住所: 〒464-8602 愛知県名古屋市千種区不老町

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Undergraduate / Graduate 理学部 専門科目 数学研究 Undergraduate Seminar JAERISCH Johannes Klaus B JAERISCH Johannes Klaus Bernhard 6 春 水曜日 3時限春 水曜日 4時限 セミナー 数理学科 選択必修
授業の目的【日本語】Goals of the Course(JPN) Thema: Fractal geometry and dynamical systemsThe 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 measure) and dynamical systems (entropy, Lyapunov exponents).A further aim is to enable the students to improve their ability to communicate in English. Thema: Fractal geometry and dynamical systemsThe 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 measure) and dynamical systems (entropy, Lyapunov exponents).A further aim is to enable the students to improve their ability to communicate in English. 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. 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. We study notions from geometric measure theory such as fractal dimension (e.g., Hausdorff dimension or box-counting dimension) and study its basic properties. We consider dynamical systems, that is, the iteration of self-maps on spaces or group actions on spaces. The central topic is then to study dynamically defined fractal sets. Here, we focus mainly on self-similar 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 self-similar sets: continuity and measurability, Ergodic Theory Dynamical systems, 2008.) Basics from analysis and linear algebra, set topology. Knownledge of measure theory, functional analysis and complex analysis will be helpful. Courses on elementary set topology, real analysis and probability theory. 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. (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. K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. 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. To prepare carefully for seminar presentation and discussion. This seminar is in English. By email. 不可 不可 2 Fractals, Hausdorff dimension, Dynamical systems, Chaos, Iterated function systems, Thermodynamic formalism Enjoy maths. There are many topics. I recommend you to look through the references given below. We meet in classroom if the pandemic situation is acceptable. Otherwise, we meet online using zoom/skype etc., and communicate by email.