授業の目的 【日本語】
Goals of the Course(JPN) | | Title: Ergodic theory and fractal geometry for conformal dynamical systems
Dynamical systems provide mathematical models to analyse the time dependence of systems in basically all sciences and our daily life. In this seminar, we study chaotic conformal dynamical systems from the view point of ergodic theory with applications to fractal geometry.
The aim is to enable the students to perform research in the area of ergodic theory and dynamcal systems with applications to fractal geometry. We provide the necessary guidance so that the student can enter this research area and develop its own ideas. A further aim is to enable the students to improve their ability to communicate in English. |
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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)) | | | To develop a deeper understanding and knowledge of the mathematics related to ergodic theory and dynamical systems. Further, the capability to contribute to the research on topics related to this seminar. A thesis should be written at the end of the seminar course. |
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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 | | A dynamical systems is given by a self-map on a (state) space. Chaos refers to the sensitive dependence of orbits on the initial value. For chaotic systems it is difficult to predict and to analyse the individual behavior of orbit points. The same applies to large particle systems. Ergodic theory aims to analyse the average long-term behavior of such systems from the viewpoint of probability measures. We focus in particular on the so-called thermodynamic formalism originating from statistical physics, which allows us to determine meaningful measures for a given dynamical system.
Chaotic behavior of dynamical systems is often related to highly irregular sets (such as attractors of conformal iterated functions systems, limit sets of Kleinian groups, or Julia sets of complex polynomials) which are called fractal sets. In this seminar, we study the interplay of ergodic theory, dynamical systems and fractal geometry for conformal dynamical systems.
The seminar is used to discuss research problems and ideas how to deal with them. Moreover, the seminar helps to monitor progress and timeline of the research project. For the main part of this course, the student is expected to take the responsibility to organise his research activities, and to monitor and manage his time. |
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履修条件
Course Prerequisites | | | Background in analysis, measure theory, probability theory, functional analysis is required. |
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関連する科目
Related Courses | | | Lectures on probability and measure (in particular, Lebesgue integration and measures on topological spaces), and functional analysis (spectral theory for bounded linear operators). |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading is based on seminar performance (attendance, presentation, discussion) and Master's Thesis. |
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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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参考書
Reference Book | | R. Bowen. Equilibrium states and the ergodic theory of Anosov di?eomorphisms. Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975.
F. Dal’Bo. Geodesic and horocyclic trajectories: Universitext. Springer-Verlag London, Ltd.,London; EDP Sciences, Les Ulis, 2011.
K. Falconer. Mathematical foundations and applications. Third edition. John Wiley and Sons,Ltd., Chichester, 2014
A. Katok, B. Hasselblatt. A ?rst course in dynamics. With a panorama of recent developments. Cambridge University Press, New York, 2003.
D. Mauldin, M. Urbanski. Graph directed Markov systems. Geometry and dynamics of limitsets. Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003.
Y. Pesin. Dimension theory in dynamical systems. Contemporary views and applications.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997.
F. Przytycki, M. Urbanski. Conformal fractals: ergodic theory methods. London MathematicalSociety Lecture Note Series, 371. Cambridge University Press, Cambridge, 2010.
P. Walters. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. SpringerVerlag, New York-Berlin, 1982. |
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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) | | |
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注意事項
Notice for Students | | | This seminar 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 of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Fractals, Hausdorff dimension, Dynamical systems, Ergodic theory, Thermodynamic formalism |
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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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