Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

Hisao Kato

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 2, page 153-165
  • ISSN: 0016-2736

Abstract

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We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions W S ( x ) | x ∈ X and W ( u ) ( x ) | x ∈ X of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, W σ ( x ) contains a nondegenerate subcontinuum A x containing x with d i a m A x ≥ ϱ , and if x,y ∈ C and x ≠ y, then W σ ( x ) ≠ W σ ( y ) . For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), W u ( ˜ x ) is equal to the arc component of (G,f) containing ˜x, and d i m W s ( W x ) = 0 .

How to cite

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Kato, Hisao. "Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets." Fundamenta Mathematicae 143.2 (1993): 153-165. <http://eudml.org/doc/211998>.

@article{Kato1993,
abstract = {We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions $\{W^S(x)|x ∈ X\}$ and $\{W^(u)(x)| x ∈ X\}$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, $W^σ(x)$ contains a nondegenerate subcontinuum $A_x$ containing x with $diam A_x ≥ ϱ$, and if x,y ∈ C and x ≠ y, then $W^σ(x) ≠ W^σ(y)$. For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), $W^u(˜x)$ is equal to the arc component of (G,f) containing ˜x, and $dim W^s(W^x)=0$.},
author = {Kato, Hisao},
journal = {Fundamenta Mathematicae},
keywords = {stable sets; decompositions; unstable sets; Cantor set},
language = {eng},
number = {2},
pages = {153-165},
title = {Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets},
url = {http://eudml.org/doc/211998},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Kato, Hisao
TI - Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 2
SP - 153
EP - 165
AB - We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions ${W^S(x)|x ∈ X}$ and ${W^(u)(x)| x ∈ X}$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, $W^σ(x)$ contains a nondegenerate subcontinuum $A_x$ containing x with $diam A_x ≥ ϱ$, and if x,y ∈ C and x ≠ y, then $W^σ(x) ≠ W^σ(y)$. For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), $W^u(˜x)$ is equal to the arc component of (G,f) containing ˜x, and $dim W^s(W^x)=0$.
LA - eng
KW - stable sets; decompositions; unstable sets; Cantor set
UR - http://eudml.org/doc/211998
ER -

References

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