On indecomposability and composants of chaotic continua
On indecomposability and composants of chaotic continua
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混沌连续体的不可分解性及其组成
DOI：
10.4064/fm-150-3-245-253
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发表时间：
1996
影响因子：
0.6
通讯作者：
H. Kato
中科院分区：
文献类型：
--
作者：
H. Kato
一个同胚f：度量为d的紧X的X → X是可扩的，如果存在c > 0使得x，y ∈ X且x6 = y，则存在整数n ∈ Z使得d（fn（x），fn（y））> c.一个同胚f：X → X是连续可扩张的，如果存在c > 0，使得如果A是X的非退化子连续统，则存在一个整数n ∈ Z，使得直径fn（A）> c。显然，每个扩张同胚都是连续可扩张的，但匡威的断言是不正确的。在[6]中，我们定义了同胚的混沌连续统的概念，并证明了连续扩张同胚的混沌连续统的存在性。研究了混沌连续系统的不可分解性。本文进一步研究了混沌连续统及其复合体的不可分解性。特别地，我们证明了：如果f：X → X是平面紧空间X <$R的连续扩张同胚，且dimX > 0，则存在f的σ-混沌连续统Z（σ = s或u），使得Z是X的不可分解子连续统，且对任意z ∈ Z，Z的包含z的合成函数c（z）与连续σ-稳定集V σ（z;Z）重合.
A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(fn(x), fn(y)) > c. A homeomorphism f : X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ Z such that diam fn(A) > c. Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua of homeomorphisms and proved the existence of chaotic continua of continuum-wise expansive homeomorphisms. Also, we studied indecomposability of chaotic continua. In this paper, we investigate further more properties of indecomposability of chaotic continua and their composants. In particular, we prove that if f : X → X is a continuum-wise expansive homeomorphism of a plane compactum X ⊂ R with dimX > 0, then there exists a σ-chaotic continuum Z (σ = s or u) of f such that Z is an indecomposable subcontinuum of X and for each z ∈ Z the composant c(z) of Z containing z coincides with the continuum-wise σ-stable set V σ(z;Z).
影响因子：
0.6
作者：
H. Kato
通讯作者：
H. Kato