Homeomorphisms of the Circle Without Periodic Points

Abstract

Homeomorphisms of the circle were first considered by Poincare who used them to obtain qualitative results for a class of differential equations on the torus. He classified those which have a dense orbit by showing that they are topologically equivalent to a rotation through an angle incommensurable with IT. However, Denjoy showed that there exist homeomorphisms of the circle without periodic points and without dense orbits. This established the existence of a class of homeomorphisms of the circle without periodic points which are not topologically equivalent to rotations through an angle incommensurable with TT. But since then this class has been largely ignored. The purpose of this paper is to present a classification scheme for all homeomorphisms of the circle without periodic points. Our method depends upon analysing those points which are distal from all other points. (Two points x and y are distal if there exists e> 0 such that d(<p n (x), <p n (y))> s for all integers n where d is a metric and <p is a homeomorphism.)