Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke
Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke
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DOI：
10.4064/fm-145-3-261-279
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发表时间：
1994
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
紧空间X的同胚f：X → X是可扩张的（或连续可扩），如果存在c > 0使得如果x，y ∈ X且x 6= y（分别为若A是X）非退化子连续统，则存在n ∈ Z使得d（f（x），f（y））> c（分别为直径f（A）> c）。证明了如下定理：如果f是紧X的连续扩张同胚，且X的覆盖维数为正（dimX > 0），则存在f的σ-混沌连续统Z = Z（σ）（σ = s或σ = u），即Z是X的一个非退化子连续统，满足：（i）对每个x ∈ Z，V（x;（ii）存在τ > 0使得对于每个x ∈ Z和x在X中的每个邻域U，存在y ∈ U <$Z使得如果σ = s，则lim infn→∞ d（f（x），f（y））≥ τ，如果σ = u，则lim infn→∞ d（f-n（x），f-n（y））≥ τ;特别地，W（x）6= W（y）。这里V（x;Z）= {z ∈ Z|存在Z的一个子连续统A，使得x，z ∈ A且lim n→∞ diam f（A）= 0}，V（x;Z）= {z ∈ Z|存在Z的一个子连续统A，使得x，z ∈ A且lim n→∞ diam f−n（A）= 0}，W（x）= {x′ ∈ X| lim n→∞ d（f（x），fn（x′））= 0}，W（x）= {x′ ∈ X| lim n→∞ d（f−n（x），f−n（x′））= 0}.作为推论，如果f是紧X的连续扩张同胚，且dimX > 0，Z是f的σ-混沌连续统，则对几乎所有的康托集C <$Z，f或f−1在Li和Yorke意义下是C上的混沌，根据σ = s或u）。证明了：如果f是紧X的连续扩张同胚，且dimX > 0，且存在有限图族F使得X是F-似的，则f的每个混沌连续统都是不可分解的.注意每个扩张同胚都是连续可扩张的。1991年数学科目分类：小学54 H20、54 F50;中学54 E40、54 B20。
A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x 6= y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ Z such that d(f(x), f(y)) > c (resp. diam f(A) > c). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dimX > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum of X satisfying: (i) for each x ∈ Z, V (x;Z) is dense in Z, and (ii) there exists τ > 0 such that for each x ∈ Z and each neighborhood U of x in X, there is y ∈ U ∩ Z such that lim infn→∞ d(f(x), f(y)) ≥ τ if σ = s, and lim infn→∞ d(f−n(x), f−n(y)) ≥ τ if σ = u; in particular, W(x) 6= W(y). Here V (x;Z) = {z ∈ Z | there is a subcontinuum A of Z such that x, z ∈ A and lim n→∞ diam f(A) = 0}, V (x;Z) = {z ∈ Z | there is a subcontinuum A of Z such that x, z ∈ A and lim n→∞ diam f−n(A) = 0}, W (x) = {x′ ∈ X | lim n→∞ d(f(x), fn(x′)) = 0}, and W(x) = {x′ ∈ X | lim n→∞ d(f−n(x), f−n(x′)) = 0}. As a corollary, if f is a continuum-wise expansive homeomorphism of a compactum X with dimX > 0 and Z is a σ-chaotic continuum of f , then for almost all Cantor sets C ⊂ Z, f or f−1 is chaotic on C in the sense of Li and Yorke according as σ = s or u). Also, we prove that if f is a continuum-wise expansive homeomorphism of a compactum X with dimX > 0 and there is a finite family F of graphs such that X is F-like, then each chaotic continuum of f is indecomposable. Note that every expansive homeomorphism is continuum-wise expansive. 1991 Mathematics Subject Classification: Primary 54H20, 54F50; Secondary 54E40, 54B20.