On smooth mappings of the circle into itself

In this article is constructed the set $\mathfrak M=\mathfrak M_1\cup\mathfrak M_2$, open and everywhere dense in $C^1(S^1,S^1)$, of $\Omega$-stable mappings. $\Omega(f)$ is totally disconnected and $f/\Omega(f)$ is topologically conjugate to the topological Markov chain with a finite number of states; for $f\in\mathfrak M_2$ we have $\Omega(f)=S^1$ and $f/S^1$ topologically conjugate to $z^n/S^1$. For $f\in\mathfrak M$ there exists a hyperbolic structure onЁ$\Omega(f)$.
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