TITLE: Denjoy counterexamples $C^1$ close to conjugates of rotations. 
QUESTION [3 upvotes]: It is well known and easy to prove that any Denjoy counterexample in the circle is approximated by homeomorphisms of the circle which have the same rotation number and are transitive (in particular, they are conjugated to the rotation). 
My question is the following. Given a $C^1$ denjoy counterexample $f:S^1\to S^1$, does there exists a sequence $f_n$ of $C^1$ diffeomorphisms of the circle such that:
1-the rotation number of $f_n$ is the same as the one of $f$.
2-the diffeomorphisms $f_n$ are conjugated to the rotation.
3-$f_n \rightarrow f$ in the $C^1$ topology.
If the diffeomorphisms $f_n$ are of class $C^2$, there is no need to ask for the second hypothesis by the Theorem of Denjoy. 
I would guess that for diophantine rotation number, one could even get that the $f_n$ are $C^1$ conjugated to the rotation. I believe that this must be known, but I could not find any reference.

REPLY [5 votes]: Dear Raphael,
perhaps I misunderstood the question, but in order to get your sequence $f_n$, one can $C^1$-approximate $f$ by $C^2$-diffeomorphisms $g_n$ and then compose each $g_n$ with an adequate small rotations $R_{\epsilon_n}$ to adjust the rotation number (so that $f_n=R_{\epsilon_n}\circ g_n$ do the job), right?
Best,
Matheus
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