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

Fundamenta Mathematicae (1994)

• Volume: 145, Issue: 3, page 261-279
• ISSN: 0016-2736

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## Abstract

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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 $d\left({f}^{n}\left(x\right),{f}^{n}\left(y\right)\right)>c$ (resp. $diam{f}^{n}\left(A\right)>c$). 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, ${V}^{\sigma }\left(x;Z\right)$ 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 $limin{f}_{n\to \infty }d\left({f}^{n}\left(x\right),{f}^{n}\left(y\right)\right)$ ≥ τ if σ = s, and $limin{f}_{n\to \infty }d\left({f}^{-n}\left(x\right),{f}^{-n}\left(y\right)\right)$ ≥ τ if σ = u; in particular, ${W}^{\sigma }\left(x\right)\ne {W}^{\sigma }\left(y\right)$. Here   ${V}^{s}\left(x;Z\right)=z\in Z|thereisasubcontinuumAofZsuchthat\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}x,z\in Aand$limn → ∞ diam fn(A) = 0$,$${V}^{u}\left(x;Z\right)=z\in Z|thereisasubcontinuumAofZsuchthat\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}x,z\in Aand$limn → ∞ diam f-n(A) = 0$,$   ${W}^{s}\left(x\right)={x}^{\text{'}}\in X|li{m}_{n\to \infty }d\left({f}^{n}\left(x\right),{f}^{n}\left({x}^{\text{'}}\right)\right)=0$, and    ${W}^{u}\left(x\right)={x}^{\text{'}}\in X|li{m}_{n\to \infty }d\left({f}^{-n}\left(x\right),{f}^{-n}\left({x}^{\text{'}}\right)\right)=0$. 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 $𝔽$ of graphs such that X is $𝔽$-like, then each chaotic continuum of f is indecomposable. Note that every expansive homeomorphism is continuum-wise expansive.

## How to cite

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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 -

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