Denjoy theorem on rotation number

Remarks

By Poincaré's classification, $f$ is semiconjugate to $R_\rho$, and it is conjugate unless $f$ has a wandering interval, an open interval $J$ whose images $f^n(J)$, $n \in \Z$, are pairwise disjoint. The proof rules out wandering intervals. Let $q_n$ be the denominators of the continued fraction of $\rho$. A combinatorial argument finds intervals whose images under $f^{q_n}$ overlap little, and the Denjoy inequality bounds $\abs{\log (f^{q_n})'}$ by the total variation of $\log f'$. For a wandering interval $J$ this gives a constant $c > 0$ with $\abs{f^{q_n}(J)} + \abs{f^{-q_n}(J)} \ge c \abs{J}$ for all $n$. But the images of $J$ are pairwise disjoint, so their lengths tend to $0$, which is a contradiction.

The smoothness hypothesis cannot be dropped. Denjoy also built $C^1$ diffeomorphisms with irrational rotation number that have a wandering interval. For them the omega-limit set of every point is a Cantor set and $f$ is not conjugate to a rotation. Such examples exist even of class $C^{1 + \alpha}$ for every $\alpha < 1$.

The conjugacy $h$ is only a homeomorphism in general; whether it is smooth depends on arithmetic properties of $\rho$, a question studied by Arnold, Herman and Yoccoz.

Going deeper

For a student

The theorem completes Poincaré's classification for smooth circle diffeomorphisms: up to topological conjugacy, a $C^2$ diffeomorphism with irrational rotation number is determined by that number. It implies that every orbit of such a map is dense and that the map is minimal and uniquely ergodic, since the irrational rotation has these properties. The key tool, control of the distortion of $(f^n)'$ through bounded variation of $\log f'$, reappears in many places in one-dimensional dynamics. Denjoy published the theorem in 1932.