Orbit equivalence, flow equivalence and ordered cohomology
The authors study the self-homeomorphisms of zero-dimensional metrizable compact Hausdorff spaces by means of the ordered first cohomology group, particularly in the light of recent work of \textit{R. H. Herman, I. F. Putnam}, and \textit{C. F. Skau} on minimal homeomorphisms [Int. J. Math. 3, No. 6, 827-864 (1992; Zbl 0786.46053)]. Let \(T\) be a homeomorphism of a compact metrizable zero-dimensional space \(X\). Let \(C(X,\mathbb{Z})\) be the continuous integer-valued functions on \(X\). Denote the subgroup of coboundaries by \(\text{cobdy}(T)=\{f-(f\circ T)\mid f\in C(X,\mathbb{Z})\}\). Let \(G^T=C(X,\mathbb{Z})/\text{cobdy}(T)\). Let \(G^T_t=\{[f]\in G^T\mid\) there exists non-negative \(f_0:X\to\mathbb{Z}\) such that \([f_0]= [f]\}\). Theorem 1.12: Suppose \(S\) and \(T\) are irreducible shifts of finite type. Then the following are equivalent. (a) \(S\) and \(T\) are flow equivalent; (b) \((G^S,G^S_t)\cong(G^T,G^T_t)\); (c) \(Z(S)=Z(T)\), \(Z(S)\) and \(Z(T)\) are the zeta functions associated with \(S\) and \(T\), respectively. If \(S\) and \(T\) are orbit equivalent, then they are flow equivalent.
zero-dimensional metrizable compact Hausdorff space
scientific article; zbMATH DE number 952563
scientific article