# 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 ${S}^{1}=\mathbb{R}/\mathbb{Z}$. For $\alpha \in \mathbb{R}$, define ${R}_{\alpha}:{S}^{1}\to {S}^{1}$ by ${R}_{\alpha}(x)=x+\alpha +\mathbb{Z}$.

We say that a homeomorphism $f:{S}^{1}\to {S}^{1}$ is orientation preserving if it lifts to an increasing homeomorphism $F:\mathbb{R}\to \mathbb{R}$: $\pi \circ F=f\circ \pi $.

The rotation number of an orientation preserving homeomorphism $f$ is defined by

$$\rho (f)=\underset{n\to \mathrm{\infty}}{lim}\frac{{F}^{n}(x)-x}{n}.$$ |

One proves that this is independent both of the lift $F$ of $f$ and the point $x\in \mathbb{R}$. Some facts about the rotation number: it is an invariant of topological conjugacy, and $\rho (f)$ is rational if and only if $f$ has a periodic point. A periodic point is $x\in {S}^{1}$ such that ${f}^{n}(x)=x$ for some $n\ge 1$.

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 $f:{S}^{1}\to {S}^{1}$ is an orientation preserving homeomorphism that is topologically transitive with irrational rotation number $\rho (f)$, then $f$ is topologically conjugate to ${R}_{\rho (f)}$. 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.

###### Lemma 1.

Let $f\mathrm{:}{S}^{\mathrm{1}}\mathrm{\to}{S}^{\mathrm{1}}$ be a ${C}^{\mathrm{1}}$ diffeomorphism and let $J$ be an interval in ${S}^{\mathrm{1}}$. Let $g\mathrm{=}\mathrm{log}\mathit{}{f}^{\mathrm{\prime}}$. If the interiors of $J\mathrm{,}f\mathit{}\mathrm{(}J\mathrm{)}\mathrm{,}\mathrm{\dots}\mathrm{,}{f}^{n\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}J\mathrm{)}$ are pairwise disjoint, then for any $x\mathrm{,}y\mathrm{\in}J$ and any $n\mathrm{\in}\mathrm{Z}$ we have

$$\mathrm{Var}(g)\ge |\mathrm{log}({({f}^{n})}^{\prime}(x))-\mathrm{log}({({f}^{n})}^{\prime}(y))|.$$ |

###### Proof.

The intervals $[x,y],[f(x),f(y)],\mathrm{\dots},[{f}^{n-1}(x),{f}^{n-1}(y)]$ are pairwise disjoint, so they are part of a partition of $[0,1]$. The total variation of $g$ is defined as a supremum over all partitions, so in particular it will be $\ge $ the sum coming from any particular partition or a subset of that partition.

$\mathrm{Var}(g)$ | $\ge $ | $\sum _{k=0}^{n-1}}|g({f}^{k}(y))-g({f}^{k}(x))|$ | ||

$\ge $ | $\left|{\displaystyle \sum _{k=0}^{n-1}}g({f}^{k}(y))-g({f}^{k}(x))\right|$ | |||

$=$ | $\left|\mathrm{log}{\displaystyle \prod _{k=0}^{n-1}}{f}^{\prime}({f}^{k}(y))-\mathrm{log}{\displaystyle \prod _{k=0}^{n-1}}{f}^{\prime}({f}^{k}(x))\right|$ | |||

$=$ | $|\mathrm{log}({({f}^{n})}^{\prime}(x))-\mathrm{log}({({f}^{n})}^{\prime}(y))|.$ |

∎

Now we can prove Denjoy’s theorem.

###### Theorem 2.

If $f\mathrm{:}{S}^{\mathrm{1}}\mathrm{\to}{S}^{\mathrm{1}}$ is a ${C}^{\mathrm{1}}$ diffeomorphism that is orientation preserving, that has irrational rotation number $\rho \mathit{}\mathrm{(}f\mathrm{)}$, and whose derivative ${f}^{\mathrm{\prime}}\mathrm{:}{S}^{\mathrm{1}}\mathrm{\to}\mathrm{R}$ has bounded variation, then $f$ is topologically conjugate to ${R}_{\rho \mathit{}\mathrm{(}f\mathrm{)}}$.

