On indecomposability and composants of chaotic continua, II

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x , y ∈ X and x ≠ y , then there is an integer n ∈ Z such that d ( f n ( x ) , f n ( y ) ) > c . A homeomorphism f : X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ Z such that diam f n ( A ) > c . Note that every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In this paper, we define the notion of closed subsets having uncountable handles and we prove that if f : X → X is a continuum-wise expansive homeomorphism of a continuum X and X does not contain any subcontinuum having uncountable handles, then each minimal chaotic continuum of f is indecomposable. As a corollary, we obtain that if X is a k-cyclic continuum and X admits a continuum-wise expansive homeomorphism f, then each minimal chaotic continuum of f is indecomposable.