Expansive homeomorphisms on surfaces with holes

A homeomorphism f: X → X of a metric space X with metric d is expansive if there is c > 0 such that if x, y ϵ X and x ≠ y, then there is an integer n ϵ Z such that d(fn(x), fn(y)) > c. In this paper, we investigate expansive homeomorphisms on (noncompact) surfaces with n holes (n ⩾ 1). Let n be a natural number. We prove that if M is a closed 2-manifold (= surface) with M ≠ S2, P2, K2, then the surface M(n) with n holes (n ⩾ 1) admits an expansive homeomorphism, where S2, P2 and K2 are the 2-sphere, the projective plane and Klein bottle, respectively, and M(n) is the (noncompact) surface obtained by deleting from M n disjoint (closed) 2-cells and we assume that M(n) has the restricted metric of that for M. For the cases M = K2, P2, S2 we obtain the following: (1) if M = P2 or K2 and n ≠ 1,2, then M(n) admits an expansive homeomorphism; (2) if M = S2 and n ≠ 3, then M(n) admits an expansive homeomorphism.