Homeomorphisms of the Circle without Periodic Points!

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( s for all integers n where d is a metric and <p is a homeomorphism.) In the case of a rotation every point is distal from every other point. On the other hand, when there is no dense orbit there is a Cantor minimal set and at least one pair of doubly asymptotic orbits. We will see that these discrete flows have considerable topological variety. For example, there are infinitely many topologically distinct ones with the same rotation number and exactly n pairs of doubly asymptotic orbits. The classification of homeomorphisms of the circle without periodic points is the same as the classification of their minimal sets, since each one has a unique minimal set. This leads to the problem of determining when an arbitrary minimal set comes from a homeomorphism of the circle. In §4 we characterize these minimal sets completely in terms of dynamical properties. The proximal relation is the dynamical object that plays the key role in this characterization. Finally, the invariants which we use in our classification contain information about homomorphisms, endomorphisms, and automorphisms. Theorems of this nature are presented in § 3.