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On the construction of the natural extension of the Hurwitz complex continued fraction map

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Abstract

We consider the Hurwitz complex continued fraction map associated to the Gaussian field \({\mathbb {Q}}(i)\). We characterize the density function of the absolutely continuous invariant measure for the map associated to the Hurwitz continued fractions. For this reason, we construct a representation of its natural extension map (in the sense of an ergodic measure preserving map) on a subset of \({\mathbb {C}} \times {\mathbb {C}}\). This subset is constructed by the closure of pairs of the n-th iteration of a complex number by the Hurwitz complex continued fraction map and \(-\frac{Q_{n}}{Q_{n-1}}\), where \(Q_{n}\) is the denominator of the n-th convergent of the Hurwitz continued fractions. The absolutely continuous invariant measure for the natural extension map is induced from the invariant measure for Möbius transformations on the set of geodesics over three dimension upper-half space. Then the absolutely continues invariant measure for the Hurwitz continued fraction map is given by its marginal measure.

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Acknowledgements

The authors would like to thank referees for their helpful comments. They also would like to thank Masahiro Mizutani, who carried out the computer simulation of \(\{ \frac{Q_{n-1}}{Q_{n}}\}\) (Fig. 10) based on the second author’s idea in the early ’80th. The first author was partially supported by JSPS Grants No. 23540141. The third author was partially supported by JSPS Grants No. 16K13766 and JSPS Core-to-core program, “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”. The fourth author was partially supported by JSPS Grants No. 15K17559.

Author information

Authors and Affiliations

  1. Graduate School of Science and Technology, Hirosaki University, Bunkyo-cho, Hirosaki, Aomori, Japan

    Hiromi Ei

  2. Higashimurayama-shi, Tokyo, Japan

    Shunji Ito

  3. Department of Mathematics, Keio University, Kohoku-ku, Yokohama, Japan

    Hitoshi Nakada

  4. Department of Mathematics and Physics, Japan Women’s University, Bunkyo-ku, Tokyo, Japan

    Rie Natsui

Authors
  1. Hiromi Ei
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  2. Shunji Ito
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  3. Hitoshi Nakada
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  4. Rie Natsui
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Corresponding author

Correspondence to Hitoshi Nakada.

Additional information

Communicated by H. Bruin.

Appendices

Appendix A

We define

$$\begin{aligned} \Lambda _{1} = \{ n - m i \, | \, n \ge 1, \, m \ge 1 \} {\setminus } \left( \Lambda _{2} \cup \{ 1 - i \} \right) \end{aligned}$$

and

$$\begin{aligned} \Lambda _{2} = \{ 1-2i, \, 2 - 2i, \, 2 - i \} . \end{aligned}$$

Then the partition \(\{ V_{\cdot , \cdot } \}\) can be decomposed by the fundamental partition of U as follows:

$$\begin{aligned} V_{1,1}= & {} \bigcup \nolimits _{a \in {{\mathbb {Z}}}, a \ge 2} \langle a \rangle \\ V_{2, 1}= & {} \bigcup \nolimits _{a \in \Lambda _1} \langle a \rangle \quad \cup \quad \bigcup \nolimits _{a \in \Lambda _2} \left( \langle a \rangle \cap V_{2,1} \right) \\ V_{3, 1}= & {} \bigcup \nolimits _{a \in \Lambda _2} \left( \langle a \rangle \cap V_{3,1} \right) \cup \langle 1-i \rangle \\ V_{1,2}= & {} \bigcup \nolimits _{a \in {{\mathbb {Z}}}, a \ge 2} \langle -ai \rangle \\ V_{2, 2}= & {} \bigcup \nolimits _{a \in \Lambda _1} \langle -ai \rangle \quad \cup \quad \bigcup \nolimits _{a \in \Lambda _2} \left( \langle -ai \rangle \cap V_{2,2} \right) \\ V_{3, 2}= & {} \bigcup \nolimits _{a \in \Lambda _2} \left( \langle -ai \rangle \cap V_{3,2} \right) \cup \langle -1-i \rangle \\ V_{1,3}= & {} \bigcup \nolimits _{a \in {{\mathbb {Z}}}, a \ge 2} \langle -a \rangle \\ V_{2, 3}= & {} \bigcup \nolimits _{a \in \Lambda _1} \langle -a \rangle \quad \cup \quad \bigcup \nolimits _{a \in \Lambda _2} \left( \langle -a \rangle \cap V_{2,3} \right) \\ V_{3, 3}= & {} \bigcup \nolimits _{a \in \Lambda _2} \left( \langle -a \rangle \cap V_{3,3} \right) \cup \langle -1+i \rangle \\ V_{1,4}= & {} \bigcup \nolimits _{a \in {{\mathbb {Z}}}, a \ge 2} \langle ai \rangle \\ V_{2, 4}= & {} \bigcup \nolimits _{a \in \Lambda _1} \langle ai \rangle \quad \cup \quad \bigcup \nolimits _{a \in \Lambda _2} \left( \langle ai \rangle \cap V_{2,4} \right) \\ V_{3, 4}= & {} \bigcup \nolimits _{a \in \Lambda _2} \left( \langle ai \rangle \cap V_{3,4} \right) \cup \langle 1+i \rangle \end{aligned}$$

