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. |
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授業の目的 【英語】
Goals of the Course | | |
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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. |
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到達目標 【英語】
Objectives of the Course | | |
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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 |
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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) |
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関連する科目
Related Courses | | | Basic courses on analysis, measure theory and functional analysis |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading is based on written reports. |
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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. |
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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. |
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教科書・テキスト
Textbook | | | Recommended books will be introduced on an individual basis. |
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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. |
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注意事項
Notice for Students | | | The course is in English. |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions for Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | |
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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) | | |
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