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Wyświetlanie 1-2 z 2
Tytuł:
On indecomposability and composants of chaotic continua
Autorzy:
Kato, Hisao
Powiązania:
https://bibliotekanauki.pl/articles/1205482.pdf
Data publikacji:
1996
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
expansive homeomorphism
continuum-wise expansive homeomorphism
indecomposable
composant
chaotic continuum
plane compactum
stable and unstable sets
Opis:
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 $V^σ(z;Z)$.
Źródło:
Fundamenta Mathematicae; 1996, 150, 3; 245-253
0016-2736
Pojawia się w:
Fundamenta Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke
Autorzy:
Kato, Hisao
Powiązania:
https://bibliotekanauki.pl/articles/1208450.pdf
Data publikacji:
1994
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
expansive homeomorphism
continuum-wise expansive homeomorphism
stable and unstable sets
scrambled set
chaotic in the sense of Li and Yorke
independent
indecomposable continuum
Opis:
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.
Źródło:
Fundamenta Mathematicae; 1994, 145, 3; 261-279
0016-2736
Pojawia się w:
Fundamenta Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
    Wyświetlanie 1-2 z 2

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