授業の目的 【日本語】
Goals of the Course(JPN) | | Thema: Fractal geometry and dynamical systems
The term "fractal" was coined by Mandelbrot in 1975 to describe highly irregular subsets of Euclidean space. Unlike d-dimensional manifolds, which locally look like an open set in R^d, fractals often have a complicated fine structure. For instance, if one zooms into the middle-third Cantor set, then more and more gaps become visible, and it is therefore difficult to give a geometric description or to make a meaningful picture of the Cantor set. Fractals are often defined by the iteration of a fixed rule (such as the removal of the middle-third for the middle third Cantor set), which indicates the close relationship between fractals and dynamical systems (i.e., iteration of maps). The goal of this course is to study the relationship between geometric properties of fractal sets (such as, fractal dimension) and dynamical systems.
A further aim is to enable the students to improve their ability to communicate in English. |
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授業の目的 【英語】
Goals of the Course | | |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | To learn the basics of fractals, tools to analyse the geometry of fractals, and the relation between fractals and dynamical systems. |
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到達目標 【英語】
Objectives of the Course | | |
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授業の内容や構成
Course Content / Plan | | We introduce notions of fractal dimension, such as Hausdorff dimension, and study its basic properties. We consider various examples of dynamically defined fractals: interval constructions, Julia sets of rational maps, limit sets of Kleinian groups, horseshoes. We study techniques to analyze the fractal dimension such as mass distribution principles, and ideas from ergodic theory.
One direction is to look closer into self-similar fractals defined by "iterated function systems". After studying the construction principle, one can consider fractal dimensions. Questions about the "continuity" of fractal sets, their dimension, or even their measure will lead us quickly to interesting recent research topics.
(see for example, L. OLSEN: Hausdorff and packing measure functions of self-similar sets: continuity and measurability, Ergodic Theory Dynamical systems, 2008.) |
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履修条件
Course Prerequisites | | | Basics from analysis and linear algebra, set topology. Knownledge of measure theory and functional analysis is helpful. Background in complex analysis is necessary to work on topics related to Julia sets or Kleinian groups. |
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関連する科目
Related Courses | | | Courses on elementary set topology, real analysis and probability theory. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading is based on the student's seminar performance, that is attendance, presentation and discussion, as well as written reports on topics of this seminar. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | |
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参考書
Reference Book | | K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014.
K. Falconer, Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997.
Y. Pesin, V. Climenhaga, Lectures on fractal geometry and dynamical systems.
Student Mathematical Library, 52. American Mathematical Society, Providence, RI, 2009.
Y. Pesin, Dimension theory in dynamical systems. Contemporary views and applications. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997.
B. Hasselblatt, A. Katok, A first course in dynamics. With a panorama of recent developments. Cambridge University Press, New York, 2003. |
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教科書・テキスト
Textbook | | | K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | |
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注意事項
Notice for Students | | | This seminar is in English. |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions for Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Fractals, Hausdorff dimension, Dynamical systems, Chaos, Iterated function systems, Thermodynamic formalism |
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履修の際のアドバイス
Advice | | | Enjoy maths. There are many interesting topics. I recommend you to look through the references given below. |
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授業開講形態等
Lecture format, etc. | | |
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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
Additional measures for remote class (on-demand class) | | |
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