Next we decompose \(\{ U_{\cdot , \cdot } \}\) by \(V_{\cdot , \cdot }\):

$$\begin{aligned} U_{1,1}= & {} V_{1,1} \cup V_{1,2} \cup V_{1,4} \cup V_{2,1} \cup V_{2,4} \cup V_{3,1} \cup V_{3,4} \\ U_{1,2}= & {} V_{1,1} \cup V_{1,2} \cup V_{1,3} \cup V_{2,1} \cup V_{2,2} \cup V_{3,1} \cup V_{3,2} \\ U_{1,3}= & {} V_{1,2} \cup V_{1,3} \cup V_{1,4} \cup V_{2,2} \cup V_{2,3} \cup V_{3,2} \cup V_{3,3} \\ U_{1,4}= & {} V_{1,1} \cup V_{1,3} \cup V_{1,4} \cup V_{2,3} \cup V_{2,4} \cup V_{3,3} \cup V_{3,4} \\ U_{2,1}= & {} V_{1,1} \cup V_{1,2} \cup V_{2,1} \cup V_{3,1} \\ U_{2,2}= & {} V_{1,2} \cup V_{1,3} \cup V_{2,2} \cup V_{3,2} \\ U_{2,3}= & {} V_{1,3} \cup V_{1,4} \cup V_{2,3} \cup V_{3,3} \\ U_{2,4}= & {} V_{1,1} \cup V_{1,4} \cup V_{2,4} \cup V_{3,4} \\ U_{3,1}= & {} U{\setminus }V_{3,3} \\ U_{3,2}= & {} U{\setminus }V_{3,4} \\ U_{3,3}= & {} U{\setminus }V_{3,1} \\ U_{3,4}= & {} U{\setminus }V_{3,2} \end{aligned}$$

Finally we write the image of \(V_{\cdot , \cdot } \times V_{\cdot , \cdot }^{*}\) by \({\hat{T}}\).

