Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets
Volume 143 / 1993
Abstract
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 dim X > 0, then the decompositions ${W^S(x)|x ∈ X}$ and ${W^(u)(x)| x ∈ X}$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, $W^σ(x)$ contains a nondegenerate subcontinuum $A_x$ containing x with $diam A_x ≥ ϱ$, and if x,y ∈ C and x ≠ y, then $W^σ(x) ≠ W^σ(y)$. 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 ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), $W^u(˜x)$ is equal to the arc component of (G,f) containing ˜x, and $dim W^s(W^x)=0$.
