Construction of a Denjoy Counterexample

Question

John Milnor proofed the existence of a Denjoy counterexample in http://www.math.sunysb.edu/~jack/DYNOTES/dn15.pdf page 15-4. I could follow his arguments until "It is now reasonably straightforward to check that the map f [...] is $C^1$-smooth with derivative equal to +1 [...]."

I tried calculating the Difference quotient for all x outside the constructed intervals $$ \frac{f(x+h)-f(x)}{h}\leq \frac{f(x)+h-f(x)}{h}=1$$ as f is a contraction. But how do I show $\geq$ ?

Or maybe do I need a different approach?

Answer 1

I think the idea is that you choose a point $x$ and a really small interval $[x,x+h]$ or $[x-h,x]$ on one side of it. This interval will be partitioned into some $I_{n_i}$'s (these are dense in the circle) and some other stuff. There may also be a piece of some $I_k$. The length of the other stuff won't change because you started with a circle rotation. Since $h$ was chosen really small, the $I_{n_i}$'s necessarily have really large indices and hence the ratio of their lengths before and after applying $f$ is close to 1. All that's left to control is the segment of $I_k$. Now you can check that for any $\epsilon>0$, there's a $\delta$ such that if you pick an initial segment of any $I_j$ of length at most $\delta$, then the length of the image of the segment is the length of the segment times $1\pm\epsilon$. (To see this, you're done automatically for large $|j|$'s (the derivative was supposed to be uniformly close to 1 for these guys); and for small $|j|$'s, the derivative is 1 at the endpoints and you just use continuity.