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 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】 Goals of the Course   

到達目標 【日本語】 Objectives of the Course(JPN))   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】 Objectives of the Course   

授業の内容や構成 Course Content / Plan   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 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 Kleinian groups. 4年生対象科目 


関連する科目 Related Courses   Courses on elementary set topology, real analysis and probability theory. 


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. 


不可(F)と欠席(W)の基準 Criteria for "Fail (F)" & "Absent (W)" grades   (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. 


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. 


教科書・テキスト Textbook   K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. 


課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)   To prepare carefully for seminar presentation and discussion. 


注意事項 Notice for Students   This seminar is in English. 


他学科聴講の可否 Propriety of Other department student's attendance   

他学科聴講の条件 Conditions for Other department student's attendance   

レベル Level   

キーワード Keyword   Fractals, Hausdorff dimension, Dynamical systems, Chaos, Iterated function systems, Thermodynamic formalism 


履修の際のアドバイス Advice   Enjoy maths. There are many topics. I recommend you to look through the references given below. 


授業開講形態等 Lecture format, etc.   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 Additional measures for remote class (ondemand class)   
