toward contradiction that neither (b1),(b2) nor (b3) holds. Thus, for each \(N\in }\), there are \(n_1,n_2\ge N\) such that \(\alpha _<\frac\le \alpha _\) or \(\alpha _\le \frac<\alpha _\). Both cases imply by Proposition 3.1 (b) that \(d_F(\alpha _,\alpha _)>\frac\). This contradicts that \((\alpha _n)_}}\) is Cauchy with respect to \(d_F\). \(\square \)

As a direct consequence, the Farey space can be represented as the interval [0, 1] where

each rational point \(r\in (0,1)\cap }\) is split up into three points \(r_-,r_+\) and r, and

0 is split up into two points 0 and \(0_+\), and

1 is split up into two points 1 and \(1_-\).

This is formulated in the following corollary and sketched in Fig. 3.

Corollary 3.4

For \(r\in }\cap [0,1]\), let \(r_+:=\lim _F r+\frac\) and \(r_-=\lim _F r-\frac\). Then,

$$\begin \overline_F = [0,1]\cup \}\cap [0,1)\} \cup \}\cap (0,1]\}. \end$$

Proof

This is an immediate consequence of Proposition 3.3. \(\square \)

We emphasize that the definition of \(r_+\) and \(r_-\) is independent of the specific sequence \(\alpha _n:=r\pm \frac\).

Remark

We note that for irrational \(\alpha \in [0,1]\), the associated subshift \(\Omega _\alpha \) can be factorized through a circle \(}/ }\) where the circle splits at each \(n\alpha \mod 1, n\in },\) into two points, see, e.g., [21, prop. 5.1]. This cutting is different to the representation of the subshifts \(\overline}}}\) developed here via \(\overline_F\) where the interval splits at the rational points into three points.

3.2 The Dynamical Representation

In this section, we show that the Farey space \(\overline_F\) and the mechanical dynamical systems \(\overline}}}\) equipped with \(d_}}\) are isometrically isomorphic.

For a set A, let \(\sharp A\) be its cardinality and define the complexity function \(\wp \) by

$$\begin \wp :}\times }}\rightarrow }, \quad \wp (n,\Omega ) := \sharp }}(\Omega )\cap }}^n. \end$$

The following statement is well known, see, e.g., [9, 22].

Lemma 3.5

Let \(\alpha \in [0,1]\). Then, the following assertions hold.

(a)

Let \(m\in }\) and \(\alpha \in [0,1] F_m\), then \(\wp (m,\Omega _\alpha )=m+1\). In particular, if \(\alpha \in [0,1]}\), then \(\wp (m,\Omega _\alpha )=m+1\) holds for all \(m\in }\).

(b)

If \(\alpha =\frac\in [0,1]\) with p and q coprime, then \(\wp (m,\Omega _\alpha )=q\) for \(m\ge q\) and \(\wp (j,\Omega _\alpha )=j+1\) for \(1\le j\le q-1\).

The key connection of the mechanical systems and Farey numbers are the following results. Those are consequences from a work by Mignosi [30], see also the recent elaboration on that [41].

Lemma 3.6

Let \(m \in }\) and \(\alpha ,\beta \in F_m\) be different, then \(}}_\alpha \cap }}^m \ne }}_\beta \cap }}^m\).

Proof

Without loss of generality, suppose \(\alpha < \beta \). For the induction base \(m=1\), observe that

$$\begin F_1=\ \qquad \text \qquad }}_0 \cap }}^1 = \ \ne \ = }}_0 \cap }}^1. \end$$

For the induction step, suppose that for all different \(\alpha ,\beta \in F_m\), we have \(}}_\alpha \cap }}^m \ne }}_\beta \cap }}^m\). Note that \(F_j\subseteq F_m\) for \(j\le m\) and \(}}_\alpha \cap }}^m \ne }}_\beta \cap }}^m\) implies \(}}_\alpha \cap }}^n \ne }}_\beta \cap }}^n\) for \(n\ge m\).

