Papers - Nakada, Hitoshi

On the dynamics of a complex continued fraction map which contains the Gauss map as its real number section

Ei H., Nakada H., Natsui R.

Advances in Mathematics 472 2025.06

ISSN  00018708

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We consider the complex continued fraction map T defined by R. Kaneiwa, I. Shiokawa, and J. Tamura (1975) associated with the Eisenstein field Q(−3). A significant aspect of their continued fraction map is that the real number part of this map T is exactly the simple continued fraction map (Gauss map). In this paper we characterize the set of strictly periodic expansions of continued fraction expansions associated to this map in terms of quadratic extensions of Q(−3) in connection with the closure of [Formula presented], where q<inf>n</inf>, n≥0, is the denominator of the nth principal convergent of the continued fraction expansion. Moreover, we show that the closure of [Formula presented] has positive Lebesgue measure on the complex plane C though it has infinitely many holes. This gives us that the construction of the natural extension of T on a subset of C²∖{diagonal} is equivalent to the geodesics over the hyperbolic space H³. Then the invariant measure for the natural extension map is given by the hyperbolic measure. Hence its projection to the complex plane is obviously the invariant measure for T.

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ON THE ERGODIC THEORY OF MAPS ASSOCIATED WITH THE NEAREST INTEGER COMPLEX CONTINUED FRACTIONS OVER IMAGINARY QUADRATIC FIELDS

Ei H., Nakada H., Natsui R.

Discrete and Continuous Dynamical Systems- Series A (Discrete and Continuous Dynamical Systems- Series A)  43 ( 11 ) 3883 - 3924 2023.11

ISSN  10780947

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We consider the nearest integer complex continued fraction map associated to the Euclidean field Q( p -d) for each d = 1; 2; 3; 7; 11. For each map, we see that there is an absolutely continuous ergodic invariant probability measure. We construct the natural extension of each map on a subset of C×C. Then the invariant measure for this extension is derived from the hyperbolic measure on H3 and the density function of the absolutely continuous invariant measure for the nearest integer map is given as its marginal. For each case, we have a fundamental domain with a fractal boundary for the lattice generated by the set of integers of the imaginary quadratic fields Q( p -d), d = 1; 2; 3; 7; 11, through the construction of the natural extension except for only one case.

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On the existence of the Legendre constants for some complex continued fraction expansions over imaginary quadratic fields

Ei H., Nakada H., Natsui R.

Journal of Number Theory (Journal of Number Theory)  238   106 - 132 2022.09

ISSN  0022314X

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We consider some types of complex continued fraction expansions and their associated maps of the Euclidean fields Q(−d), d=1,2,3,7 and 11 introduced by Hurwitz (1887) [7], Lakein (1973) [10] and Shiokawa et al. (1975) [20]. We show that there exists the Legendre constant for every type of continued fraction expansions introduced by them improving the proof by Ei et al. (2019) [4] in the case of the one by A. Hurwitz with Q(−1).

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Orbits of N-expansions with a finite set of digits

de Jonge J., Kraaikamp C., Nakada H.

Monatshefte fur Mathematik (Monatshefte fur Mathematik)  198 ( 1 ) 79 - 119 2022.05

ISSN  00269255

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For N∈ N≥ 2 and α∈ R such that 0<α≤N-1, we define Iα: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: Iα→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N≥ 7, for certain values of α, open intervals (a, b) ⊂ Iα exist such that for almost every x∈ Iα there is an n∈ N for which Tαn(x)∉(a,b) for all n≥ n. These gaps (a, b) are investigated using the square Υα:=Iα×Iα-, where the orbitsTαk(x),k=0,1,2,… of numbers x∈ Iα are represented as cobwebs. The squares Υα are the union of fundamental regions, which are related to the cylinder sets of the map Tα, according to the finitely many values of d in Tα. In this paper some clear conditions are found under which Iα is gapless. If Iα consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of Iα with regard to the fixed points of Iα under Tα.

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AN ENTROPY PROBLEM OF THE α-CONTINUED FRACTION MAPS

Nakada H.

Osaka Journal of Mathematics (Osaka Journal of Mathematics)  59 ( 2 ) 453 - 464 2022.04

ISSN  00306126

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We show that the entropy of the α-continued fraction map w.r.t the absolutely continuous invariant probability measure is strictly less than that of the nearest integer continued fraction map when 0 < α <3−√5 2. This answers a question by C. Kraaikamp, T. A. Schmidt, and W. Steiner (2012). To prove this result we make use of the notion of the geodesic continued fractions introduced by A. F. Beardon, M. Hockman, and I. Short (2012).

On the Ergodic Theory of Tanaka–Ito Type α-continued Fractions

Nakada H., Steiner W.

Tokyo Journal of Mathematics (Tokyo Journal of Mathematics)  44 ( 2 ) 451 - 465 2021.12

ISSN  03873870

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We show the ergodicity of Tanaka–Ito type α-continued fraction maps and construct their natural extensions. We also discuss the relation between entropy and the size of the natural extension domain.

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Analysis of generalized continued fraction algorithms over polynomials

Berthé V., Nakada H., Natsui R., Vallée B.

Finite Fields and their Applications (Finite Fields and their Applications)  73 2021.08

ISSN  10715797

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We study and compare natural generalizations of Euclid's algorithm for polynomials with coefficients in a finite field. This leads to gcd algorithms together with their associated continued fraction maps. The gcd algorithms act on triples of polynomials and rely on two-dimensional versions of the Brun, Jacobi–Perron and fully subtractive continued fraction maps, respectively. We first provide a unified framework for these algorithms and their associated continued fraction maps. We then analyse various costs for the gcd algorithms, including the number of iterations and two versions of the bit-complexity, corresponding to two representations of polynomials (the usual and the sparse one). We also study the associated two-dimensional continued fraction maps and prove the invariance and the ergodicity of the Haar measure. We deduce corresponding estimates for the costs of truncated trajectories under the action of these continued fraction maps, obtained thanks to their transfer operators, and we compare the two models (gcd algorithms and their associated continued fraction maps). Proving that the generating functions appear as dominant eigenvalues of the transfer operator allows indeed a fine comparison between the models.

