Abstract
We consider the irrational Aubry-Mather sets of an exact symplectic monotone $C^1$ twist map of the two-dimensional annulus, introduce for them a notion of “$C^1$-regularity” (related to the notion of Bouligand paratingent cone) and prove that
- $\bullet$ a Mather measure has zero Lyapunov exponents if and only if its support is $C^1$-regular almost everywhere;
- $\bullet$ a Mather measure has nonzero Lyapunov exponents if and only if its support is $C^1$-irregular almost everywhere;
- $\bullet$ an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere;
- $\bullet$ the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be $C^1$-irregular, are not “too irregular” (i.e., have small paratingent cones).
The main tools that we use in the proofs are the so-called Green bundles.
-
[Arna1]
M. Arnaud, "Three results on the regularity of the curves that are invariant by an exact symplectic twist map," Publ. Math. Inst. Hautes Études Sci., iss. 109, pp. 1-17, 2009.@article {Arna1, MRKEY = {2511585},
AUTHOR = {Arnaud, Marie-Claude},
TITLE = {Three results on the regularity of the curves that are invariant by an exact symplectic twist map},
JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
NUMBER = {109},
YEAR = {2009},
PAGES = {1--17},
ISSN = {0073-8301},
MRCLASS = {37E40 (37J10)},
MRNUMBER = {2511585},
MRREVIEWER = {Sonia Pinto de Carvalho},
DOI = {10.1007/s10240-009-0017-8},
ZBLNUMBER = {1177.53070},
} -
[Arna2] M. Arnaud, "Fibrés de Green et régularité des graphes $C^0$-lagrangiens invariants par un flot de Tonelli," Ann. Henri Poincaré, vol. 9, iss. 5, pp. 881-926, 2008.
@article {Arna2, MRKEY = {2438501},
AUTHOR = {Arnaud, Marie-Claude},
TITLE = {Fibrés de {G}reen et régularité des graphes {$C\sp 0$}-lagrangiens invariants par un flot de {T}onelli},
JOURNAL = {Ann. Henri Poincaré},
FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
VOLUME = {9},
YEAR = {2008},
NUMBER = {5},
PAGES = {881--926},
ISSN = {1424-0637},
MRCLASS = {37J45 (37C50)},
MRNUMBER = {2438501},
DOI = {10.1007/s00023-008-0375-7},
ZBLNUMBER = {1143.70009},
} -
[Arno] V. I. Arnol$’$d, "Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian," Uspehi Mat. Nauk, vol. 18, iss. 5 (113), pp. 13-40, 1963.
@article {Arno, MRKEY = {0163025},
AUTHOR = {Arnol$'$d, V. I.},
TITLE = {Proof of a theorem of {A}. {N}. {K}olmogorov on the preservation of conditionally periodic motions under a small perturbation of the {H}amiltonian},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {18},
YEAR = {1963},
NUMBER = {5 (113)},
PAGES = {13--40},
ISSN = {0042-1316},
MRCLASS = {34.65 (57.48)},
MRNUMBER = {0163025},
MRREVIEWER = {J. Moser},
ZBLNUMBER={0129.16606},
} -
[A-F] J. Aubin and H. Frankowska, Set-Valued Analysis, Boston, MA: Birkhäuser, 1990, vol. 2.
@book {A-F, MRKEY = {1048347},
AUTHOR = {Aubin, Jean-Pierre and Frankowska, H{é}l{è}ne},
TITLE = {Set-Valued Analysis},
SERIES = {Systems Control Found. App.},
VOLUME = {2},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1990},
PAGES = {xx+461},
ISBN = {0-8176-3478-9},
MRCLASS = {49-02 (26-02 28-02 46-02 47H15 49J52 58C06 90C48)},
MRNUMBER = {1048347},
MRREVIEWER = {Leszek Zaremba},
ZBLNUMBER = {0713.49021},
} -
[ALD] S. Aubry and P. Y. Le Daeron, "The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states," Phys. D, vol. 8, iss. 3, pp. 381-422, 1983.
