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

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 6= y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ Z such that d(f(x), f(y)) > c (resp. diam f(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 (dimX > 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 (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 infn→∞ d(f(x), f(y)) ≥ τ if σ = s, and lim infn→∞ d(f−n(x), f−n(y)) ≥ τ if σ = u; in particular, W(x) 6= W(y). Here V (x;Z) = {z ∈ Z | there is a subcontinuum A of Z such that x, z ∈ A and lim n→∞ diam f(A) = 0}, V (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 (x) = {x′ ∈ X | lim n→∞ d(f(x), fn(x′)) = 0}, and W(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 dimX > 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 dimX > 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. 1991 Mathematics Subject Classification: Primary 54H20, 54F50; Secondary 54E40, 54B20.