# Structural stability on the circle

How can i proove that on the space of circle diffeomorphisms with the C-r topology, Morse Smale diffeomorphisms and Structurally stable diffeomorphisms are the same set?

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It is not hard to show that to be $C^r$-structurally stable, the diffeomorphism must be Morse-Smale: Indeed, if it has irrational rotation number one can easily perturb (by composing with a small rotation) in order to change the rotation number (conjugacy invariant, see for example Proposition 11.1.9 in Katok-Hasselblatt's book). If it has a periodic point whose derivative is of absolute value $1$, it is also easy to perturb in order to create either an interval of periodic points or an isolated periodic point (thus change the orbit structure).

Now, to see the converse, notice that if the homeomorphism is Morse-Smale, the set of periodic points varies continuously (it is thus $\Omega$-Stable). To see that the conjugacy extends to the whole circle, notice that any pair of maps of the interval which fixes the boundaries and whose orbits are increasing are conjugated by a homeomorphism (just define an injective map from one fundamental domain to the other and glue in order to obtain conjugacy). By gluing these homeomorphisms one can construct a conjugacy with any small perturbation.

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