@article {ALD, MRKEY = {0719634},
AUTHOR = {Aubry, S. and Le Daeron, P. Y.},
TITLE = {The discrete {F}renkel-{K}ontorova model and its extensions. {I}. {E}xact results for the ground-states},
JOURNAL = {Phys. D},
FJOURNAL = {Physica D. Nonlinear Phenomena},
VOLUME = {8},
YEAR = {1983},
NUMBER = {3},
PAGES = {381--422},
ISSN = {0167-2789},
CODEN = {PDNPDT},
MRCLASS = {58F05 (82A60)},
MRNUMBER = {0719634},
MRREVIEWER = {Dieter H. Mayer},
DOI = {10.1016/0167-2789(83)90233-6},
} -
[Bi-Ma] M. L. Bialy and R. S. MacKay, "Symplectic twist maps without conjugate points," Israel J. Math., vol. 141, pp. 235-247, 2004.
@article {Bi-Ma, MRKEY = {2063035},
AUTHOR = {Bialy, M. L. and MacKay, R. S.},
TITLE = {Symplectic twist maps without conjugate points},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {141},
YEAR = {2004},
PAGES = {235--247},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {37E40 (37J10)},
MRNUMBER = {2063035},
MRREVIEWER = {Christophe Gol{é}},
DOI = {10.1007/BF02772221},
ZBLNUMBER = {1055.37060},
} -
[Bir2] G. D. Birkhoff, "Proof of Poincaré’s geometric theorem," Trans. Amer. Math. Soc., vol. 14, iss. 1, pp. 14-22, 1913.
@article {Bir2, MRKEY = {1500933},
AUTHOR = {Birkhoff, George D.},
TITLE = {Proof of {P}oincaré's geometric theorem},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {14},
YEAR = {1913},
NUMBER = {1},
PAGES = {14--22},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {37C27 (34C99 54H25)},
MRNUMBER = {1500933},
JFMNUMBER={44.0761.01},
DOI = {10.2307/1988766},
} -
[Ch1] A. Chenciner, "La dynamique au voisinage d’un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather," in Seminar Bourbaki, 1983/84, , 1985, vol. 121–122, pp. 147-170.
@incollection {Ch1, MRKEY = {0768958},
AUTHOR = {Chenciner, Alain},
TITLE = {La dynamique au voisinage d'un point fixe elliptique conservatif: de {P}oincaré et {B}irkhoff à {A}ubry et {M}ather},
BOOKTITLE = {Seminar Bourbaki, 1983/84},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {121--122},
YEAR = {1985},
PAGES = {147--170},
ISSN = {0303-1179},
MRCLASS = {58F05 (58F22)},
MRNUMBER = {0768958},
MRREVIEWER = {A. Vanderbauwhede},
ZBLNUMBER = {0582.58013},
} -
[C-I] G. Contreras and R. Iturriaga, "Convex Hamiltonians without conjugate points," Ergodic Theory Dynam. Systems, vol. 19, iss. 4, pp. 901-952, 1999.
@article {C-I, MRKEY = {1709426},
AUTHOR = {Contreras, Gonzalo and Iturriaga, Renato},
TITLE = {Convex {H}amiltonians without conjugate points},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {19},
YEAR = {1999},
NUMBER = {4},
PAGES = {901--952},
ISSN = {0143-3857},
MRCLASS = {37J99 (37D25 37D40 37J50)},
MRNUMBER = {1709426},
MRREVIEWER = {Thierry Barbot},
DOI = {10.1017/S014338579913387X},
ZBLNUMBER = {1044.37046},
} -
[F-L] C. Falcolini and R. de la Llave, "A rigorous partial justification of Greene’s criterion," J. Statist. Phys., vol. 67, iss. 3-4, pp. 609-643, 1992.
@article {F-L, MRKEY = {1171146},
AUTHOR = {Falcolini, Corrado and de la Llave, Rafael},
TITLE = {A rigorous partial justification of {G}reene's criterion},
JOURNAL = {J. Statist. Phys.},
FJOURNAL = {Journal of Statistical Physics},
VOLUME = {67},
YEAR = {1992},
NUMBER = {3-4},
PAGES = {609--643},
ISSN = {0022-4715},
CODEN = {JSTPSB},
MRCLASS = {58F27 (70H05)},
MRNUMBER = {1171146},
MRREVIEWER = {Alessandra Celletti},
ZBLNUMBER={0892.58045},
} -
[F-M] J. N. Mather and G. Forni, "Action minimizing orbits in Hamiltonian systems," in Transition to Chaos in Classical and Quantum Mechanics, New York: Springer-Verlag, 1994, vol. 1589, pp. 92-186.
