| 授業の目的 【日本語】 | | | The aim is to introduce mathematical concepts and ideas to deal with highly irregular fractal sets. We study relations between topological, metric and measure-theoretic properties of subsets of Euclidean space R^n. A further goal is to enable the students to improve their ability to communicate in English. |
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| 授業の目的 【英語】 | | |
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| 到達目標 【日本語】 | | | To get an idea of irregularity and self-similarity of fractal sets. To study the mathematical framework to define fractals sets, and to develop the mathematical tools to study properties of fractals sets, such as dimension and measure. |
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| 到達目標 【英語】 | | |
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| 授業の内容や構成 | | Preliminaries from Measure theory and Topology.
Box-counting dimension (Definition, Examples, Properties)
Hausdorff dimension (Definition, Examples, Properties)
Calculation of Hausdorff dimension using Measure theory
Iterated function systems (Banach fixed point theorem, Attractors, Bowen's formula for the Hausdorff dimension)
Interplay between Fractal Geometry and Dynamical Systems (Chaos, Repellers, Julia sets) |
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| 履修条件 | | | Analysis (epsilon-delta method), point set topology |
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| 関連する科目 | | | Lectures on analysis, (point set) topology and measure theory. |
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| 成績評価の方法と基準 | | | Grading based on reports and quizzes. |
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| 教科書・テキスト | | | Recommended books will be introduced on an individual basis. |
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| 参考書 | | M. Barnsley. Fractals everywhere. Academic Press, 1988.
K. Falconer. Mathematical foundations and applications. Third edition. John Wiley and Sons,Ltd., Chichester, 2014
Y. Pesin, V. Climenhaga. Lectures on Fractal Geometry and Dynamical Systems, AMS 2009.
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| 課外学習等（授業時間外学習の指示） | | |
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| 注意事項 | | | The lectures are in English. |
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| 他学科聴講の可否 | | |
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| 他学科聴講の条件 | | | Please contact me for details. |
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| レベル | | |
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| キーワード | | | Fractals, irregular sets, self-similar set, fractal dimension |
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| 履修の際のアドバイス | | | To review the lectures carefully. |
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| 授業開講形態等 | | |
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| 遠隔授業（オンデマンド型）で行う場合の追加措置 | | |
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