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

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)