@incollection {F-M, MRKEY = {1323222},
AUTHOR = {Mather, John N. and Forni, Giovanni},
TITLE = {Action minimizing orbits in {H}amiltonian systems},
BOOKTITLE = {Transition to Chaos in Classical and Quantum Mechanics},
VENUE={{M}ontecatini {T}erme, 1991},
SERIES = {Lecture Notes in Math.},
VOLUME = {1589},
PAGES = {92--186},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
MRCLASS = {58F05 (58F27)},
MRNUMBER = {1323222},
MRREVIEWER = {F. Cardin},
DOI = {10.1007/BFb0074076},
ZBLNUMBER = {0822.70011},
} -
[Fo1] P. Foulon, "Estimation de l’entropie des systèmes lagrangiens sans points conjugués," Ann. Inst. H. Poincaré Phys. Théor., vol. 57, iss. 2, pp. 117-146, 1992.
@article {Fo1, MRKEY = {1184886},
AUTHOR = {Foulon, P.},
TITLE = {Estimation de l'entropie des systèmes lagrangiens sans points conjugués},
NOTE = {With an appendix, ``About Finsler geometry'', in English},
JOURNAL = {Ann. Inst. H. Poincaré Phys. Théor.},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Physique Théorique},
VOLUME = {57},
YEAR = {1992},
NUMBER = {2},
PAGES = {117--146},
ISSN = {0246-0211},
CODEN = {AHPAAO},
MRCLASS = {58F17 (58F05 58F11 58F15)},
MRNUMBER = {1184886},
MRREVIEWER = {Viviane Baladi},
URL = {http://www.numdam.org/item?id=AIHPA_1992__57_2_117_0},
ZBLNUMBER = {0806.58029},
} -
[Fur] A. Furman, "On the multiplicative ergodic theorem for uniquely ergodic systems," Ann. Inst. H. Poincaré Probab. Statist., vol. 33, iss. 6, pp. 797-815, 1997.
@article {Fur, MRKEY = {1484541},
AUTHOR = {Furman, Alex},
TITLE = {On the multiplicative ergodic theorem for uniquely ergodic systems},
JOURNAL = {Ann. Inst. H. Poincaré Probab. Statist.},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Probabilités et Statistiques},
VOLUME = {33},
YEAR = {1997},
NUMBER = {6},
PAGES = {797--815},
ISSN = {0246-0203},
CODEN = {AHPBAR},
MRCLASS = {28D05},
MRNUMBER = {1484541},
MRREVIEWER = {Oliver Knill},
DOI = {10.1016/S0246-0203(97)80113-6},
ZBLNUMBER = {0892.60011},
} -
[Go1] C. Golé, Symplectic Twist Maps. Global Variational Techniques, River Edge, NJ: World Scientific Publishing Co., 2001, vol. 18.
@book {Go1, MRKEY = {1992005},
AUTHOR = {Gol{é},
Christophe},
TITLE = {Symplectic Twist Maps. Global Variational Techniques},
SERIES = {Adv. Ser. Nonlinear Dynam.},
VOLUME = {18},
PUBLISHER = {World Scientific Publishing Co.},
ADDRESS = {River Edge, NJ},
YEAR = {2001},
PAGES = {xviii+305},
ISBN = {981-02-0589-9},
MRCLASS = {37J10 (37E40 37E45 37J45 53D05 70H08)},
MRNUMBER = {1992005},
MRREVIEWER = {Hansj{ö}rg Geiges},
DOI = {10.1142/9789812810762},
ZBLNUMBER = {01710304},
} -
[Gr] L. W. Green, "A theorem of E. Hopf," Michigan Math. J., vol. 5, pp. 31-34, 1958.
@article {Gr, MRKEY = {0097833},
AUTHOR = {Green, L. W.},
TITLE = {A theorem of {E}. {H}opf},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {5},
YEAR = {1958},
PAGES = {31--34},
ISSN = {0026-2285},
MRCLASS = {53.00},
MRNUMBER = {0097833},
MRREVIEWER = {W. M. Boothby},
DOI = {10.1307/mmj/1028998009},
ZBLNUMBER = {0134.39601},
} -
[Gre] J. M. Greene, "A method for determining a stochastic transition," J. Math. Phys., vol. 20, pp. 1183-1201, 1979.
@article{Gre,
author={Greene, J. M.},
TITLE={A method for determining a stochastic transition},
JOURNAL={J. Math. Phys.},
VOLUME={20},
PAGES={1183--1201},
YEAR={1979},
DOI = {10.1063/1.524170},
} -
[H-K] B. Hasselblatt and A. Katok, A First Course in Dynamics. With a Panorama of Recent Developments, New York: Cambridge Univ. Press, 2003.
