学部・大学院区分
Undergraduate / Graduate
多・博前
時間割コード
Registration Code
3213183
科目区分
Course Category
C類(実習)
Category C
科目名 【日本語】
Course Title
確率論実習3
科目名 【英語】
Course Title
Practical Class on Probability Theory 3
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
1
開講期・開講時間帯
Term / Day / Period
春集中 その他 その他
Intensive(Spring) Other Other
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


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 and 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 students can enter this research area and develop their own ideas. A further aim is to enable the students to improve their ability to communicate in English.
授業の目的 【英語】
Goals of the Course
到達目標 【日本語】
Objectives of the Course(JPN))
The students will develop deeper understanding and knowledge of mathematics especially related to ergodic theory and fractal geometry for conformal dynamical systems. Through study and research in this area, the students will develop their learning and problem-solving skills. Further, the students will acquire the capability to contribute to the research on topics of this course. The students will improve their communication skills as well as written and oral presentation skills by giving regular reports on their study and research progress. A thesis is written in the second half of the course.
到達目標 【英語】
Objectives of the Course
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 students are expected to take the responsibility to organise their research activities, and to monitor and manage their time.
履修条件
Course Prerequisites
Background in analysis, measure theory, probability theory, functional analysis is required. Knowledge of ergodic theory is desirable.
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).
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.
教科書・テキスト
Textbook
Recommended books will be introduced on an individual basis.
参考書
Reference Book
R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. 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 first 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.
課外学習等(授業時間外学習の指示)
Study Load(Self-directed Learning Outside Course Hours)
To prepare carefully for seminar presentation and discussion.
注意事項
Notice for Students
This seminar 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
Every student with strong motivation to explore mathematics is welcome.
レベル
Level
2
キーワード
Keyword
Fractals, Hausdorff dimension, Dynamical systems, Ergodic theory, Thermodynamic formalism
履修の際のアドバイス
Advice
Enjoy maths.
授業開講形態等
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.
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)