Homeomorphisms of manifolds with zero-dimensional sets of nonwandering points

Lawrence Stanislaus Husch, Jr. and Ping-Fun Lam

Vol. 50 (1974), No. 1, 109–124

DOI: 10.2140/pjm.1974.50.109

Abstract

Let h be a self-homeomorphism of a compact n-dimensional manifold M, which is not homeomorphic to an odd dimensionaI sphere, such that the set N of irregular points of h is closed in M and the set of nonwandering points of h is zero-dimensional. One of the main results of this paper is that N is either connected or consists of the two fixed points of h. In the latter case, N is homeomorphic to the n-sphere. In the former case when n = 2, it is shown that each component of M − N is an open 2-cell. If M is open and N is compact, then it is shown that M is homeomorphic to Euclidean n-space and N consists of a single fixed point of h.

Mathematical Subject Classification

Primary: 57A15

Secondary: 58F10

Milestones

Received: 14 July 1972

Revised: 11 July 1973

Published: 1 January 1974