Nakada, Hitoshi

写真a

Affiliation

Faculty of Science and Technology (Mita)

Position

Professor Emeritus

External Links
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Career 【 Display / hide 】

  • 1978.04
    -
    1981.03

    慶應義塾大学（工学部数理工学科） ,助手

  • 1981.04
    -
    1985.03

    慶應義塾大学（理工学部数理科学科） ,助手

  • 1985.04
    -
    1990.03

    慶應義塾大学（理工学部数理科学科） ,専任講師

  • 1988.10
    -
    1990.09

    慶應義塾大学理工学部（数理科学科） ,学習指導副主任

  • 1990.04
    -
    1999.03

    慶應義塾大学（理工学部数理科学科） ,助教授

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Academic Background 【 Display / hide 】

  • 1973.03

    Keio University, Faculty of Engineering, 管理工学科

    University, Graduated

  • 1975.03

    Keio University, Graduate School, Division of Engineering, 管理工学専攻

    Graduate School, Completed, Master's course

  • 1978.03

    Keio University, Graduate School, Division of Engineering, 管理工学専攻

    Graduate School, Withdrawal after completion of doctoral course requirements, Doctoral course

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Academic Degrees 【 Display / hide 】

  • 工学　, Keio University, 1981.03

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Research Areas 【 Display / hide 】

  • Natural Science / Basic analysis

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Papers 【 Display / hide 】

  • 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

    　View Summary

    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.

  • 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

    　View Summary

    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).

  • 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.

  • 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

    　View Summary

    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

    　View Summary

    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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Papers, etc., Registered in KOARA 【 Display / hide 】

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Presentations 【 Display / hide 】

  • エルゴード的変換の同型問題とその不変量としてのエントロピー--- old and new

    NAKADA HITOSHI

    電子回路設計と力学系 (京都大学数理解析研究所) , 

    2015.09
    , 

    Oral presentation (invited, special)

    　View Summary

    エルゴード理論における同型問題と符号化理論の発展に関する講演を行った。

  • On the dual map of Rauzy induction

    NAKADA HITOSHI

    Workshop on Measurable and Topological Dynamical Systems (National Institute of Mathematical Scioences, Daejon, Korea) , 

    2015.07
    , 

    Oral presentation (general)

    　View Summary

    Rauzy induction の dual map の構成について解説した。

  • How to find the absolutely continuous invariant measures for continued fraction maps

    NAKADA HITOSHI

    Ergodic Theory and Combinatrics (University of Agder, Kristiansand, Norway) , 

    2015.06
    , 

    Oral presentation (invited, special)

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    連分数変換の絶対連続不変測度がどのように導かれるかについて講演を行った。

  • On the natural extension of the Rauzy-Veech induction

    NAKADA HITOSHI

    Ergodic Theory and Dynamical Systems -- Torun 2014 (Nicolaus Copernicus University, Torun, Poland) , 

    2014.05
    , 

    Oral presentation (invited, special)

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    Rauzy induction の逆に対応するzippered rectanglesに対するinductionについて講演を行った。

  • On the notion of normal numbers for alpha continued fractions

    NAKADA HITOSHI

    Probability,Ergodic Theory, Dynamical Systems (Tel Aviv University, Tel Aviv, Israel) , 

    2014.04
    , 

    Oral presentation (invited, special)

    　View Summary

    α連分数に対する正規数の概念がαに独立に定まることを紹介した。

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • Ergodic theory of transformations arising from Euclidean algorithm

    2024.04
    -
    2027.03

    基盤研究(C), Principal investigator

  • グラフ構造の幾何学的表現の解析による数論的変換のエルゴード理論

    2020.04
    -
    2023.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

  • 無限大不変測度を持つエルゴード的変換の多重再帰性とエルデシ予想

    2016.04
    -
    2019.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Challenging Exploratory Research, Principal investigator

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Memberships in Academic Societies 【 Display / hide 】

  • Mathematical Society of Japan

    　
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Committee Experiences 【 Display / hide 】

  • 2015.06
    -
    2015.07

    organizer, Workshop on Measurable and Topological Dynamical Systems

  • 2014.01

    organizer, Warwick-Keio Seminar in Ergodic Theory

  • 2013.06

    organizer of the specialsession "Measurable and Topological Dynamics", Pacific Rim Mathematical Association Congress 2013

  • 2010.04
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    2011.03

    プロジェクト研究組織委員, 京都大学数理解析研究所

  • 2010.03

    2010年度年会実行委員長, 日本数学会

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