Let \(\alpha = \frac,\beta = \frac\in F_\) be different and irreducible. If \(q\ne q'\), then Lemma 3.5 (b) implies \(}}_\alpha \cap }}^n \ne }}_\beta \cap }}^n\) for \(n=\max \\le m+1\) since their cardinalities differ. Thus, there is no loss of generality in assuming \(q=q'\). If \(q=q'<m+1\), then \(\alpha ,\beta \in F_m\) and so \(}}_\alpha \cap }}^ \ne }}_\beta \cap }}^\) follows from the induction base.

It is left to treat the case \(q=q'=m+1\). Since \(m+1\ge 2\), we have \(\alpha >0\). Assume toward contradiction that \(}}_\alpha \cap }}^ = }}_\beta \cap }}^\). Lemma 2.3 asserts that there exist \(\beta _*,\beta ^*\in F_j\) for some \(j<m+1\) such that \(\beta =\beta _* \oplus \beta ^*\). Since \(j<m+1\), we conclude (by Lemma 3.5) \(\alpha \ne \beta _*\) and \(\wp (m+1,\Omega _)\le j < m+1\). Let \(\gamma \in (\beta _*, \beta )\). Then, \(}}_\beta \cap }}^ \subseteq }}_\gamma \cap }}^\) holds by [30, cor. 9]. With this at hand, [30, thm. 16] applied to \(0<\alpha<\beta _*<\gamma <\beta \) leads to

$$\begin (}}_\beta \cap }}^) }}_ \cap }}_\alpha \subseteq (}}_\gamma \cap }}^) }}_ \cap }}_\alpha = \emptyset . \end$$

Thus, our assumption \(}}_\alpha \cap }}^ = }}_\beta \cap }}^\) leads to \((}}_\beta \cap }}^) }}_ = \emptyset \). Hence, \(}}_ \cap }}^ \subseteq }}_ \cap }}^\) follows. Since \(\beta = \frac\) is irreducible, Lemma 3.5 (b) yields

$$\begin m+1 = \wp (m+1, \Omega _\beta ) \le \wp (m+1, \Omega _) \le j < m+1, \end$$

a contradiction. Therefore, \(\alpha ,\beta \in F_\) different implies \(}}_\alpha \cap }}^ \ne }}_\beta \cap }}^\). \(\square \)

Proposition 3.7

Let \(\alpha ,\beta \in [0,1]\) and \(m\in }\). Then, \(}}_\alpha \cap }}^m=}}_\beta \cap }}^m\) holds if and only if there exists an \(r\in F_m\) with \(\alpha ,\beta \in (r,r^*)\).

Proof

If \(\alpha ,\beta \in (r,r^*)\), then the equation \(}}_\alpha \cap }}^m=}}_\beta \cap }}^m\) follows from [30, cor. 10]. For the converse direction, assume without loss of generality \(\alpha <\beta \). By contraposition, we prove that if there is an \(r\in F_m\) with \(\alpha \le r\le \beta \), then \(}}_\alpha \cap }}^m\ne }}_\beta \cap }}^m\). For \(\alpha \in F_m\\) and \(\beta =r\), we get \(}}_\alpha \cap }}^m \ne }}_\beta \cap }}^m\) directly from Lemma 3.6. If \(\alpha <r=\beta \) and \(\alpha \not \in F_m\), then

$$\begin \wp (m,\Omega _\alpha )=m+1>m=\wp (m,\Omega _\beta ) \end$$

follows from Lemma 3.5 proving \(}}_\alpha \cap }}^m\ne }}_\beta \cap }}^m\). If \(\alpha =r<\beta \), then \(}}_\alpha \cap }}^m\ne }}_\beta \cap }}^m\) is similarly concluded.