$$\begin{aligned} {\hat{T}} \left( V_{1,1} \times V_{1,1}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in {{\mathbb {Z}}}, \, a \ge 3}}\left( U \times (X_{1,1}- a)\right) \\&\cup \left( V_{1,1} \times (X_{1,1} - 2) \right) \cup \left( V_{1,2} \times (X_{1,1} - 2) \right) \\&\cup \left( V_{1,4} \times (X_{1,1} - 2) \right) \cup \left( V_{2,1} \times (X_{1,1} - 2) \right) \cup \left( V_{2,4} \times (X_{1,1} - 2) \right) \\ {}{} & {} \cup \left( V_{3,1} \times (X_{1,1} - 2) \right) \cup \left( V_{3,4} \times (X_{1,1} - 2) \right) \\ {\hat{T}} \left( V_{2,1} \times V_{2,1}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in \Lambda _{1}}}\left( U \times (X_{2,1}- a)\right) \, \cup \left( V_{1,1} \times (X_{2,1} -(1- 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,1} -(1- 2i)) \right) \cup \left( V_{1,4} \times (X_{2,1} -(1- 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,1} -(1- 2i)) \right) \cup \left( V_{2,4} \times (X_{2,1} -(1- 2i)) \right) \\&\cup \left( V_{3,3} \times (X_{2,1} -(1- 2i)) \right) \cup \left( V_{3,4} \times (X_{2,1} -(1- 2i)) \right) \\&\cup \left( V_{1,1} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{1,2} \times (X_{2,1} -(2- 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{1,4} \times (X_{2,1} -(2- 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{2,2} \times (X_{2,1} -(2- 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{2,4} \times (X_{2,1} -(2- 2i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{3,3} \times (X_{2,1} -(2- 2i)) \right) \\&\cup \left( V_{3,4} \times (X_{2,1} -(2- 2i)) \right) \cup \left( V_{1,1} \times (X_{2,1} -(2- i)) \right) \\&\cup \left( V_{1,2} \times (X_{2,1} -(2- i)) \right) \cup \left( V_{1,4} \times (X_{2,1} -(2- i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,1} -(2- i)) \right) \cup \left( V_{2,4} \times (X_{2,1} -(2- i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,1} -(2- i)) \right) \cup \left( V_{3,4} \times (X_{2,1} -(2- i)) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{3,1} \times V_{3,1}^{*} \right)= & {} \left( V_{1,2} \times (X_{3,1} -(1- 2i)) \right) \cup \left( V_{2,1} \times (X_{3,1} -(1- 2i)) \right) \\&\cup \left( V_{2,2} \times (X_{3,1} -(1- 2i)) \right) \cup \left( V_{3,1} \times (X_{3,1} -(1- 2i)) \right) \\&\cup \left( V_{3,2} \times (X_{3,1} -(2- 2i)) \right) \cup \left( V_{1,3} \times (X_{3,1} -(2- i)) \right) \\&\cup \left( V_{2,2} \times (X_{3,1} -(2- i)) \right) \cup \left( V_{2,3} \times (X_{3,1} -(2- i)) \right) \\&\cup \left( V_{3,3} \times (X_{3,1} -(2- i)) \right) \cup \left( V_{1,1} \times (X_{3,1} -(1- i)) \right) \\&\cup \left( V_{1,4} \times (X_{3,1} -(1- i)) \right) \cup \left( V_{2,4} \times (X_{3,1} -(1- i)) \right) \\&\cup \left( V_{3,4} \times (X_{3,1} -(1- i)) \right) \\ {\hat{T}} \left( V_{1,2} \times V_{1,2}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in {{\mathbb {Z}}}, \, a \ge 3}}\left( U \times (X_{1,2}+ ai)\right) \\&\cup \left( V_{1,1} \times (X_{1,2} + 2i) \right) \cup \left( V_{1,3} \times (X_{1,2} + 2i) \right) \\&\cup \left( V_{1,4} \times (X_{1,2} + 2i) \right) \cup \left( V_{2,3} \times (X_{1,2} + 2i) \right) \\&\cup \left( V_{2,4} \times (X_{1,2} + 2i) \right) \\ {}{} & {} \cup \left( V_{3,3} \times (X_{1,2} + 2i) \right) \cup \left( V_{3,4} \times (X_{1,2} + 2i) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{2,2} \times V_{2,2}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in \Lambda _{1}}}\left( U \times (X_{2,2} + ai)\right) \, \cup \left( V_{1,2} \times (X_{2,2} -(-2- i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,2} -(-2 - i)) \right) \cup \left( V_{1,4} \times (X_{2,2} -(-2 - i)) \right) \\&\cup \left( V_{2,2} \times (X_{2,2} -(-2 - i)) \right) \cup \left( V_{2,3} \times (X_{2,2} -(-2 - i)) \right) \\&\cup \left( V_{3,2} \times (X_{2,2} -(-2 - i)) \right) \cup \left( V_{3,3} \times (X_{2,2} -(-2 - i)) \right) \\&\cup \left( V_{1,1} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{1,2} \times (X_{2,2} -(-2- 