Expansive homeomorphisms in continuum theory (Q1198633)

We investigate expansive homeomorphisms from a point of view of continuum theory. In Section 2, by using the notion of positively pseudo-expansive map, we give a characterization of expansiveness of shift homeomorphisms of inverse limits of graphs. We can easily see that the characterization is not true for the case of \(n\)-dimensional polyhedra \((n\geq 2)\). In Section 3, 4 and 5, we deal with more general expansive homeomorphisms which are not shift homeomorphisms. In Section 3, we prove that if \(f: X\to X\) is an expansive homeomorphism of a compact metric space \(X\) with \(\dim X\geq 1\), then there exists a closed subset \(Z\) of \(X\) such that each component of \(Z\) is nondegenerate, the space of components of \(Z\) is a Cantor set, the decomposition space of \(Z\) into components is continuous, and all components of \(Z\) are stable or unstable. In Section 4, we prove that there are no expansive homeomorphisms on hereditarily decomposable tree-like continua. In Section 5, we also prove that there are no expansive homeomorphisms on hereditarily decomposable circle-like continua.