I'm currently studying Denjoy's theorem, which says the following:
Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by $Var(f))$is bounded then $f$ is conjugated with the irrational rotation $R_{\rho}$.
The heart of the proof of this theorem is an argument of limited distortion.
Sketch of proof: By the classification theorem of Poincaré, we have the following dichotomy for $ S^1 $ homeomorphisms with irrational rotation number $\rho$, or $f$ is semi-conjugated ith the irrational rotation $R_{\rho}$ or $f$ is conjugated with the irrational rotation $R_{\rho}$. More precisely, for all $x,y\in S^1$ we have $\omega(x)=\omega(y),$ and we have only two options:
- Or $\omega(x)=S^1$ and $f$ is conjugated with the irrational rotation $R_{\rho}$
- Or $\omega(x)=S^1$ is a cantor set and $f$ is semi-conjugated ith the irrational rotation $R_{\rho}$
Our goal is to show that the (2.) can not occur. We assume by contradiction that $\omega(x)\neq S ¹$, then $S^1\setminus \omega(x)$ is a countable union of intervals. Let be $I$ one of these intervals, then $I, f(I), \ldots, f^n(I)$ are all disjoint intervals. Put $I_n=f^n(I)$ then we have
\begin{eqnarray} l(I_n)+l(I_{-n})&=&\int (f^n(t))^{'}dt+ \int (f^{-n}(t))^{'}dt \\ &=& \int [ (f^n(t))^{'}+(f^{-n}(t))^{'}]dt \\ &\geq& \int \sqrt{ (f^n(t))^{'}\cdot(f^{-n}(t))^{'} }dt \end{eqnarray}
The heart of the proof of this theorem is an argument of limited distortion:
Lemma: Put $g=\log |f^{'}|$. Let $J$ be an interval in $S^1$ , and suppose the interiors of the intervals $J, f (J ), . . . , f^{n−1}(J )$ are pairwise disjoint. Then for any $n ∈ Z$, $$ Var(g) ≥ | \log(f^n)^{'}(x) − \log(f^n)^{'}(y)|=| \log(f^n)^{'}(x)\cdot\log(f^{-n})^{'}(y)| $$ Using this lemma and some technicalities we obtain: $$ l(I_n)+l(I_{-n})\geq \exp(-{\frac{1}{2}Var(g)})l(I) $$ this is absurd because it implies that $\sum_{n\in \mathbb{Z}}l(I_n)=\infty$ contradicting the fact that the intervals $I_n$ are disjoint.
MY QUESTION: In the midst of this demonstration have a technical lemma, which enables us to work the idea of limited distortion in this proof:
Let be $R_{\alpha}$ a irrational rotation, if $x\in S^1$ there are infinitely many indices $n\in \mathbb{N}$ such that the intervals $$ I_{k}=R_{\alpha}^{k}(x,R_{\alpha}^{-n}x);~~0\leq |k|<n $$ are disjoint.
I can not prove it
Write $x_k=R_{\alpha}^k(x_0)$ and $I=(x_0,x_{-n})$, then $I_k=(x_k,x_{k-n})$. If for some $n$ we have the property that for $0<|k|<n$ no $x_k$ lies in I, then we can see that the intervals $I_k$ are disjoint. We claim that there are infinite many such $n$, or otherwise there exists $N\in\mathbb{N}$ such that the claim fails for $n\geq\mathbb{N}$. Then by induction we can show that for any $n\geq N$ there exists $M$ with $0<|m|<N$ such that $x_m$ is nearer to $x_0$ than $x_n$. So the set $\{x_n: n\in\mathbb{Z},n\neq 0\}$ has positive distance away from $x_0$, which means that $x_0$ can not be in the orbit closure of $x_0$, a contradiction to the property of the irrational rotation on the circle.