Infinite minimal sets of continuum-wise expansive homeomorphisms of 1-dimensional compacta

摘要

In [R. Mane, Trans. Amer. Math. Soc. 252 (1979) 313-319], R. Mane proved that minimal sets of expansive homeomorphisms are 0-dimensional. More generally, minimal sets of continuum-wise expansive homeomorphisms are 0-dimensional (see [H. Kato, Canad. J. Math. 45 (1993) 576-598]). Also, for each continuum-wise expansive homeomorphism f : X → X of a compactum X with dim X > 0, there is an f-invariant closed subset Y of X such that dim Y > 0 and f|Y : Y → Y is weakly chaotic in the sense of Devaney (see [H. Kato, Lecture Notes in Pure and Appl. Math., Vol. 170, Dekker, New York, 1995, pp. 265-274]). In this paper, we prove the following result: If f : X → X is a continuum-wise expansive homeomorphism of a compactum X with dim X = 1, then there is a Cantor set Z in X such that for some natural number N, f~N(Z) = Z and f~N|Z : Z → Z is semiconjugate to the shift homeomorphism σ : Σ → Σ, where Σ is the Cantor set {0, 1}~Z. As a corollary, there is a family {C_α | α ∈ Λ} of minimal sets C_α of f such that each C_α is a Cantor set, Cl(∪{C_α | α ∈ Λ}) = Y is 1-dimensional and f|Y : Y → Y is weakly chaotic in the sense of Devaney.