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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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