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I'm not very familiar with the dynamics of circle maps, and incidentally I realized that an answer to a question concerning circle diffeomorphisms can help me to solve a problem related to a discrete Schrodinger operator.

Let $f, g:\mathbb{S}^1\to\mathbb{S}^1$ be two orientation preserving diffeomorphisms of the unit circle, having rotation number $p/q$. Both $f,g$ are different from the rotation $R_{p/q}$. My question: Is it always possible to construct a conjugacy, i.e. an orientation preserving diffeomorphism $\varphi:\mathbb{S}^1\to\mathbb{S}^1$, such that $\varphi \circ f=g\circ \varphi$?

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  • $\begingroup$ Created account two days ago, asked one question, received an answer and deleted account. I hope this is not a new trend. $\endgroup$ – user53153 Feb 23 '13 at 1:11
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No, in the case of rational rotation number the periodic orbits add a lot of rigidity. For each $n$ the sets $P_n(f)=\{x\in \mathbb S^1: f^{(n)}(x)=x\}$ and $P_n(g)=\{x\in \mathbb S^1: g^{(n)}(x)=x\}$ must satisfy $\varphi(P_n(f))=\varphi^{-1}(P_n(g))$. In particular, $P_n(f)$ and $P_n(g)$ must be homeomorphic. It is easy to construct $f,g$ with the same rotation number such that $P_n(f)$ and $P_n(g)$ have different number of connected components. For example (with zero rotation number) let $f$ be the identity perturbed on one open arc (so that the points of that arc are not fixed), while $g$ is the identity similarly perturbed on two disjoint open arcs. Similar construction works for nonzero rotations numbers.

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