Expansive homeomorphisms on surfaces with holes

✍ Scribed by Hisao Kato

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 564 KB
• Volume: 82
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00069-2

✦ Synopsis

A homeomorphism f : X ---t X of a metric space X with metric d is expansive if there is c > 0 such that if x, y E X and z # y, then there is an integer n E Z such that d(f"(x), fn(y)) > c.

In this paper, we investigate expansive homeomorphisms on (noncompact) surfaces with n holes (n 3 1). Let n be a natural number. We prove that if A4 is a closed 2-manifold (= surface) with M # S2, P2, K2, then the surface M(n) with n holes (n 3 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, P', 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.