学部・大学院区分
Undergraduate / Graduate
多・博前
時間割コード
Registration Code
3211019
科目区分
Course Category
A類Ⅰ(基礎科目)
Category A-1
科目名 【日本語】
Course Title
解析学概論V
科目名 【英語】
Course Title
Introduction to Analysis V
コースナンバリングコード
Course Numbering Code
担当教員 【日本語】
Instructor
JAERISCH Johannes Klaus B ○
担当教員 【英語】
Instructor
JAERISCH Johannes Klaus Bernhard ○
単位数
Credits
2
開講期・開講時間帯
Term / Day / Period
秋 木曜日 2時限
Fall Thu 2
授業形態
Course style

学科・専攻
Department / Program
多元数理科学研究科
必修・選択
Required / Selected
選択


授業の目的 【日本語】
Goals of the Course(JPN)
Title: Ergodic theory

The goal of this course is to provide an introduction to ergodic theory. Ergodic theory deals with measure-preserving transformations of probability spaces. A central aim is to investigate the long-term behavior of typical orbits with respect to an invariant probability measure. If the system is ergodic, then the time averages of many observables are given by the space average with respect to the invariant measure. In physics this is known as the ergodic hypothesis.
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 about the basics from ergodic theory of measure-preserving transformations (recurrence, ergodicity, mixing, etc.) To introduce the spectral viewpoint on mixing properties of measure-preserving dynamical systems. For topological dynamical systems, we investigate the structure of the space of all Borel probability measures which are preserved by the dynamics.
到達目標 【英語】
Objectives of the Course
授業の内容や構成
Course Content / Plan
1) Measure-theoretic preliminaries
2) Measure-preserving transformations
3) Ergodic theorems
4) Spectral theory
5) Invariant measures for continous transformations
履修条件
Course Prerequisites
Real analysis (Measure and integration theory in general measure spaces), Functional analysis (Spectrum of bounded operators on Hilbert spaces, Riesz representation of measures on topological spaces)
関連する科目
Related Courses
Basic courses on analysis, measure theory and functional analysis.
成績評価の方法と基準
Course Evaluation Method and Criteria
Grading is based on written reports.
教科書・テキスト
Textbook
Recommended books will be introduced on an individual basis.
参考書
Reference Book
Walters, Peter. Introduction to Ergodic theory. Graduate texts in mathematics, 1982.
Petersen, Karl. Ergodic theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1983.
Brin, Michael and Stuck, Garrett. Introduction to Dynamical Systems. Cambridge, 2002.
Royden, H. and Fitzpatrick, P. Real Analysis (4th Edition), Pearson Modern Classics for Advanced Mathematics Series, 2010.
Bauer, H., Measure and integration theory, de Gruyter, 2001.
課外学習等(授業時間外学習の指示)
Study Load(Self-directed Learning Outside Course Hours)
To carefully review the lectures, and to work on the assignments independently.
注意事項
Notice for Students
The course 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 of Other department student's attendance
-
レベル
Level
2
キーワード
Keyword
-
履修の際のアドバイス
Advice
-
授業開講形態等
Lecture format, etc.
Face-to-face
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)
-