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On the Legendre and the Lenstra constants for complex continued fractions introduced by J. Hurwitz

Nakada H.

Acta Arithmetica (Acta Arithmetica)  196 ( 3 ) 269 - 289 2020

ISSN  00651036

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

Ei H., Ito S., Nakada H., Natsui R.

Monatshefte fur Mathematik (Monatshefte fur Mathematik)  188 ( 1 ) 37 - 86 2019.01

ISSN  00269255

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© 2018, Springer-Verlag GmbH Austria, part of Springer Nature. We consider the Hurwitz complex continued fraction map associated to the Gaussian field 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 C× 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 -QnQn-1, where Qn 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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A Piecewise rotation of the circle, IPR maps and their connection with translation surfaces

Inoue K., Nakada H.

Lecture Notes in Mathematics (Lecture Notes in Mathematics)  2213   393 - 431 2018

ISSN  00758434

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© Springer International Publishing AG, part of Springer Nature 2018. For given positive integers dj, 1 ⩽ j⩽ ∀ s, ∑j=1sdj even, we construct a piecewise rotation map of the circle with ∑j=1sdj+s discontinuous points such that its critical iterates generate translation surfaces with singularity orders dj, 1 ⩽ j⩽ s, and with any Rauzy class associated to this singularity orders. The construction of the piecewise rotation map is combinatorial, on the other hand, the construction of the translation surfaces is based on the idea by Cruz and da Rocha.

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Farey maps, Diophantine approximation and Bruhat-Tits tree

Kim Dong Han, Lim SeonheeNAKADA HITOSHI, Natsui Rie

Finite Fields and their Applications 30   14 - 32 2014.11

Research paper (scientific journal), Joint Work, Accepted

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形式的べき級数に関する Farey map を Bruhat-Tits tree の中で表現した。また、Farey map に関するエルゴード理論的性質をいくつか証明した。

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On the equivalence relations of α-continued fractions

NAKADA HITOSHI, Natsui Rie

Indagationes Mathematicae 25 ( 4 ) 800 - 815 2014.06

Research paper (scientific journal), Joint Work, Accepted,  ISSN  0019-3577

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α-連分数の同値関係が α が減少するにつれて群作用の同値関係とどのように異なっていくかを示した。

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Fine costs for Euclid's algorithm on polynomials and Farey maps

Valerie Berthe, NAKADA HITOSHI, Natsui Rie, Brigit Vallee

Advances in Applied Mathematics 54   27 - 65 2014.03

Research paper (scientific journal), Joint Work, Accepted

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有限体を係数をする多項式に対するユークリッドアルゴリズムに関して、fine bit の概念を導入し、コストに関する大数の法則を証明した。

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A refined Kurzweil type theorem in positive characteristic

Kim Dong Han, NAKADA HITOSHI, Natsui Rie

Finete fields and their Applications 20   64-75 2013.03

Research paper (scientific journal), Joint Work, Accepted

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非アルキメデス数体上のディオファンタス近似に対して、Kurzweil型定理を証明した。

On the Existence of a Darling-Kac Set for the Renormalized Rauzy Map

Inoue Kae, NAKADA HITOSHI

Tokyo Journal of Mathematics 34 ( 2 ) 289-302 2011

Research paper (scientific journal), Joint Work, Accepted

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interval exchange に関する renormalized Rauzy induction に関して Darling-Kac 集合が存在することを証明した。

Metric inhomogeneous Diophantine approximation on the field of formal Laurent series

Kim Dong Han, NAKADA HITOSHI,

Acta Arithmetica 150 ( 2 ) 129-142 2011

Research paper (scientific journal), Joint Work, Accepted

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非アルキメデス数体上の非斉次ディオファンタス近似の問題について、ある測度論的性質を証明した。

On Schweiger's problems on fully subtractive algorithms

Fokkink Robbert, Kraaikamp Cor, NAKADA HITOSHI

Israel Journal of Mathematics 186   285-296 2011

Research paper (scientific journal), Joint Work, Accepted

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Fully subtractive algorithm に関する Schweiger の予想を証明した。

On the Lenstra constant associated to the Rosen continued fractions

NAKADA HITOSHI

Jouranl of the European Mathematical Society 12 ( 1 ) 55-70 2010.01

Research paper (scientific journal), Single Work, Accepted

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連分数変換において Legendre 定数と Lenstra 定数が一致することを証明した。

Generalized Brjuno functions associated to α-continued fractions

LUZZI LAURA, MARMI STEFANO, NAKADA HITOSHI, NATSUI RIE

Journal of Approximation Theory 162 ( 1 ) 24-41 2010.01

Research paper (scientific journal), Joint Work, Accepted

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α-連分数に関するブルノ関数を定義し、その性質を議論した。特に、α=0の場合に semi-Brjuno数の概念を与え、Brjuno数との関係を論じた。

Metric and arithmetic properties of mediant-Rosen maps

Kraaikamp Cor,NAKADA HITOSHI, Schmidt Thomas A.

Acta Arithmetica 137 ( 4 ) 295-324 2009

Research paper (scientific journal), Joint Work, Accepted

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Rosen型連分数に関して、中間近似を与える写像のエルゴード理論に関するいくつかの新しい結果を与えた。