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On the construction of the natural extension of the Hurwitz complex continued fraction map. (English) Zbl 1414.11090

Let \[ U=\Big\{z \in \mathbb{C} : z=x+iy,-\frac{1}{2}\leq x < \frac{1}{2},\; -\frac{1}{2}\leq y < \frac{1}{2},\Big\} \] and \( [z]_2=[x]_2+i[y]_2 \) for \(z=x+iy,x,y \in \mathbb{R}\), where \([x]_2\) means the nearest integer of \(x\), that is, \([x]_2=k\) when \(x \in [k-\frac{1}{2},k+\frac{1}{2}]\) for \(k \in \mathbb{Z}\). The set of algebraic integers in \(\mathbb{Q}(\sqrt{-1})\) is denoted by \(\mathbb{Z}[i]\). Define the Hurwitz continued fraction map \(T\) by \[ T(z)=\begin{cases} \frac{1}{z}-\Big[ \frac{1}{z}\Big]_2 & \text{if } z\neq 0, \\ 0 & \text{if } z=0 \end{cases} \] for \(z\in U \). We have the Hurwitz continued fraction expansion of \(z \in U \) as \[ z=\frac{\; 1 \; |}{|\, a_1}+\frac{\; 1 \; |}{|\, a_2}+\frac{\; 1 \; |}{|\, a_3}+ \cdots \] with \( a_n=a_n(z)=\left[\frac{1}{T^{n-1}(z)}\right]_2 \; \text{for}\; n \geq 1. \) The expansion terminates in finitely many steps, i.e., \[ z=\frac{\; 1 \; |}{|\, a_1}+\frac{\; 1 \; |}{|\, a_2}+\frac{\; 1 \; |}{|\, a_3}+ \cdots+ \frac{\; 1 \; |}{|\, a_n} \] for some \(n \geq 1\) iff \(z \in \mathbb{Q}(\sqrt{-1}) \cap U\). The existence of an absolutely continuous (with respect to the Lebesgue measure) invariant probability measure for \(T\) was shown in [H. Nakada, Keio Eng. Rep. 29, 93–108 (1976; Zbl 0412.10036)] (see also [H. Nakada, Sémin. Théor. Nombres, Univ. Bordeaux I 1987–1988, Exp. No. 45, 10 p. (1988; Zbl 0714.11047)]) with its ergodicity.
In this article, the density function of the absolutely continuous invariant measure for the map associated to the Hurwitz continued fractions is characterized. A representation of its natural extension map (in the sense of an ergodic measure preserving map) on a subset of \(\mathbb{C}\times \mathbb{C}\) is constructed. 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 the three-dimensional upper-half space. Then the absolutely continues invariant measure for the Hurwitz continued fraction map is given by its marginal measure. The work contains a lot figures which illustrate the theoretical results.

MSC:

11K50 Metric theory of continued fractions
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
05B45 Combinatorial aspects of tessellation and tiling problems
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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[1] Ahlfors, L.V.: Mobius Transformations in Several Dimensions. Ordway Professorship Lectures in Mathematics. School of Mathematics, University of Minnesota, Minneapolis, MN (1981)
[2] Bandt, C.; Wang, Y., Disk-like self-affine tiles in \({\mathbb{R}}^{2}\), Discrete Comput. Geom., 26, 591-601, (2001) · Zbl 1020.52018 · doi:10.1007/s00454-001-0034-y
[3] Dani, SG; Nogueira, A., Continued fractions for complex numbers and values of binary quadratic forms, Trans. Am. Math. Soc., 366, 3553-3583, (2014) · Zbl 1392.11006 · doi:10.1090/S0002-9947-2014-06003-0
[4] Hensley, D.: Continued Fractions. World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ (2006) · Zbl 1161.11028 · doi:10.1142/5931
[5] Hurwitz, A., Über die Entwicklung complexer Grössen in Kettenbrüche, Acta Math., 11, 187-200, (1887) · JFM 20.0201.01 · doi:10.1007/BF02612324
[6] Hurwitz, J., Über die Reduction der binären quadratischen Formen mit complexen Coefficienten und Variabeln, Acta Math., 25, 231-290, (1902) · JFM 33.0221.04 · doi:10.1007/BF02419027
[7] Ito, S., Ergodic theory and Diophantine approximations: on natural extensions of algorithms, Sugaku, 39, 140-158, (1987) · Zbl 0639.10035
[8] Lakein, RB, Approximation properties of some complex continued fractions, Monatsh. Math., 77, 396-403, (1973) · Zbl 0307.10033 · doi:10.1007/BF01295317
[9] Lakein, RB, A continued fraction proof of Ford’s theorem on complex rational approximations, J. Reine Angew. Math., 272, 1-13, (1974) · Zbl 0298.10020
[10] Lakein, RB, Continued fractions and equivalent complex numbers, Proc. Am. Math. Soc., 42, 641-642, (1974) · Zbl 0278.10031
[11] Nakada, H., On the Kuzmin’s theorem for Hurwitz complex continued fractions, Keio Eng. Rep., 29, 93-108, (1976) · Zbl 0412.10036
[12] Nakada, H.: On metric Diophantine approximation of complex numbers, complex continued fractions. Seminaire de Theorie des Nombres, 1987-1988 (Talence, 1987-1988), Exp. No. 45, University of Bordeaux I, Talence (1988) · Zbl 0714.11047
[13] Nakada, H., On ergodic theory of A. Schmidt’s complex continued fractions over Gaussian field, Monatsh. Math., 105, 131-150, (1988) · Zbl 0635.10042 · doi:10.1007/BF01501166
[14] Nakada, H.: Continued fractions, geodesic flows and Ford circles. Algorithms, Fractals, and Dynamics (Okayama, Kyoto, pp. 179-191, 1995. Plenum, New York (1992) · Zbl 0868.30005
[15] Nakada, H., On the Lenstra constant associated to the Rosen continued fractions, J. Eur. Math. Soc., 12, 55-70, (2010) · Zbl 1205.11091 · doi:10.4171/JEMS/189
[16] Nakada, H.; Natsui, R., On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm, Monatsh. Math., 138, 267-288, (2003) · Zbl 1026.11066 · doi:10.1007/s00605-002-0473-4
[17] Rohlin, VA, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25, 499-530, (1961)
[18] Schmidt, A., Diophantine approximation of complex numbers, Acta Math., 134, 1-85, (1975) · Zbl 0329.10023 · doi:10.1007/BF02392098
[19] Schmidt, A., Ergodic theory of complex continued fractions, Monatsh. Math., 93, 39-62, (1982) · Zbl 0467.10038 · doi:10.1007/BF01579029
[20] Schweiger, F., Kuzmin’s theorem revisited, Ergodic Theory Dyn. Syst., 20, 557-565, (2000) · Zbl 1016.11028 · doi:10.1017/S0143385700000286
[21] Schweiger, F., A new proof of Kuzmin’s theorem, Rev. Roum. Math. Pures Appl., 56, 229-234, (2011) · Zbl 1265.11083
[22] Tanaka, S., A complex continued frcation transformation and its ergodic properties, Tokyo J. Math., 8, 191-214, (1985) · Zbl 0581.10028 · doi:10.3836/tjm/1270151579
[23] Waterman, M., Some ergodic properties of multi-dimensional f-expansions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 16, 77-103, (1970) · Zbl 0199.37102 · doi:10.1007/BF00535691
[24] Yuri, M., Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. N.S., 6, 355-383, (1995) · Zbl 0844.58048 · doi:10.1016/0019-3577(95)93202-L
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