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{1,4} \times (X_{2,2} -(-2- 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{2,2} \times (X_{2,2} -(-2- 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{2,4} \times (X_{2,2} -(-2- 2i)) \right) \\&\cup \left( V_{3,2} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{3,3} \times (X_{2,2} -(-2- 2i)) \right) \\&\cup \left( V_{3,4} \times (X_{2,2} -(-2- 2i)) \right) \cup \left( V_{1,1} \times (X_{2,2} -(-1 - 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,2} -(-1 - 2i)) \right) \cup \left( V_{1,4} \times (X_{2,2} -(-1 - 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,2} -(-1 - 2i)) \right) \cup \left( V_{2,4} \times (X_{2,2} -(-1 - 2i)) \right) \\&\cup \left( V_{3,3} \times (X_{2,2} -(-1 - 2i)) \right) \cup \left( V_{3,4} \times (X_{2,2} -(-1 - 2i)) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{3,2} \times V_{3,2}^{*} \right)= & {} \left( V_{1,1} \times (X_{3,2} -(-2- i)) \right) \cup \left( V_{2,1} \times (X_{3,2} -(-2- i)) \right) \\&\cup \left( V_{2,4} \times (X_{3,2} -(-2- i)) \right) \cup \left( V_{3,4} \times (X_{3,2} -(-2- i)) \right) \\&\cup \left( V_{3,1} \times (X_{3,2} -(-2- 2i)) \right) \cup \left( V_{1,2} \times (X_{3,2} -(-1- 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{3,2} -(-1- 2i)) \right) \cup \left( V_{2,2} \times (X_{3,2} -(-1- 2i)) \right) \\&\cup \left( V_{3,2} \times (X_{3,2} -(-1- 2i)) \right) \cup \left( V_{1,3} \times (X_{3,2} -(-1- i)) \right) \\&\cup \left( V_{1,4} \times (X_{3,2} -(-1- i)) \right) \cup \left( V_{2,3} \times (X_{3,2} -(-1- i)) \right) \\&\cup \left( V_{3,3} \times (X_{3,2} -(-1- i)) \right) \\ {\hat{T}} \left( V_{1,3} \times V_{1,3}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in {{\mathbb {Z}}}, \, a \ge 3}}\left( U \times (X_{1,3}+ a)\right) \, \cup \left( V_{1,2} \times (X_{1,3} + 2) \right) \\&\cup \left( V_{1,3} \times (X_{1,3} + 2) \right) \cup \left( V_{1,4} \times (X_{1,3} + 2) \right) \cup \left( V_{2,2} \times (X_{1,3} + 2) \right) \\&\cup \left( V_{2,3} \times (X_{1,3} + 2) \right) \cup \left( V_{3,2} \times (X_{1,3} + 2) \right) \cup \left( V_{3,3} \times (X_{1,3} + 2) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{2,3} \times V_{2,3}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in \Lambda _{1}}}\left( U \times (X_{2,3}-(-a))\right) \, \cup \left( V_{1,1} \times (X_{2,3} -(-1 + 2i)) \right) \\&\cup \left( V_{1,2} \times (X_{2,3} -(-1 + 2i)) \right) \cup \left( V_{1,3} \times (X_{2,3} - (-1 + 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,3} - (-1 + 2i)) \right) \cup \left( V_{2,2} \times (X_{2,3} - (-1 + 2i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,3} - (-1 + 2i)) \right) \cup \left( V_{3,2} \times (X_{2,3} - (-1 + 2i)) \right) \\&\cup \left( V_{1,1} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{1,2} \times (X_{2,3} - (-2 + 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{1,4} \times (X_{2,3} - (-2 + 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{2,2} \times (X_{2,3} - (-2 + 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{2,4} \times (X_{2,3} - (-2 + 2i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{3,2} \times (X_{2,3} - (-2 + 2i)) \right) \\&\cup \left( V_{3,3} \times (X_{2,3} - (-2 + 2i)) \right) \cup \left( V_{1,2} \times (X_{2,3} - (-2 + i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,3} - (-2 + i)) \right) \cup \left( V_{1,4} \times (X_{2,3} -(-2 + i)) \right) \\&\cup \left( V_{2,2} \times (X_{2,3} - (-2 + i)) \right) \cup \left( V_{2,3} \times (X_{2,3} - (-2 + i)) \right) \\&\cup \left( V_{3,2} \times (X_{2,3} - (-2 + i)) \right) \cup \left( V_{3,3} \times (X_{2,3} - (-2 + i)) \right) \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{3,3} \times V_{3,3}^{*} \right)= & {} \left( V_{1,4} \times (X_{3,3} -(-1+ 2i)) \right) \cup \left( V_{2,3} \times (X_{3,3} -(-1+ 2i)) \right) \\&\cup \left( V_{2,4} \times (X_{3,3} -(-1+ 2i)) \right) \cup \left( V_{3,3} \times (X_{3,3} -(-1+ 2i)) \right) \\&\cup \left( V_{3,4} \times (X_{3,3} -(-2+ 