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Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number. (English) Zbl 0651.58008

The author presents simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number. The mentioned theorems are proved in some previous papers. Let \({\mathbb{R}}\) be the set of real numbers and T \(1={\mathbb{R}}/{\mathbb{Z}}\) be the circle. Let \(a\in {\mathbb{R}}\) satisfy the diophantine condition, \(DC_{\beta}:\) \(\exists \beta \geq 0\), \(\exists \gamma >0\), \(\forall p/q\in {\mathbb{Q}}\) implies \(| a-(p/q)| =\gamma q^{-i-\beta}\) and \(0\leq \beta \leq 1\). By \(R_ a\) is denoted a translation of T 1 or \({\mathbb{R}}\), \(R_ a(x)=x+a.\)
The main new idea in the proofs is the use of the Schwarzian derivative of a diffeomorphism \(f\in D\) \(3(T\) \(1_ d)\), \(Sf=D\quad 2Log Df-()(DLog Df)\quad 2.\) To solve \(f\circ h=h\circ R_ a\) this equation the author takes the Schwarzian derivative \(S(f\circ h)=S(h\circ R_ a)\) which reduces to \(((Sf)\circ h)(Dh)\quad 2=(Sh)\circ R_ a-Sh.\)
The solution h belongs to \(D^{3,2}(T\) 1), \(h(0)=0\) and satisfies the following inequality: \[ \| D\quad 3h\|_{L\quad 2}\leq (c/\gamma)\| f-R_ a\|_{C\quad 5}, \] where \(\gamma\), \(c>0\) are constants.
Reviewer: V.Angelov

MSC:

58C25 Differentiable maps on manifolds
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[1] V. I. Arnold,On the mappings of the circumference onto itself, Translation of the Amer. Math. Soc.46, 2nd series, p. 213–184.
[2] M. R. Herman,Sur la conjugaison diffèrentiable des difféomorphismes du cercles à des rotations, Pub. Inst. Hautes Etudes Sci.49 (1979), p. 5–233. · Zbl 0448.58019 · doi:10.1007/BF02684798
[3] M. R. Herman,Sur les courbes invariantes par les difféomorphismes de l’anneau, vol. let Vol. 2 Vol. l=I to IV, Vol. 2=V to VIII, Vol. 1 appeared in: Astérisque Vol. 103–104 SMF (1983), Vol. 2 is a preprint of Centre de Mathématiques de l’Ecole Polytechnique (1984), to appear in Astérisque.
[4] M. R. Herman,Exemples de fractions rationnelles ayant une orbite dense sur la sphére de Riemann. Bull. Soc. Math. de France,112 (1984), p. 93–142. · Zbl 0559.58020 · doi:10.24033/bsmf.2002
[5] M.R. Herman,Majoration du nombre de cucles périodiques pour cetaines familles de difféomorphismes du cercle, preprint du Centre de Mathématiques de l’ecole Polytechnique (1984).
[6] J. Pöschel,Intebrabilitu of hamiltonian sustems on Cantor sets. Com. Pure Apl. Math., Vol. XXXV (1982), p. 653–695. · Zbl 0542.58015 · doi:10.1002/cpa.3160350504
[7] H. Rüssmann,On optimal estimates for the solutions of linear difference equations on the circle. Celestial Mech., 14 (1976), p. 33–37. · Zbl 0343.39002 · doi:10.1007/BF01247129
[8] E. Stein, Singular integrals and differentiability of functions Princeton Univ. Press, Princeton (1970).
[9] J. C. Yoccoz,Conjugaison des difféomorphismes du cercle dont le nombre de rotation véritie une condition diophantienne, Ann. Sci. Ec. Norm. Sup., 4ème série, 17 (1984), p. 333–359. · Zbl 0595.57027 · doi:10.24033/asens.1475
[10] J. C. Yoccoz,C 1-conjugaison des difféomorphismes du cercle, Lec. Notes in Math. No. 1007, Springer Verlag (1983), p. 814–827.
[11] [B] E. D. Belokolos,Quantum particle in a one-dimensional deformed lattice, Th. Math. Phys., Vol. 26 (1976), p. 21–25. · doi:10.1007/BF01038252
[12] [B 1] E. Borel,Oeuvres, Tome II, Editions du C. N. R. S., Paris (1972), p. 691–693, 773–787, 791–801, 805–807.
[13] [B 2] E. Borel,Leçons sur les fonctions monogènes uniformes d’une variable complexes, Gauthier-Villars, Paris (1917).
[14] [C] T. Carleman,Les fonctions quasi-analutiques, Gauthier-Villars (1926).
[15] [D] A. Denjoy,Notes communiquèes aux acadèes aux acadèmies, Vol. I, Gauthier-Villars, Paris (1957). · Zbl 0077.05801
[16] [G] T.W. Gamelin,Uniform algebras, Prentice-Hall, Englewood Cliffs (1969). · Zbl 0213.40401
[17] [Z] L. Zalcman,Analytic capacity and rational approximation, Lec. Notes Math. No. 50, Springer-Verlag, Berlin (1968). · Zbl 0171.03701
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