@book {H-K, MRKEY = {1995704},
AUTHOR = {Hasselblatt, Boris and Katok, Anatole},
TITLE = {A First Course in Dynamics. With a Panorama of Recent Developments},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {New York},
YEAR = {2003},
PAGES = {x+424},
ISBN = {0-521-58304-7; 0-521-58750-6},
MRCLASS = {37-01 (37-02)},
MRNUMBER = {1995704},
MRREVIEWER = {M. I. Brin},
ZBLNUMBER = {1027.37001},
} -
[He1] M. Herman, "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations," Inst. Hautes Études Sci. Publ. Math., iss. 49, pp. 5-233, 1979.
@article {He1, MRKEY = {0538680},
AUTHOR = {Herman, Michael-Robert},
TITLE = {Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {49},
YEAR = {1979},
PAGES = {5--233},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {58F11 (28D99)},
MRNUMBER = {0538680},
MRREVIEWER = {C. S. Hartzman},
URL = {http://www.numdam.org/item?id=PMIHES_1979__49__5_0},
ZBLNUMBER = {0448.58019},
} -
[He2] M. Herman, "Sur les courbes invariantes par les difféomorphismes de l’anneau," Astérisque, vol. 103–104, 1983.
@article {He2, MRKEY = {0499079},
AUTHOR = {Herman, Michael-Robert},
TITLE = {Sur les courbes invariantes par les difféomorphismes de l'anneau},
JOURNAL={Astérisque},
VOLUME={103--104},
YEAR={1983},
MRNUMBER={0728564},
ZBLNUMBER={0532.58011},
} -
[It1] R. Iturriaga, "A geometric proof of the existence of the Green bundles," Proc. Amer. Math. Soc., vol. 130, iss. 8, pp. 2311-2312, 2002.
@article {It1, MRKEY = {1896413},
AUTHOR = {Iturriaga, Renato},
TITLE = {A geometric proof of the existence of the {G}reen bundles},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {130},
YEAR = {2002},
NUMBER = {8},
PAGES = {2311--2312},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {37D40 (37D10 53D25)},
MRNUMBER = {1896413},
DOI = {10.1090/S0002-9939-02-06540-1},
ZBLNUMBER = {1067.37037},
} -
[Ko] A. N. Kolmogorov, "On conservation of conditionally periodic motions for a small change in Hamilton’s function," Dokl. Akad. Nauk SSSR, vol. 98, pp. 527-530, 1954.
@article {Ko, MRKEY = {0068687},
AUTHOR = {Kolmogorov, A. N.},
TITLE = {On conservation of conditionally periodic motions for a small change in {H}amilton's function},
JOURNAL = {Dokl. Akad. Nauk SSSR},
FJOURNAL = {Doklady Akademii Nauk SSSR},
VOLUME = {98},
YEAR = {1954},
PAGES = {527--530},
ISSN = {0002-3264},
MRCLASS = {36.0X},
MRNUMBER = {0068687},
MRREVIEWER = {Y. N. Dowker},
ZBLNUMBER={0056.31502},
} -
[Ko-Fo] A. N. Kolmogorov, S. Fomine, and V. M. Tihomirov, Eléments de la théorie des fonctions et de l’analyse fonctionnelle, Moscow: Éditions Mir, 1974.
@book {Ko-Fo, MRKEY = {0367598},
AUTHOR = {Kolmogorov, A. N. and Fomine, S. and Tihomirov, V. M.},
TITLE = {Eléments de la théorie des fonctions et de l'analyse fonctionnelle},
NOTE = {Avec un compl{é}ment sur les alg{è}bres de Banach, par V. M. Tikhomirov, Traduit du russe par Michel Dragnev},
PUBLISHER = {Éditions Mir},
ADDRESS = {Moscow},
YEAR = {1974},
PAGES = {536},
MRCLASS = {46-01 (28-01)},
MRNUMBER = {0367598},
ZBLNUMBER = {0299.46001},
} -
[Lc1] P. Le Calvez, "Propriétés dynamiques des difféomorphismes de l’anneau et du tore," Astérisque, vol. 204, p. 131, 1991.
