Expansive homeomorphisms and topological dimension

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Abstract
Let K be a compact metric space. A homeomorphism f : K ∣ f:\,K\mid is expansive if there exists ε > 0 \varepsilon \, > \,0 such that if x , y ∈ K x, y\, \in \,K satisfy d ( f n ( x ) , f n ( y ) ) > ε d\left ( {{f^n}\left ( x \right ),\,{f^n}\left ( y \right )} \right )\, > \,\varepsilon for all n ∈ Z n\, \in \,{\textbf {Z}} (where d ( ⋅ , ⋅ ) d\left ( { \cdot ,\, \cdot } \right ) denotes the metric on K) then x = y 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.

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