Expansive homeomorphisms and topological dimension

Abstract:

Let K be a compact metric space. A homeomorphism $f: K\mid$ is expansive if there exists $\varepsilon > 0$ such that if $x, y \in K$ satisfy $d\left ( {{f^n}\left ( x \right ), {f^n}\left ( y \right )} \right ) < \varepsilon$ for all $n \in {\textbf {Z}}$ (where $d\left ( { \cdot , \cdot } \right )$ denotes the metric on K) then $x = y$. We prove that a compact metric space that admits an expansive homeomorphism is finite dimensional and that every minimal set of an expansive homeomorphism is 0-dimensional.