Denjoy’s theorem on circle diffeomorphisms
In this note I’m just presenting the proof of Denjoy’s theorem in Michael Brin and Garrett Stuck’s Introduction to dynamical systems, Cambridge University Press, 2002.
Let . For , define by .
We say that a homeomorphism is orientation preserving if it lifts to an increasing homeomorphism : .
The rotation number of an orientation preserving homeomorphism is defined by
One proves that this is independent both of the lift of and the point . Some facts about the rotation number: it is an invariant of topological conjugacy, and is rational if and only if has a periodic point. A periodic point is such that for some .
There are some lemmas in Chapter 7 that I don’t want to write out. The important theorem that we’re going to use without proof is that if is an orientation preserving homeomorphism that is topologically transitive with irrational rotation number , then is topologically conjugate to . This reduces our problem to showing that a map is topologically transitive.
We will use the following lemma in the proof of Denjoy’s theorem.
Let be a diffeomorphism and let be an interval in . Let . If the interiors of are pairwise disjoint, then for any and any we have
The intervals are pairwise disjoint, so they are part of a partition of . The total variation of is defined as a supremum over all partitions, so in particular it will be the sum coming from any particular partition or a subset of that partition.
Now we can prove Denjoy’s theorem.
If is a diffeomorphism that is orientation preserving, that has irrational rotation number , and whose derivative has bounded variation, then is topologically conjugate to .
Suppose by contradiction that is not topologically transitive. It’s a fact proved in Chapter 7 of Brin and Stuck that this implies that is perfect and nowhere dense, and is independent of the point . (Recall that .) It follows that there is an interval in its complement.
The intervals , , are pairwise disjoint, for otherwise would have a periodic point. Let be Haar measure on . Then
Let . Suppose for the moment that there are infinitely such that the intervals are pairwise disjoint; we shall prove that this is true later. By applying the lemma we proved with we get
To see the equality in the above line it helps to write out what is.
Then for infinitely many we have
Since this implies that , a contradiction. Therefore is topologically transitive, and so it is topologically conjugate to the . ∎
It is indeed necessary that has bounded variation. Brin and Stuck give an example on p. 161 that they attribute to Denjoy: for any irrational number , there is a nontransitive orientation preserving diffeomorphism of with rotation number . The only condition of Denjoy’s theorem that isn’t satisfied here is that have bounded variation.