The last case to treat is when \(\alpha \) nor \(\beta \) are in \(F_m\). Since there is still \(r \in F_m\) with \(\alpha< r < \beta \), we can choose r such that \(\beta \in (r,r^*)\). For this choice, [30, thm. 16] applies and yields

$$\begin \emptyset = \big ((}}_\beta \cap }}^m) }}_\big )\cap }}_\alpha = \big ((}}_\beta \cap }}^m) }}_\big )\cap (}}_\alpha \cap }}^m). \end$$

Assume by contradiction that \(}}_\alpha \cap }}^m=}}_\beta \cap }}^m\). Then, the previous considerations imply \(}}_\alpha \cap }}^m\subseteq }}_r\cap }}^m\). On the other hand,

$$\begin \wp (m,\Omega _\alpha )=m+1>m=\wp (m,\Omega _r) \end$$

holds by Lemma 3.5, a contradiction. \(\square \)

The latter statement is the crucial ingredient to prove that the following map is isometric.

Theorem 3.8

There is a unique surjective isometry \(\Phi :(\overline_F,d_F) \rightarrow (\overline}}},d_}}})\) such that \(\Phi (\alpha )=\Omega _\alpha \) for \(\alpha \in [0,1]\). In particular, \(d_F(x,y)=d_}}(\Phi (x),\Phi (y))\) for all \(x,y\in \overline_F\).

Proof

First note that it suffices to show that the map \([0,1]_F\rightarrow }},\alpha \mapsto \Omega _\alpha ,\) is bijective and isometric. By definition of \(}}\), this map is surjective. Hence, it is sufficient to prove \(d_F(\alpha ,\beta )=d_}}(\Omega _\alpha ,\Omega _\beta )\) for all different \(\alpha ,\beta \in [0,1]_F\).

If \(\alpha \in \\) or \(\beta \in \\), then \(d_F(\alpha ,\beta )=1\). Since \(\alpha \ne \beta \), \(}}_1\cap }}^1=\\) and \(}}_0\cap }}^1=\\), we conclude \(d_}}(\Omega _\alpha ,\Omega _\beta )=1=d_F(\alpha ,\beta )\).

Next, let \(\alpha ,\beta \in (0,1)\) and \(m\in }\) be such that \(\alpha <\beta \) and \(d_F(\alpha ,\beta )=\frac\). Hence, there is an \(r\in F_m\) and an \(s\in F_\) such that \(\alpha ,\beta \in (r,r^*)\) and \(\alpha \le s\le \beta \). Thus, Proposition 3.7 leads to

$$\begin }}_\alpha \cap }}^m = }}_\beta \cap }}^m \qquad \text \qquad }}_\alpha \cap }}^ \ne }}_\beta \cap }}^. \end$$

Hence, \(d_}}(\Omega _\alpha ,\Omega _\beta )=\frac\) is concluded from Proposition 2.1, proving \(d_}}(\Omega _\alpha ,\Omega _\beta )=d_F(\alpha ,\beta )\). \(\square \)

Corollary 3.9

The Farey space \(\overline_F\) is compact.

Proof

The space \(}}\) is compact [4, 8]. Therefore, \(\overline}}} \subseteq }}\) is compact in the induced topology. The statement then follows from Theorem 3.8, as \(\Phi \) is a homeomorphism. \(\square \)

Corollary 3.10

Let \(\Phi :(\overline_F,d_F) \rightarrow (\overline}}},d_}}})\) be the surjective isometry from Theorem 3.8.

(a)

If \(x\in \overline_F([0,1]\cap })\), then \(\wp (n,\Phi (x))=n+1\) for all \(n\in }\).

(b)

If \(x\in [0,1]\cap }\), then there exists an \(n_x\in }\) such that \(\wp (n,\Phi (x))=n\) for all \(n\ge n_x\).

Proof

If \(x\in [0,1]\), then the statement follows from Lemma 3.5. Let \(x\in \overline_F[0,1]\) and \(n\in }\). Then, there is a sequence \((x_k)_}}\subseteq [0,1]}\) such that \(\lim _F x_k=x\). Since \(x_k\) is irrational, we have \(\wp (n,\Phi (x_k))=n+1\). Then, Proposition 2.1 implies that the complexity function is continuous in \(}}\), namely \(\wp (n,\Phi (x))=\lim _\wp (n,\Phi (x_k))=n+1\). \(\square \)

3.3 The Graph Representation

In this section, we isometrically represent the Farey space as the boundary of a tree which is closely related to the Farey tree.