@article {Lc1, MRKEY = {1183304},
AUTHOR = {Le Calvez, Patrice},
TITLE = {Propriétés dynamiques des difféomorphismes de l'anneau et du tore},
JOURNAL = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {204},
YEAR = {1991},
PAGES = {131},
ISSN = {0303-1179},
MRCLASS = {58F12 (58F27)},
MRNUMBER = {1183304},
MRREVIEWER = {Jaroslav Stark},
ZBLNUMBER = {0784.58033},
} -
[LC2] P. Le Calvez, "Les ensembles d’Aubry-Mather d’un difféomorphisme conservatif de l’anneau déviant la verticale sont en général hyperboliques," C. R. Acad. Sci. Paris Sér. I Math., vol. 306, iss. 1, pp. 51-54, 1988.
@article {LC2, MRKEY = {0929278},
AUTHOR = {Le Calvez, Patrice},
TITLE = {Les ensembles d'{A}ubry-{M}ather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques},
JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {306},
YEAR = {1988},
NUMBER = {1},
PAGES = {51--54},
ISSN = {0249-6291},
CODEN = {CASMEI},
MRCLASS = {58F15},
MRNUMBER = {0929278},
MRREVIEWER = {Hans G. Bothe},
ZBLNUMBER={0639.58005},
} -
[Mac1] R. S. MacKay, "Hyperbolic cantori have dimension zero," J. Phys. A, vol. 20, iss. 9, p. l559-l561, 1987.
@article {Mac1, MRKEY = {0895050},
AUTHOR = {MacKay, R. S.},
TITLE = {Hyperbolic cantori have dimension zero},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics. A. Mathematical and General},
VOLUME = {20},
YEAR = {1987},
NUMBER = {9},
PAGES = {L559--L561},
ISSN = {0305-4470},
CODEN = {JPHAC5},
MRCLASS = {58F13},
MRNUMBER = {0895050},
URL = {http://stacks.iop.org/0305-4470/20/L559},
ZBLNUMBER = {0623.58031},
} -
[Mac2] R. S. MacKay, "Greene’s residue criterion," Nonlinearity, vol. 5, iss. 1, pp. 161-187, 1992.
@article {Mac2, MRKEY = {1148794},
AUTHOR = {MacKay, R. S.},
TITLE = {Greene's residue criterion},
JOURNAL = {Nonlinearity},
FJOURNAL = {Nonlinearity},
VOLUME = {5},
YEAR = {1992},
NUMBER = {1},
PAGES = {161--187},
ISSN = {0951-7715},
CODEN = {NONLE5},
MRCLASS = {58F27 (34C99 58F22 70H05 70K99)},
MRNUMBER = {1148794},
URL = {http://stacks.iop.org/0951-7715/5/161},
ZBLNUMBER = {0749.58036},
} -
[MMP] R. S. MacKay, J. D. Meiss, and I. C. Percival, "Resonances in area-preserving maps," Phys. D, vol. 27, iss. 1-2, pp. 1-20, 1987.
@article {MMP, MRKEY = {0912848},
AUTHOR = {MacKay, R. S. and Meiss, J. D. and Percival, I. C.},
TITLE = {Resonances in area-preserving maps},
JOURNAL = {Phys. D},
FJOURNAL = {Physica D. Nonlinear Phenomena},
VOLUME = {27},
YEAR = {1987},
NUMBER = {1-2},
PAGES = {1--20},
ISSN = {0167-2789},
CODEN = {PDNPDT},
MRCLASS = {58F20 (58F08 58F15)},
MRNUMBER = {0912848},
DOI = {10.1016/0167-2789(87)90002-9},
ZBLNUMBER = {0622.58016},
} -
[Man1] R. Mañé, "Quasi-Anosov diffeomorphisms and hyperbolic manifolds," Trans. Amer. Math. Soc., vol. 229, pp. 351-370, 1977.
@article {Man1, MRKEY = {0482849},
AUTHOR = {Ma{ñ}{é},
Ricardo},
TITLE = {Quasi-{A}nosov diffeomorphisms and hyperbolic manifolds},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {229},
YEAR = {1977},
PAGES = {351--370},
ISSN = {0002-9947},
MRCLASS = {58F15},
MRNUMBER = {0482849},
MRREVIEWER = {Zbigniew Nitecki},
DOI = {10.2307/1998515},
ZBLNUMBER = {0356.58009},
} -
[Man2] R. Mañé, Ergodic Theory and Differentiable Dynamics, New York: Springer-Verlag, 1987, vol. 8.
