| 授業の目的 【日本語】 | | Title: Ergodic theory and Thermodynamic formalism
The goal of this course is to provide an introduction to ergodic theory and thermodynamic formalism. Ergodic theory deals with measure-preserving transformations on probability spaces. The 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 observables are given by the space average with respect to the invariant measure. In physics this is known as the ergodic hypothesis.
We introduce entropy which is an important quantity to classify measure-preserving transformations. Moreover, we address the question how to determine meaningful measures for a given topological dynamical system. We explain central notions from thermodynamic formalism, such as topological pressure and equilibrium states.
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| 授業の目的 【英語】 | | |
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| 到達目標 【日本語】 | | | To study the basics from ergodic theory of measure-preserving transformations (recurrence, ergodic theorems, mixing, spectral theory). To give an introduction to entropy theory for measure preserving transformations. To determine meaningful invariant measures for topological dynamical sytems using central notions from thermodynamic formalism. |
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| 到達目標 【英語】 | | |
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| 授業の内容や構成 | | 1) Measure-preserving transformations
2) Ergodic theorems and Spectral theory
3) Entropy
4) Invariant measures for continous transformations
5) Topological pressure and equilibium states |
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| 履修条件 | | | Measure and integration theory, Functional analysis |
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| 関連する科目 | | | Lectures on measure theory, functional analysis and dynamical systems. |
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| 成績評価の方法と基準 | | | Grading is based on written reports. |
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| 教科書・テキスト | | | Recommended books will be introduced on an individual basis. |
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| 参考書 | | 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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| 課外学習等（授業時間外学習の指示） | | |
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| 注意事項 | | | The lectures are in English. |
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| 他学科聴講の可否 | | |
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| 他学科聴講の条件 | | | All students with a strong background in mathematics are welcome. |
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| レベル | | |
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| キーワード | | | Measure preserving transformations, ergodic theorems, entropy, topological pressure, equilibrium states |
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| 履修の際のアドバイス | | | To review the lectures carefully. |
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| 授業開講形態等 | | |
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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 | | |
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