ファイル更新日：2022年11月25日
教育・就職
■卒業研究シラバス■
●J. イェーリッシュ
学部・大学院区分

理学部 
時間割コード


科目区分

専門科目 
科目名【日本語】

数学研究 
科目名【英語】

Undergraduate Seminar 
コースナンバリングコード


担当教員【日本語】

JAERISCH Johannes Klaus B 
担当教員【英語】

JAERISCH Johannes Klaus Bernhard 
単位数

6 
開講期・開講時間帯

春 水曜日 3時限 春 水曜日 4時限 
授業形態

セミナー 
学科・専攻

数理学科 
必修・選択

選択必修 
授業の目的【日本語】

Thema: Fractal geometry and dynamical systems The term "fractal" was coined by Mandelbrot in 1975 to describe highly irregular subsets of Euclidean space. Unlike ddimensional 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 middlethird 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 middlethird 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 measure) and dynamical systems (entropy, Lyapunov exponents). A further aim is to enable the students to improve their ability to communicate in English. 
授業の目的【英語】

Thema: Fractal geometry and dynamical systems The term "fractal" was coined by Mandelbrot in 1975 to describe highly irregular subsets of Euclidean space. Unlike ddimensional 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 middlethird 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 middlethird 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 measure) and dynamical systems (entropy, Lyapunov exponents). A further aim is to enable the students to improve their ability to communicate in English. 
到達目標【日本語】

The student will learn fundamental techniques from fractal geometry to analyse the geometry of fractals. Moreover, the student will acquire knowledge about the connection between the geometry of fractals and complexity of dynamical systems. 
到達目標【英語】

The student will learn fundamental techniques from fractal geometry to analyse the geometry of fractals. Moreover, the student will acquire knowledge about the connection between the geometry of fractals and complexity of dynamical systems. 
授業の内容や構成

We study notions from geometric measure theory such as fractal dimension (e.g., Hausdorff dimension or boxcounting dimension) and study its basic properties. We consider dynamical systems, that is, the iteration of selfmaps on spaces or group actions on spaces. The central topic is then to study dynamically defined fractal sets. Here, we focus mainly on selfsimilar fractals defined by Iterated Function Systems (see e.g., Falconer's textbook for an introduction). The idea of Iterated function system appears in many other dynamical systems such as iteration of rational maps or Fuchsian groups. We learn how fractal dimension is related to properties of the functions defining the Iterated Function System. We study mass distribution principles as an important technique. After getting familar with the basics, one project is to investigate continuity properties of fractal constructions and their properties. For instance, to study the dependence of Hausdorff dimension (or Hausdorff measure) on the iterated function system. This will lead us quickly to interesting recent research topics (see for example, L. OLSEN: Hausdorff and packing measure functions of selfsimilar sets: continuity and measurability, Ergodic Theory Dynamical systems, 2008.) 
履修条件

Basics from analysis and linear algebra, set topology. Knownledge of measure theory, functional analysis and complex analysis will be helpful. 
関連する科目

Courses on elementary set topology, real analysis and probability theory. 
成績評価の方法と基準

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. 
不可(F)と欠席(W)の基準

(W) is for students who are absent excessively, or who do not complete the required work for evaluation. (F) is for students who fail to achieve the minimally acceptable performance. 
教科書・テキスト

K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. 
参考書

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. 
課外学習等 (授業時間外学習の指示)

To prepare carefully for seminar presentation and discussion. 
注意事項

This seminar is in English. 
質問への対応方法

By email. 
他学科聴講の可否

不可 
他学科聴講の条件

不可 
レベル

2 
キーワード

Fractals, Hausdorff dimension, Dynamical systems, Chaos, Iterated function systems, Thermodynamic formalism 
履修の際のアドバイス

Enjoy maths. There are many topics. I recommend you to look through the references given below. 
授業開講形態等

We meet in classroom if the pandemic situation is acceptable. Otherwise, we meet online using zoom/skype etc., and communicate by email. 
遠隔授業(オンデマンド型)で行う場合の追加措置


