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數學學院學術講座  (2019054)

報告人: Kawamura Shinzo (河村新蔵)

單位: 日本山形大學（Yamagata University）

職務: 教授

報告時間: 2019年9月16日 上午10:00—11:00

報告地點: 廣州大學理學實驗樓314

Title：Chaos on symbolic dynamical systems

ABSTRACT：Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. ”Chaos” is an interdisciplinary theory stating within the apparent randomness of chaotic complex systems such as f(z)=z^2+C and the deterministic nonlinear system which can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas.

Nowadays, in common usage, ”chaos” means ”a state of disorder”. However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition originally formulated by Robert L. Devaney for a continuous map f:X—>X on some metric space X as follows [2]: the dynamical system Σ=(X,f) is said to be chaotic if f has the following three properties called chaotic properties.

(1) The set of all periodic points of f is dense in X. (2) f is topologically transitive. (3) f has sensitive dependence on initial conditions.

We here note that five mathematicians [1] show that if a dynamical system (X; f) satisfies Properties (1) and (2) and the cardinal number of X is infinite, then Property (3) automatically holds. Namely two topological properties implies a property of metric space. It was a surprising result.

Now, we restrict the compact metric space X to the compact metric Cantor space Σ_n consisting of all infinite sequences of integers between 1 and n, and the function f to the backward shiftσ_n. It is well-known that the dynamical system (Σ_n,σ_n) is chaotic in the sense of Devaney. In this talk, we consider a kind of dynamical systems (Σ_A,σ_A) associated with n×n matrix A with all entries belonging to {0,1}, whereΣ_A is a compact and σ-invarinat subset ofΣ_n andσ_A is the restriction of σ toΣ_A. We show a necessary and sufficient condition for the dynamical system (Σ_A,σ_A) to be chaotic in term of the propery of the following matrix [3]: A+A^2+…+A^n

[1] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s definition of Chaos, Amer. Math. Monthly, 99(1992), 332-334.

[2] R. L. Devaney, An intorodunction to chaotic dynamical systems, Second Edition, Addision-Wesley, Redwood City, 1989

[3] S. Kawamura, H.Takaegara and A.Uchiyam, Chaotic conditions of dubshift on symbolic dynamical systems, preprint.

報告人簡介:河村新蔵，日本山形大學數理科學部教授。1983年畢業于北海道大學，獲得博士學位。1974—2014間，日本山形大學講師，副教授，教授。1988年Wales 大學（United Kingdom）留學，2012年-至今，北京林業大學客座教授。

主要研究內容：泛函分析，代數算子，模糊理論，動力系統等，分別在Tohoku Math.J.，J.Math.Soc.Japan,，Proc.Amer.Math.Soc.，Math. Scand等學術雜志上發表學術論文60余篇。