2i)) \right) \cup \left( V_{1,1} \times (X_{3,3} -(-2 + i)) \right) \\&\cup \left( V_{2,1} \times (X_{3,3} -(-2+ i)) \right) \cup \left( V_{2,4} \times (X_{3,3} -(-2+ i)) \right) \\&\cup \left( V_{3,1} \times (X_{3,3} -(-2 + i)) \right) \cup \left( V_{1,2} \times (X_{3,3} -(-1+ i)) \right) \\&\cup \left( V_{1,3} \times (X_{3,3} -(-1+ i)) \right) \cup \left( V_{2,2} \times (X_{3,3} -(-1+ i)) \right) \\&\cup \left( V_{3,2} \times (X_{3,3} -(-1+ i)) \right) \\ {\hat{T}} \left( V_{1,4} \times V_{1,4}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in {{\mathbb {Z}}}, \, a \ge 3}}\left( U \times (X_{1,4} - ai)\right) \, \cup \left( V_{1,1} \times (X_{1,4} - 2i) \right) \\&\cup \left( V_{1,2} \times (X_{1,4} - 2i) \right) \\&\cup \left( V_{1,3} \times (X_{1,4} - 2i) \right) \cup \left( V_{2,1} \times (X_{1,4} - 2i) \right) \cup \left( V_{2,2} \times (X_{1,4} - 2i) \right) \\ {}{} & {} \cup \left( V_{3,1} \times (X_{1,4} - 2i) \right) \cup \left( V_{3,2} \times (X_{1,4} - 2i) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{2,4} \times V_{2,4}^{*} \right)= & {} {\mathop {\bigcup }\nolimits _{a \in \Lambda _{1}}}\left( U \times (X_{2,4} - ai)\right) \, \cup \left( V_{1,1} \times (X_{2,4} -(2 + i)) \right) \\&\cup \left( V_{1,2} \times (X_{2,4} -(2 + i)) \right) \cup \left( V_{1,4} \times (X_{2,4} -(2 + i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,4} -(2 + i)) \right) \cup \left( V_{2,4} \times (X_{2,4} -(2 + i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,4} -(2 + i)) \right) \cup \left( V_{3,4} \times (X_{2,4} -(2 + i)) \right) \\&\cup \left( V_{1,1} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{1,2} \times (X_{2,4} -(2 + 2i)) \right) \\&\cup \left( V_{1,3} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{1,4} \times (X_{2,4} -(2 + 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{2,2} \times (X_{2,4} -(2 + 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{2,4} \times (X_{2,4} -(2 + 2i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{3,2} \times (X_{2,4} -(2 + 2i)) \right) \\&\cup \left( V_{3,4} \times (X_{2,4} -(2 + 2i)) \right) \cup \left( V_{1,1} \times (X_{2,4} -(1 + 2i)) \right) \\&\cup \left( V_{1,2} \times (X_{2,4} -(1 + 2i)) \right) \cup \left( V_{1,3} \times (X_{2,4} -(1 + 2i)) \right) \\&\cup \left( V_{2,1} \times (X_{2,4} -(1 + 2i)) \right) \cup \left( V_{2,2} \times (X_{2,4} - (1 + 2i)) \right) \\&\cup \left( V_{3,1} \times (X_{2,4} - (1 + 2i)) \right) \cup \left( V_{3,2} \times (X_{2,4} - (1 + 2i)) \right) \\ \end{aligned}$$
$$\begin{aligned} {\hat{T}} \left( V_{3,4} \times V_{3,4}^{*} \right)= & {} \left( V_{1,3} \times (X_{3,4} -(2+ i)) \right) \cup \left( V_{2,2} \times (X_{3,4} -(2 + i)) \right) \\&\cup \left( V_{2,3} \times (X_{3,4} -(2+ i)) \right) \cup \left( V_{3,2} \times (X_{3,4} -(2+ i)) \right) \\&\cup \left( V_{3,3} \times (X_{3,4} -(2+ 2i)) \right) \cup \left( V_{1,4} \times (X_{3,4} -(1+ 2i)) \right) \\&\cup \left( V_{2,3} \times (X_{3,4} -(1+ 2i)) \right) \cup \left( V_{2,4} \times (X_{3,4} -(1+ 2i)) \right) \\&\cup \left( V_{3,4} \times (X_{3,4} -(1 + 2i)) \right) \cup \left( V_{1,1} \times (X_{3,4} -(1+ i)) \right) \\&\cup \left( V_{1,2} \times (X_{3,4} -(1+ i)) \right) \cup \left( V_{2,1} \times (X_{3,4} -(1+ i)) \right) \\&\cup \left( V_{3,1} \times (X_{3,4} -(1+ i)) \right) \end{aligned}$$

Appendix B

See Figs. 13, 14 and 15.

Fig. 13
figure 13

\(X_{1,\ell }\) and \(V^*_{1,\ell }\) for \(\ell =1,2,3,4\)

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Fig. 14
figure 14

\(X_{2,\ell }\) and \(V^*_{2,\ell }\) for \(\ell =1,2,3,4\)

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Fig. 15
figure 15

\(X_{3,\ell }\) and \(V^*_{3,\ell }\) for \(\ell =1,2,3,4\)

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Ei, H., Ito, S., Nakada, H. et al. On the construction of the natural extension of the Hurwitz complex continued fraction map. Monatsh Math 188, 37–86 (2019). https://doi.org/10.1007/s00605-018-1229-0

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