A tuple \(}}= (}}, }})\) is called a directed graph where \(}}\) is a countable set and \(}}\subseteq }}\times }}\). We call \(}}\) the vertex set and \(}}\) the edge set of \(}}\). Then, \(u\in }}\) is connected to \(v\in }}\) if \((u,v)\in }}\). A finite path is a tuple \((v_0, \dots v_n) \in }}^\) if \((v_,v_i)\in }}\) for \(1\le i\le n\). If \(v_0=v_n\) and \(n\in }\), we call the path a cycle. Finite paths define a (possibly strict, i.e., irreflexive) partial order relation on the vertex set \(}}\), namely \(u\rightarrow v\) holds for \(u,v\in }}\), if there is a finite path \((v_0,\ldots ,v_n)\) with \(v_0=u\) and \(v_n=v\). A directed graph \(}}\) is called a rooted directed tree, if \(}}\) has no cycles and there is a vertex \(o\in }}\) such that for each \(v\in }}\\), we have \(o\rightarrow v\). Note that this vertex o is unique (as the graph admits no cycles) and it is called the root.

We recursively define a rooted directed tree \(}}_F=(}}_F,}}_F)\), where each vertex gets a label assigned; namely, we have a map \(L:}}_F\rightarrow }}\) where

$$\begin }}:= \big \ \,:\, r\in }\cap [0,1]\big \} \cup \bigcup _}} \big \ \cup \. \end$$

For convenience, the building blocks for this recursive definition are plotted in Fig. 2. The root \(o\in }}_F\) gets the label \([0,1]\in }}\), and it belongs to level 0. The root o is connected to exactly three vertices \(u_1,u_2,u_3\) in level 1, i.e., \((o,u_i)\in }}_F\), such that \(L(u_1):=\\), \(L(u_2):=(0,1)\) and \(L(u_3):=\\). We continue recursively defining the tree \(}}_F\). Let \(u\in }}_F\) be in level \(k\ge 1\).

If \(L(u)=(r,r^*)\in }}\) for some \(r\in F_m\), set \(s:=r\oplus r^*\in F_n\) where \(n>m\) by Lemma 2.3. Then, u is connected to exactly three vertices \(v_1,v_2,v_3\) in level \(k+1\), i.e., \((u,v_i)\in }}_F\), such that \(L(v_1):=(r,s)\in }}\), \(L(v_2):=\\in }}\) and \(L(v_3):=(s,r^*)\in }}\).

If \(L(u)=\\in }}\) for some \(r\in F_m\), then there is exactly one edge \((u,v)\in }}_F\) where \(v\in }}_F\) is in level \(k+1\) and \(L(v):=\\).

Fig. 2

The building blocks of the interval-Farey tree are plotted

Definition 3.11

The previously defined rooted directed tree \(}}_F=(}}_F,}}_F)\) is called interval-Farey tree and \(L:}}_F\rightarrow }}\) is called the label map.

Note that the label map is surjective but not injective, see Fig. 3. We emphasize that the interval-Farey tree is closely related to the classical Farey tree. The Farey tree was used for the Kohmoto model in [31]. There the authors construct a renormalization group formulation using the Farey address defined by the Farey tree and not the continued fraction expansion.

Fig. 3

The first levels of the Farey tree \(}}_F\), its boundary \(\partial }}_F\) and (in the gray box) the Farey space \(\overline_F\) are sketched. Each rational point splits up into three points (except 0 and 1), confer Proposition 3.3

All vertices in level k have graph combinatorial distance k to the root o; namely, these are the vertices in the sphere of radius k around o. For convenience, we define the k-sphere by

$$\begin }}_F(k) := \}}_F \,:\, v \text k\},\qquad k\in }_0. \end$$

Lemma 3.12

Let \(}}_F= (}}_F,}}_F)\) be the interval-Farey tree.

(a)

If \(u\rightarrow v\) for \(u,v\in }}_F\), then \(L(v)\subseteq L(u)\).

(b)

We have \([0,1]=\bigsqcup _}}_F(k)} L(v)\) for all \(k \in }_0\).

