Abstract
We show that a finite number of commuting diffeomorphisms with simultaneously Diophantine rotation numbers are smoothly conjugated to rotations. This solves a problem raised by Moser.
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@article {A, MRKEY = {0140699},
AUTHOR = {Arnold, V. I.},
TITLE = {Small denominators. {I}. On the mapping of a circle into itself},
JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {25},
YEAR = {1961},
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ISSN = {0373-2436},
MRCLASS = {32.44 (57.48)},
MRNUMBER = {25 \#4113},
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} -
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Danijela and Katok, Anatole},
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@misc{FKT,
author={Fayad, B and Khanin, K and Teplinsky, A},
TITLE={Differentiability of the conjugation of commuting diffeomorphisms of the circle},
NOTE={work in progress},
} -
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@article {S, MRKEY = {662606},
AUTHOR = {Hawkins, Jane and Schmidt, Klaus},
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} -
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@article {H, MRKEY = {538680},
AUTHOR = {Herman, Michael-Robert},
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JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {49},
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CODEN = {PMIHA6},
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URL = {http://www.numdam.org/item?id=PMIHES_1979__49__5_0},
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@book {KH, MRKEY = {1326374},
AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
SERIES = {Encycl. Math. Appl.},
NUMBER = {54},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1995},
PAGES = {xviii+802},
ISBN = {0-521-34187-6},
MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
MRNUMBER = {96c:58055},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER = {0878.58020},
} -
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@article {KO1, MRKEY = {1036902},
AUTHOR = {Katznelson, Y. and Ornstein, D.},
TITLE = {The differentiability of the conjugation of certain diffeomorphisms of the circle},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {9},
YEAR = {1989},
NUMBER = {4},
PAGES = {643--680},
ISSN = {0143-3857},
MRCLASS = {58F27 (58F08)},
MRNUMBER = {91i:58121},
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ZBLNUMBER = {0819.58033},
doi={10.1017/S0143385700005277},
} -
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@article {KO2, MRKEY = {1036903},
AUTHOR = {Katznelson, Y. and Ornstein, D.},
TITLE = {The absolute continuity of the conjugation of certain diffeomorphisms of the circle},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {9},
YEAR = {1989},
NUMBER = {4},
PAGES = {681--690},
ISSN = {0143-3857},
MRCLASS = {58F27 (58F08)},
MRNUMBER = {91i:58122},
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doi={10.1017/S0143385700005289},
} -
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@article {SK, MRKEY = {904139},
AUTHOR = {Khanin, K. M. and Sina{\u\i},
Ya. G.},
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CODEN = {CMPHAY},
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MRNUMBER = {88j:58114},
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URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104159810},
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} -
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AUTHOR = {Khanin, K. M. and Sina{\u\i},
Ya. G.},
TITLE = {Smoothness of conjugacies of diffeomorphisms of the circle with rotations},
JOURNAL = {Uspekhi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {44},
YEAR = {1989},
NUMBER = {1(265)},
PAGES = {57--82, 247},
ISSN = {0042-1316},
MRCLASS = {58F99},
MRNUMBER = {90i:58183},
MRREVIEWER = {Pawe{\l}G{ó}ra},
DOI = {10.1070/RM1989v044n01ABEH002008},
ZBLNUMBER = {0701.58053},
} -
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@article {M, MRKEY = {1069487},
AUTHOR = {Moser, J{ü}rgen},
TITLE = {On commuting circle mappings and simultaneous {D}iophantine approximations},
JOURNAL = {Math. Z.},
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YEAR = {1990},
NUMBER = {1},
PAGES = {105--121},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {58F11 (11J13 26A18 58F18 58F30)},
MRNUMBER = {92e:58120},
MRREVIEWER = {Autorreferat},
DOI = {10.1007/BF02571227},
ZBLNUMBER = {0689.58031},
} -
[Y] J. -C. Yoccoz, "Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne," Ann. Sci. École Norm. Sup., vol. 17, iss. 3, pp. 333-359, 1984.
@article {Y, MRKEY = {777374},
AUTHOR = {Yoccoz, J.-C.},
TITLE = {Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
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YEAR = {1984},
NUMBER = {3},
PAGES = {333--359},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {58F11},
MRNUMBER = {86j:58086},
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URL = {http://www.numdam.org/item?id=ASENS_1984_4_17_3_333_0},
ZBLNUMBER = {0595.57027},
} -
[Y3] J. Yoccoz, "Analytic linearization of circle diffeomorphisms," in Dynamical Systems and Small Divisors, New York: Springer-Verlag, 2002, pp. 125-173.
@incollection {Y3, MRKEY = {1924912},
AUTHOR = {Yoccoz, Jean-Christophe},
TITLE = {Analytic linearization of circle diffeomorphisms},
BOOKTITLE = {Dynamical Systems and Small Divisors},
VENUE={Cetraro, 1998},
SERIES = {Lecture Notes in Math.},
NUMBER = {1784},
PAGES = {125--173},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2002},
MRCLASS = {37E10 (37C15 37E20 37E45 37F25)},
MRNUMBER = {2004c:37073},
MRREVIEWER = {Stefano Marmi},
}
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