###### Proof.

Suppose by contradiction that $f$ is not topologically transitive. It’s a fact proved in Chapter 7 of Brin and Stuck that this implies that $\omega (x)$ is perfect and nowhere dense, and is independent of the point $x$. (Recall that $\omega (x)={\bigcap}_{n\ge 1}\overline{{\bigcup}_{i\ge n}{f}^{i}(x)}$.) It follows that there is an interval $I=(a,b)$ in its complement.

The intervals ${f}^{n}(I)$, $n\in \mathbb{Z}$, are pairwise disjoint, for otherwise $f$ would have a periodic point. Let $\mu $ be Haar measure on ${S}^{1}$. Then

$$\sum _{n\in \mathbb{Z}}\mu ({f}^{n}(I))\le 1.$$ |

Let $x\in {S}^{1}$. Suppose for the moment that there are infinitely $n\ge 1$ such that the intervals $(x,{f}^{-n}(x)),(f(x),{f}^{1-n}(x)),\mathrm{\dots},({f}^{n}(x),x)$ are pairwise disjoint; we shall prove that this is true later. By applying the lemma we proved with $y={f}^{-n}(x)$ we get

$$\mathrm{Var}(g)\ge \left|\mathrm{log}\frac{{({f}^{n})}^{\prime}(x)}{{({f}^{n})}^{\prime}(y)}\right|=|\mathrm{log}({({f}^{n})}^{\prime}(x){({f}^{-n})}^{\prime}(x)|.$$ |

To see the equality in the above line it helps to write out what ${({f}^{-n})}^{\prime}(x)$ is.

Then for infinitely many $n$ we have

$\mu ({f}^{n}(I))+\mu ({f}^{-n}(I))$ | $=$ | ${\int}_{I}}{({f}^{n})}^{\prime}(x)\mathit{d}x+{\displaystyle {\int}_{I}}{({f}^{-n})}^{\prime}(x)\mathit{d}x$ | ||

$=$ | ${\int}_{I}}({({f}^{n})}^{\prime}(x)+{({f}^{-n})}^{\prime}(x))\mathit{d}x$ | |||

$\ge $ | ${\int}_{I}}\sqrt{{({f}^{n})}^{\prime}(x){({f}^{-n})}^{\prime}(x)}\mathit{d}x$ | |||

$=$ | ${\int}_{I}}\sqrt{\mathrm{exp}\mathrm{log}({({f}^{n})}^{\prime}(x){({f}^{-n})}^{\prime}(x))}\mathit{d}x$ | |||

$\ge $ | ${\int}_{I}}\sqrt{\mathrm{exp}(-|\mathrm{log}({({f}^{n})}^{\prime}(x){({f}^{-n})}^{\prime}(x))|)}\mathit{d}x$ | |||

$\ge $ | ${\int}_{I}}\sqrt{\mathrm{exp}(-\mathrm{Var}(g))}\mathit{d}x$ | |||

$=$ | $\mathrm{exp}\left(-{\displaystyle \frac{1}{2}}\mathrm{Var}(g)\right)\mu (I).$ |

Since $\mu (I)>0$ this implies that ${\sum}_{n\in \mathbb{Z}}\mu ({f}^{n}(I))=\mathrm{\infty}$, a contradiction. Therefore $f$ is topologically transitive, and so it is topologically conjugate to the ${R}_{\rho (f)}$. ∎

It is indeed necessary that ${f}^{\prime}$ has bounded variation. Brin and Stuck give an example on p. 161 that they attribute to Denjoy: for any irrational number $\rho \in (0,1)$, there is a nontransitive orientation preserving ${C}^{1}$ diffeomorphism of ${S}^{1}$ with rotation number $\rho $. The only condition of Denjoy’s theorem that isn’t satisfied here is that ${f}^{\prime}$ have bounded variation.