Proof

Both assertion follow inductively from the definition of the interval-Farey tree. \(\square \)

Recall that we seek to isometrically identify the Farey space with the boundary of the interval-Farey tree. Here, the boundary of the tree \(}}_F\) is defined by

$$\begin \partial }}_F := \big \}_0\rightarrow }}_F \,:\, \big (\gamma (i-1),\gamma (i)\big )\in }}_F \text \gamma (0)=o \big \}. \end$$

Then, the boundary \(\partial }}_F\) is a compact space if equipped with the product topology. We turn this compact space into a metric space following the Michon correspondence [29, 32]. A map \(\kappa : }}_F \rightarrow (0,\infty )\), such that \(v\rightarrow w\) implies \(\kappa (w) < \kappa (v)\) and \(\lim _\kappa (\gamma (n))=0\) for all \(\gamma \in \partial }}\), is called a weight on \(}}_F\). Moreover, given \(\gamma , \eta \in \partial }}_F\) different, we define \(\gamma \wedge \eta :=\gamma (n)\), where \(n\in }_0\) is unique number such that \(\gamma (n)=\eta (n)\) and \(\gamma (n+1)\ne \eta (n+1)\). Note that either \(L(\gamma (n))=[0,1]\) or \(L(\gamma (n))=(r,r^*)\) must hold for some \(r\in F_m\) with \(m\in }\). This is because if \(L(\gamma (n))=\\), then \(\gamma (m)=\eta (m)\) follows for all \(m\ge n\) from the construction of the interval-Farey tree.

For the interval-Farey tree, define \(\kappa _F: }}_F \rightarrow [0,\infty )\) by

$$\begin \kappa _F(v) := 1,\qquad & v=o,\\ \frac,\qquad & L(v)=(r,r^*), r\in F_m \text r\oplus r^*\in F_ F_m,\\ \frac\kappa _F(u),\qquad & L(v)=\ \text u\in }}_F \text (u,v)\in }}_F. \end\right. } \end$$

Note that \(m\in }\) in the previous definition is chosen to be the largest integer so that r and \(r^*\) are Farey neighbors in \(F_m\) but not in \(F_\), compare with Lemma 2.3 (b).

Example 3.13

If the label of \(v\in }}_F\) is \((\frac,\frac)\), then \(\frac\oplus \frac=\frac\) and so \(\kappa _F(v)=\frac\). If \(v\in }}_F\) has label \((0,\frac)\), then \(0\oplus \frac=\frac\) and so \(\kappa _F(v)=\frac\). Also if (u, v, w) is the path labeled by \(L(u) =(0,1), L(v) = L(w)= \\}\), then we have \(\kappa _F(u)= \frac,\kappa _F(v) = \frac\) and \(\kappa _F(w) = \frac\).

Proposition 3.14

The map \(\kappa _F: }}_F \rightarrow [0,\infty )\) is a weight on the interval-Farey tree \(}}_F\) such that \(\kappa _F(v)\le \frac\) for all \(v\in }}_F(k)\) and \(k\in }\). The map \(d_}}_F}: \partial }}_F \times \partial }}_F \rightarrow [0,\infty )\) defined by

$$\begin d_}}_F}(\gamma , \eta ) := 0,\qquad & \gamma =\eta ,\\ \kappa _F(\gamma \wedge \eta ),\qquad & \gamma \ne \eta \\ \end\right. } \end$$

is an ultrametric inducing the product topology, and \((\partial }}_F,d_}}_F})\) is a compact metric space.

Proof

We prove inductively over the level \(k\in }\) that \(\kappa _F(v)\le \frac\) for all \(v\in }}_F(k)\), and if \((u,v)\in }}_F\), then \(\kappa _F(u)<\kappa _F(v)\). For the induction base \(k\in \\), observe that \(\kappa _F(o)=1\) and \(\kappa _F(u)=\frac\) for \(u\in }}_F(1)\). Now, suppose the statement is true for \(k\in }\). Let \(v\in }}_F(k+1)\). By construction of the interval-Farey tree, there is a unique \(u\in }}_F(k)\) such that \((u,v)\in }}_F\). By the induction hypothesis, we have \(\kappa (u)\le \frac\). If \(L(u)=\\), then \(L(v)=\\) and

$$\begin \kappa _F(v)=\frac\kappa _F(u)< \min \left\,\kappa _F(u)\right\} . \end$$

