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

Hisao Kato¹

¹

Fundamenta Mathematicae, Tome 145 (1994) no. 3, pp. 261-279

Résumé

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 $lim_{n → ∞} diam f^n(A) = 0}$, $V^u(x; Z) = {z ∈ Z| there is a subcontinuum A of Z such that       x, z ∈ A and $lim_{n → ∞} diam f^{-n}(A) = 0}$,    $W^s(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^n(x), f^n(x')) = 0}$, and    $W^u(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^{-n}(x), f^{-n}(x'))=0}$. 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.

Détail

DOI : 10.4064/fm-145-3-261-279

Keywords: expansive homeomorphism, continuum-wise expansive homeomorphism, stable and unstable sets, scrambled set, chaotic in the sense of Li and Yorke, independent, indecomposable continuum