Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Kato, Hisao. "Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke." Fundamenta Mathematicae 145.3 (1994): 261-279. <http://eudml.org/doc/212046>.

@article{Kato1994,
abstract = {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 $limn → ∞ diam fn(A) = 0$, $$V^u(x; Z) = \{z ∈ Z| there is a subcontinuum A of Z such that \}~~~~~ x, z ∈ A and $limn → ∞ diam f-n(A) = 0$, $   $W^s(x) = \{x^\{\prime \} ∈ X| lim_\{n → ∞\} d(f^n(x), f^n(x^\{\prime \})) = 0\}$, and    $W^u(x) = \{x^\{\prime \} ∈ X| lim_\{n → ∞\} d(f^\{-n\}(x), f^\{-n\}(x^\{\prime \}))=0\}$. 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. },
author = {Kato, Hisao},
journal = {Fundamenta Mathematicae},
keywords = {expansive homeomorphism; continuum-wise expansive homeomorphism; stable and unstable sets; scrambled set; chaotic in the sense of Li and Yorke; independent; indecomposable continuum; chaos},
language = {eng},
number = {3},
pages = {261-279},
title = {Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke},
url = {http://eudml.org/doc/212046},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Kato, Hisao
TI - Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 3
SP - 261
EP - 279
AB - 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 $limn → ∞ diam fn(A) = 0$, $$V^u(x; Z) = {z ∈ Z| there is a subcontinuum A of Z such that }~~~~~ x, z ∈ A and $limn → ∞ diam f-n(A) = 0$, $   $W^s(x) = {x^{\prime } ∈ X| lim_{n → ∞} d(f^n(x), f^n(x^{\prime })) = 0}$, and    $W^u(x) = {x^{\prime } ∈ X| lim_{n → ∞} d(f^{-n}(x), f^{-n}(x^{\prime }))=0}$. 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.
LA - eng
KW - expansive homeomorphism; continuum-wise expansive homeomorphism; stable and unstable sets; scrambled set; chaotic in the sense of Li and Yorke; independent; indecomposable continuum; chaos
UR - http://eudml.org/doc/212046
ER -