﻿﻿ 学术报告（Kawamura Shinzo 教授，2019.9.16）-广州大学2019-09-01-线上赌博平台，tt娱乐场，信誉赌场-Welcome-默认标题模板  TitleChaos on symbolic dynamical systems

ABSTRACTChaos 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 : the dynamical system Σ=(X,f) is said to be chaotic if has the following three properties called chaotic properties.

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

We here note that five mathematicians  show that if a dynamical system (X; f) satisfies Properties (1) and (2) and the cardinal number of 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 to the compact metric Cantor space Σ_n consisting of all infinite sequences of integers between 1 and n, and the function 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&times;n matrix 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 : A+A^2+…+A^n

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

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

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