In 1930, M. H. A. Newman proved a rather remarkable theorem which has become one of the classical theorems in topology. It has many important applications. A special case of Newman's Theorem is that a periodic homeomorphism of period n > 1 of a sphere S onto itself must have some orbit which is not contained in a “cap” smaller than a hemisphere. The general theorem is as follows:
THEOREM (). Suppose that Mn is a connected (metric) n-manifold, U is a domain in Mn, andp is an integer greater than 1. Then there is a positive number d such that no uniformly continuous homeomorphism h of Mn onto itself of period p moves every point of U a distance < d. That is, there is x ∊ U so that the orbit of x under h has diameter ≦ d.