On indecomposability and composants of chaotic continua

• Autorzy: Hisao Kato
• Języki publikacji: EN

Abstrakty

EN

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 $V^σ(z;Z)$.

Słowa kluczowe

EN

expansive homeomorphism; continuum-wise expansive homeomorphism; indecomposable; composant; chaotic continuum; plane compactum; stable and unstable sets

Kategorie tematyczne

• 54H20: Topological dynamics
• 54F50: Spaces of dimension ≤ 1 ; curves, dendrites
• 54E50: Complete metric spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 150
• Numer: 3
• Strony: 245-253

Daty

wydano

1996

otrzymano

1995-06-20

Twórcy

autor

Hisao Kato

• Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan

Bibliografia

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