Transitive homeomorphisms of the circle

In this note we present a new p r o o f o f the classical t heo rem that a transitive h o m e o m o r p h i s m o f the circle is topologically equ iva len t to a rotat ion t h rough an angle incommensurab le with ~'. T h e p r o o f depends upon some o f the concepts and theorems belonging to topological dynamics. Let ~0 be a se l f -homeomorph ism of a topological space X, T h e n ~0 generates a discrete flow on X which we deno te by (X, ~o). Given (X, ~o) a n d x ~X we deno te the orbit o f x u n d e r ~o by dT(x, ~0) = {~o"(x): n is an integer}. I f ~0"(x) = x for some non-zero in teger n, then x is called a periodic point. Two discrete flows (X, ~0) and (Y, to) are isomorphic i f there exists a homeomor phism 0 o f X onto Y such that 0 o ~0 = tO ° 0. A subset M of X is a minimal set of (X, ~o) i f M is closed, ~(M) = M , and no p r o p e r subset o f M has these properties. Let S deno te the circle with the usual topology and g roup structure. It is a monothe t ic topological group. For a e S let La deno te t he self-homeomorph i sm o f S def ined by left multiplication by a (La(z)= az). An e lement g o f S is a generator of S if CI [d?(g, Lo)] = S.