Geodesic
On smooth mappings of the circle into itself
Sbornik. Mathematics, Tome 14 (1971) no. 2, pp. 161-185 
M. V. Jakobson. On smooth mappings of the circle into itself. Sbornik. Mathematics, Tome 14 (1971) no. 2, pp. 161-185. http://geodesic.mathdoc.fr/item/SM_1971_14_2_a0/
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     author = {M. V. Jakobson},
     title = {On smooth mappings of the circle into itself},
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     volume = {14},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_2_a0/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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$\Omega(f)$. Figures: 1 Bibliography: 9 titles.

[1] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 ; Uspekhi matem. nauk, XXV:1(151) (1970), 113–185 | DOI | MR | MR

[2] A. Puankare, O krivykh, opredelyaemykh differentsialnymi uravneniyami, Gostekhizdat, Moskva–Leningrad, 1947

[3] A. G. Maier, “Gruboe preobrazovanie okruzhnosti v okruzhnost”, Uchenye zap. Gorkovskogo un-ta, 12 (1939), 215–225

[4] Z. Nitecki, On $\Omega$ for immersion, Summer Institute on Global Analisis, Preprint, Bercley, 1968

[5] P. Fatou, “Sur les equations fonctionnelles I”, Bull. Soc. math. France, 47 (1919), 161–271 | MR

[6] Ch. Pugh, “An improved closing lemma and a general density theorem”, Amer. J. Math., 89:4 (1967), 1010–1021 ; Matematika, 12:6 (1968), 136–146 | DOI | MR | Zbl

[7] M. Shub, “Endomorphisms of compact differentiable manifolds”, Amer. J. Math., 91:1 (1969), 175–199 | DOI | MR | Zbl

[8] Z. Nitecki, “Nonsingular endomorphisms of the circle”, Proc. of Symposia in Pure Mathematics, 14 (1970), 203–220 | MR | Zbl

[9] J. Guckenheimer, “Endomorphisms of the Niemann sphere”, Proc. of Symposia in Pure Mathematics, 14 (1970), 95–123 | MR | Zbl