| 研究課題/領域番号 |
21K03269
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分12010:基礎解析学関連
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| 研究機関 | 名古屋大学 |
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研究代表者 |
イェーリッシュ ヨハネス 名古屋大学, 多元数理科学研究科, 准教授 (90741869)
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| 研究期間 (年度) |
2021-04-01 – 2024-03-31
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| 研究課題ステータス |
交付 (2022年度)
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| 配分額 *注記 |
4,030千円 (直接経費: 3,100千円、間接経費: 930千円)
2023年度: 1,300千円 (直接経費: 1,000千円、間接経費: 300千円)
2022年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
2021年度: 1,300千円 (直接経費: 1,000千円、間接経費: 300千円)
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| キーワード | Fuchsian groups / Hausdorff dimension / Large deviations / Ergodic theory / Fractal geometry / Multifractal analysis / Non-uniformly hyperbolic / Ergodic Theory / Fractal Geometry |
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| 研究開始時の研究の概要 |
We develop the ergodic theory and its applications to fractal geometry for conformal dynamical systems which are non-uniformly hyperbolic or whose state space is not compact. The main examples are Kleinian groups, semigroups of rational maps on the Riemann sphere, and Markov interval maps.
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| 研究実績の概要 |
We studied the interplay of dynamical, geometric and stochastic properties of conformal dynamical systems from the viewpoint of ergodic theory. We focused mainly on the dynamics of Fuchsian groups admitting a Dirichlet fundamental domain with even corners. For the associated hyperbolic surface, we obtained new results on the Hausdorff dimension spectrum of homological growth rates associated with oriented geodesics. In particular, we are able to express the dimension in terms of a generalized Poincare exponent associated with a given inverse temperature.
As a stochastic counterpart we studied the probability to observe a certain homological growth rate. It turns out that the growth rate satisfies large deviations with a rate function closely related to the Hausdorff dimension spectrum. We have combined distortion arguments based on some geometric properties of the geodesic flow with the symbolic thermodynamic formalism for countable Markov shits. This is a joint work with Hiroki Takahasi (Keio U.).
We also had some progress on transient dynamics on the real line with a reflective boundary. In particular, we obtained formulas for the topological pressure function which indicate a possible phase transition which arises from the reflective boundary.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Although some travel plans had to be cancelled because of covid-19 restrictions in summer, we had overall substantial progress on topics of this project.
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| 今後の研究の推進方策 |
The large deviation results for Fuchsian groups shall be completed within a few weeks. The project on transient dynamics on the real line with a reflective boundary shall be completed when visiting University Bremen in summer 2023. A first preprint on thermodynamic formalism for infinitely generated Schottky groups shall be completed in autumn 2023.
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