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On indecomposability and composants of chaotic continua

100%

Kato H.

Fundamenta Mathematicae

1996

tom 150

nr 3

245-253

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 ≠ y, then there is an integer n ∈ ℤ such that $d(f^n(x),f^n(y)) > c$. 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 $diami f^n(A) > c$. 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 $X ⊂ ℝ^2$ with dim X > 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 $V^σ(z;Z)$.

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

100%

Kato H.

Fundamenta Mathematicae

1994

tom 145

nr 3

261-279

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 ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d(f^n(x), f^n(y)) > c$ (resp. $diam f^n(A) > c$). 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 (dim X > 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, $V^σ(x; Z)$ 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 $lim inf_{n → ∞} d(f^n(x), f^n(y))$ ≥ τ if σ = s, and $lim inf_{n → ∞} d(f^{-n}(x), f^{-n}(y))$ ≥ τ if σ = u; in particular, $W^σ(x) ≠ W^σ(y)$. Here $V^s(x; Z) = {z ∈ Z|$ there is a subcontinuum A of Z such that x, z ∈ A and $lim_{n → ∞} diam f^n(A) = 0}$, $V^u(x; Z) = {z ∈ Z| there is a subcontinuum A of Z such that x, z ∈ A and $lim_{n → ∞} diam f^{-n}(A) = 0}$, $W^s(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^n(x), f^n(x')) = 0}$, and $W^u(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^{-n}(x), f^{-n}(x'))=0}$. As a corollary, if f is a continuum-wise expansive homeomorphism of a compactum X with dim X > 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 dim X > 0 and there is a finite family $\mathbb{F}$ of graphs such that X is $\mathbb{F}$-like, then each chaotic continuum of f is indecomposable. Note that every expansive homeomorphism is continuum-wise expansive.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Kato H.

1 1996

1 1994

Wyniki wyszukiwania

On indecomposability and composants of chaotic continua

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke