Ergodic theory of number theoretical transformations based on geometric analysis of structure of graphs

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03661
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Keio University
• Principal Investigator: Nakada Hitoshi 慶應義塾大学, 理工学部(矢上), 名誉教授 (40118980)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 連分数変換 / エルゴード理論 / Farey グラフ / 虚二次体

Outline of Final Research Achievements

In this project, we studied some ergodic and probabilistic properties of various types of continued fraction maps concerning to the sturucture of the Farey graph. In particular, we constructed the natural extensions of complex continued fraction maps of the nearest integer type associated with the Euclidean imaginary quadratic fields. The extension maps are defined on the set of geodesics on the upper-half space. Concerning the real number case, we showed that the range of alpha of the maximun value of the entropy of the alpha-continued fraction maps. This gives the answer to the question by C. Kraaikamp, T. Schmidt and W. Steiner.

Free Research Field

連分数のエルゴード理論

Academic Significance and Societal Importance of the Research Achievements

連分数の数論的側面と双曲空間上の測地線の関係はある程度知られていたが、本研究成果によりさらに Fareyグラフとその上の測地線の概念を利用することにより、この分野の研究に新たな展開を見せることに成功した。これにより、連分数の研究における数論、双曲幾何、グラフ理論など様々な側面の関連が見通せるようになった。今後、複素連分数の研究の進展に新たな道筋を示すことができた。