Goals of the Course(JPN)
|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.
Goals of the Course
Objectives of the Course(JPN))
|The student will learn about the concept of self-similarity of fractal sets, which often implies high irregularity of the sets. The student will understand that such sets can not be investigated using the classical concepts of length and area. New ideas will be introduced. The student will learn about the framework to define fractals sets, and to develop the mathematical tools to study properties of fractals sets, such as dimension and measure.
Objectives of the Course
Course Content / Plan
|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)
|Analysis (epsilon-delta method), point set topology
|Lectures on analysis, (point set) topology and measure theory.
Course Evaluation Method and Criteria
|Grading based on reports and quizzes.
|Recommended books will be introduced on an individual basis.
|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.
Study Load(Self-directed Learning Outside Course Hours)
|To prepare carefully for seminar presentation and discussion.
Notice for Students
|The lectures are in English.
Propriety of Other department student's attendance
Conditions of Other department student's attendance
|Please contact me for details.
|Fractals, irregular sets, self-similar set, fractal dimension
|To review the lectures carefully.
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.
Additional measures for remote class (on-demand class)