On indecomposability and composants of chaotic continua

Hisao Kato

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 3, page 245-253
  • ISSN: 0016-2736

Abstract

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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 d i a m i 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 ) .

How to cite

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Kato, Hisao. "On indecomposability and composants of chaotic continua." Fundamenta Mathematicae 150.3 (1996): 245-253. <http://eudml.org/doc/212175>.

@article{Kato1996,
abstract = {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 $V^σ(z;Z)$.},
author = {Kato, Hisao},
journal = {Fundamenta Mathematicae},
keywords = {expansive homeomorphism; continuum-wise expansive homeomorphism; indecomposable; composant; chaotic continuum; plane compactum; stable and unstable sets; stable sets; unstable sets; indecomposability; indecomposable subcontinuum},
language = {eng},
number = {3},
pages = {245-253},
title = {On indecomposability and composants of chaotic continua},
url = {http://eudml.org/doc/212175},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Kato, Hisao
TI - On indecomposability and composants of chaotic continua
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 3
SP - 245
EP - 253
AB - 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 $V^σ(z;Z)$.
LA - eng
KW - expansive homeomorphism; continuum-wise expansive homeomorphism; indecomposable; composant; chaotic continuum; plane compactum; stable and unstable sets; stable sets; unstable sets; indecomposability; indecomposable subcontinuum
UR - http://eudml.org/doc/212175
ER -

References

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