Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities
Подождите немного. Документ загружается.
Bibliography 341
[36] J. Franks and M. Misiurewicz, Topological methods in dynamics. In Handbook of
dynamical systems, Vol. 1A, ed. B. Hasselblatt and A. Katok, North-Holland, Ams-
terdam, 2002, 547–599. 181
[37] H. Fürstenberg, Recurrence in ergodic theory and combinatorial number theory. M.B.
Porter lectures. Princeton University Press, Princeton, NJ, 1981. 45
[38] A. M. Garsia, Topics in almost everywhere convergence. Lectures in Adv. Math. 4,
Markham Publishing, Chicago, Ill., 1970.
[39] C. Genecand, Transversal homoclinic orbits near elliptic fixed points of area-
preserving diffeomorphisms of the plane. In Dynamics reported, Dynam. Report.
Expositions Dynam. Systems (N.S.) 2, Springer-Verlag, Berlin, 1993, 1–30. 238
[40] V. L. Ginzburg, Hamiltonian dynamical systems without periodic orbits. In Northern
California symplectic geometry seminar, ed. by Ya. Eliashberg et al., Amer. Math.
Soc. Transl. Ser. 196, Amer. Math. Soc., Providence, RI, 1999, 35–48. 317
[41] V. L. Ginzburg, The Hamiltonian Seifert conjecture: examples and open problems. In
Proceedings of the third European Congress of Mathematics (Barcelona 2000), Vol.
II, Progr. Math. 202, Birkhäuser, Basel, 2001, 547–555. 338
[42] V. L. Ginzburg, The Weinstein conjecture and theorems of nearby and almost exis-
tence. In The breath of symplectic and Poisson geometry, Festschrift in Honor of Alan
Weinstein, ed. by J. E. Marsden and T. S. Ratiu. Birkhäuser, Boston, Mass., 2005,
139–172. 338
[43] V. L. Ginzburg and B. Z. Gürel, A C
2
-smooth counterexample to the Hamiltonian
Seifert conjecture in R
4
. Ann. of Math. (2) 158 (2003), 953–976. 317
[44] V. L. Ginzburg and B. Z. Gürel, Relative Hofer-Zehnder capacity and periodic orbits
in twisted cotangent bundles. Duke Math. J. 123 (1) (2004), 1–47.
[45] C. Godbillon, Géométrie différentielle et mécanique analytique. Collection Méthodes,
Hermann, Paris, 1969. 238
[46] L. Grafakos, Classical and modern Fourier analysis. Prentice Hall, Upper Saddle
River, NJ, 2004. 35
[47] J. Hadamard, Sur l’iteration et les solutions asymptotiques des équations differen-
tielles. Bull. Soc. Math. France 29 (1901), 224–228. 80
[48] B. Hasselblatt, Hyperbolic dynamical systems. Chapter 5 in Handbook of dynamical
systems, Vol. 1 A, ed. by B. Hasselblatt and A. Katok, North-Holland, Amsterdam,
2002. 126
[49] B. Hasselblatt and A. Katok, Introduction to the modern theory of dynamical systems.
Encyclopedia Math. Appl. 54, Cambridge University Press, Cambridge, 1995. 45,
120, 123, 126
[50] B. Hasselblatt and A. Katok (eds.), Handbook of dynamical systems. Vol. 1A, North-
Holland, Amsterdam, 2002. 45
[51] H. Hofer and E. Zehnder, A new capacity for symplectic manifolds. In Analysis, et
cetera, ed. by P. Rabinowitz and E. Zehnder, Academic Press, Boston, Mass., 1990,
405–427 260
342 Bibliography
[52] H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics. Birk-
häuser Adv. Texts, Birkhäuser, Basel, 1994. vii, 244, 255, 262, 297, 299, 306, 314,
317, 338
[53] M. C. Irwin, Smooth dynamical systems. Academic Press, 1980. 80
[54] M.Y. Jiang, Hofer–Zehnder symplectic capacity for two dimensional manifolds. Proc.
Roy. Soc. Edinburgh Sect. A 123 (1993), 945–950. 262
[55] J. Jost, Riemannian geometry and geometric analysis. 3rd edition, Universitext, Sprin-
ger-Verlag, Berlin, 2002. 177, 180
[56] R. Jost, Poisson brackets. An unpedagogical lecture. Rev. Mod. Phys. 36 (1964),
572–579. 214
[57] R. Jost, Winkel- und Wirkungsvariable für allgemeine mechanische Systeme. Hel-
vetica Physica Acta 41 (1968), 965–968. 227
[58] A. Katok, Dynamical systems with hyperbolic structure. Three papers on dynamical
systems. Amer. Math. Soc. Transl. 116 (1981), 43–95. 126
[59] U. Kirchgraber and D. Stoffer, Chaotic behaviour in simple dynamical systems. SIAM
Review 32 (1990), 424-452. 115
[60] U. Krengel, