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

In [R. Mañé, Trans. Amer. Math. Soc. 252 (1979) 313–319], R. Mañé 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 dimX>0, there is an f-invariant closed subset Y of X such that dimY>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 dimX=1, then there is a Cantor set Z in X such that for some natural number N, fN(Z)=Z and fN|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 {Cα∣α∈Λ} of minimal sets Cα of f such that each Cα is a Cantor set, Cl(⋃{Cα∣α∈Λ})=Y is 1-dimensional and f|Y :Y→Y is weakly chaotic in the sense of Devaney.