The case \(L(u)=(r,r^*)\) for \(r\in F_m\) and \(s:=r\oplus r^*\in F_ F_m\) still needs to be treated. Since then \(\frac=\kappa _F(u)\le \frac\), we have \(m\ge k\). Furthermore, u is connected to exactly three vertices \(v_1,v_2,v_3\in }}_F(k+1)\) such that \(L(v_1):=(r,s)\in }}\), \(L(v_2):=\\in }}\) and \(L(v_3):=(s,r^*)\in }}\). Thus, \(v\in \\). By construction, we have \(\kappa _F(v_j)\le \frac<\kappa _F(u)\) for \(j\in \\) and \(\kappa _F(v_2)=\frac\kappa _F(u)<\min \,\kappa _F(u)\}\). Thus, the induction step is finished by using \(m\ge k\).

The previous considerations imply that \(\kappa _F\) is a weight. Hence, \(d_}}_F}\) defines an ultrametric on \(\partial }}_F\) inducing the product topology, see [32, Proof of prop. 6]. Since \(\partial }}_F\) is compact in the product topology, we conclude \((\partial }}_F,d_}}_F})\) is a compact metric space.

\(\square \)

Lemma 3.15

Let \(\gamma \in \partial }}_F\). Then, there is a unique \(x(\gamma )\in \overline_F\) such that for any sequence \((\alpha _k)_}_0}\subseteq [0,1]_F\) with \(\alpha _k\in L(\gamma (k))\) for all \(k\in }_0\), we have \(\lim _F\alpha _k =x(\gamma )\).

Proof

Let \(\gamma \in \partial }}_F\) and \(\alpha _k\in L(\gamma (k))\) for all \(k\in }_0\). For \(k\ge 1\), we either have \(L(\gamma (k))=\\) or \(L(\gamma (k))=(r_k,r_k^*)\) with \(r_k\in F_\) and \(m_k\in }\). Then, \(\gamma (k)\in }}_F(k)\) implies \(\kappa \big (\gamma (k)\big )\le \frac\) by Proposition 3.14. Hence, \(m_k\ge k\) follows by definition of \(\kappa \). Moreover, Lemma 3.12 states \(L(\gamma (m))\subseteq L(\gamma (k))\) if \(m\ge k\). Thus, \(d_F(\alpha _k,\alpha _m)\le \frac\) follows for all \(k_0\in }\) and \(k,m\ge k_0\). Thus, \((\alpha _k)_}_0}\) is a Cauchy sequence with respect to \(d_F\) and so the limit \(\lim _F \alpha _k=:x(\gamma )\) exists.

Let \((\beta _k)_}_0}\) be another sequence satisfying \(\beta _k\in L(\gamma (k))\) for all \(k\in }_0\). Then, the previous considerations imply \(\alpha _k,\beta _k\in L(\gamma (k))\subseteq L(\gamma (k_0))\) and so \(d_F(\alpha _k,\beta _k)\le \frac\) for all \(k_0\in }\) and \(k\ge k_0\). Hence, \(\lim _F\beta _k=\lim _F\alpha _k=x(\gamma )\) is concluded. \(\square \)

The previous lemma allows us to define a map \(\Psi :\partial }}_F\rightarrow \overline_F, \, \gamma \mapsto x(\gamma )\).

Theorem 3.16

The map \(\Psi : \partial }}_F \rightarrow \overline_F, \, \gamma \mapsto x(\gamma ),\) is a surjective isometry. In particular, the metric spaces \((\partial }}_F, d_}}_F})\) and \((\overline_F,d_F)\) are isometric isomorphic.

Proof

We first prove that \(\Psi \) is isometric. Let \(\gamma , \eta \in \partial }}_F\) be different and \(v:=\gamma \wedge \eta =\gamma (n)\in }}_F\) for some \(n\in }_0\)