Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets
نویسنده
چکیده
We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dimX > 0, then the decompositions {W (x) | x ∈ X} and {W (x) | x ∈ X} of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ̺ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, Wσ(x) contains a nondegenerate subcontinuum Ax containing x with diamAx ≥ ̺, and if x, y ∈ C and x 6= y, then W σ(x) 6= Wσ(y). For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map f̃ : (G, f) → (G, f) of f is expansive, then for each x̃ ∈ (G, f), W (x̃) is equal to the arc component of (G, f) containing x̃, and dimW (x̃) = 0.
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