The Nonexistence of Expansive Z^d Actions on Graphs

It is well known that if X is an arc or a circle, then there is no expansive homeomorphism on X. In this paper we prove that there is no expansive .d action on X, which answers the two questions raised by us before. In 1979, Mañé proved that there is no expansive homeomorphism on infinite dimensional spaces. Contrary to this result, we construct an expansive .2 action on an infinite dimensional space. We also construct an expansive .2 action on a zero dimensional space but no element in .2 is expansive.