@book {Man2, MRKEY = {0889254},
AUTHOR = {Ma{ñ}{é},
Ricardo},
TITLE = {Ergodic Theory and Differentiable Dynamics},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {8},
NOTE = {translated from the Portuguese by Silvio Levy},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1987},
PAGES = {xii+317},
ISBN = {3-540-15278-4},
MRCLASS = {58F11 (28Dxx)},
MRNUMBER = {0889254},
ZBLNUMBER = {0616.28007},
} -
[Mat1] J. N. Mather, "Existence of quasi-periodic orbits for twist homeomorphisms of the annulus," Topology, vol. 21, iss. 4, pp. 457-467, 1982.
@article {Mat1, MRKEY = {0670747},
AUTHOR = {Mather, John N.},
TITLE = {Existence of quasi-periodic orbits for twist homeomorphisms of the annulus},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {21},
YEAR = {1982},
NUMBER = {4},
PAGES = {457--467},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {58F15},
MRNUMBER = {0670747},
MRREVIEWER = {Robert L. Devaney},
DOI = {10.1016/0040-9383(82)90023-4},
ZBLNUMBER={0506.58032},
} -
[Mo] J. Moser, "On invariant curves of area-preserving mappings of an annulus," Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, vol. 1962, pp. 1-20, 1962.
@article {Mo, MRKEY = {0147741},
AUTHOR = {Moser, J.},
TITLE = {On invariant curves of area-preserving mappings of an annulus},
JOURNAL = {Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II},
FJOURNAL = {Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse},
VOLUME = {1962},
YEAR = {1962},
PAGES = {1--20},
ISSN = {0065-5295},
MRCLASS = {34.65 (57.48)},
MRNUMBER = {0147741},
MRREVIEWER = {H. Ehrmann},
ZBLNUMBER = {0107.29301},
} -
[Po] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge: Cambridge Univ. Press, 1993, vol. 180.
@book {Po, MRKEY = {1215938},
AUTHOR = {Pollicott, Mark},
TITLE = {Lectures on Ergodic Theory and {P}esin Theory on Compact Manifolds},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {180},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1993},
PAGES = {x+162},
ISBN = {0-521-43593-5},
MRCLASS = {58F11},
MRNUMBER = {1215938},
MRREVIEWER = {Nicola{\u\i} T. A. Haydn},
ZBLNUMBER = {0772.58001},
} -
[Ru] H. Rüssmann, "On the existence of invariant curves of twist mappings of an annulus," in Geometric Dynamics, New York: Springer-Verlag, 1983, vol. 1007, pp. 677-718.
@incollection {Ru, MRKEY = {0730294},
AUTHOR = {R{ü}ssmann, Helmut},
TITLE = {On the existence of invariant curves of twist mappings of an annulus},
BOOKTITLE = {Geometric Dynamics},
VENUE={{R}io de {J}aneiro, 1981},
SERIES = {Lecture Notes in Math.},
VOLUME = {1007},
PAGES = {677--718},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
MRCLASS = {58F05 (58F10)},
MRNUMBER = {0730294},
MRREVIEWER = {Dietrich Flockerzi},
DOI = {10.1007/BFb0061441},
ZBLNUMBER={0531.58041},
} -
[Shu] M. Shub, Stabilité Globale des Systèmes Dynamiques, Paris: Société Mathématique de France, 1978, vol. 56.
@book {Shu, MRKEY = {0513592},
AUTHOR = {Shub, Michael},
TITLE = {Stabilité Globale des Systèmes Dynamiques},
SERIES = {Astérisque},
VOLUME = {56},
NOTE = {with an English preface and summary},
PUBLISHER = {Société Mathématique de France},
ADDRESS = {Paris},
YEAR = {1978},
PAGES = {iv+211},
MRCLASS = {58F10 (34C35 58F20)},
MRNUMBER = {0513592},
MRREVIEWER = {John Franke},
ZBLNUMBER = {0396.58014},
} -
[Sib] K. F. Siburg, The principle of Least Action in Geometry and Dynamics, New York: Springer-Verlag, 2004, vol. 1844.
@book {Sib, MRKEY = {2076302},
AUTHOR = {Siburg, Karl Friedrich},
TITLE = {The principle of Least Action in Geometry and Dynamics},
SERIES = {Lecture Notes in Math.},
VOLUME = {1844},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {xii+128},
ISBN = {3-540-21944-7},
MRCLASS = {37J50 (37E40 37J10 53D35 58E30)},
MRNUMBER = {2076302},
MRREVIEWER = {Renato Iturriaga},
ZBLNUMBER={1060.37048},
}
Print version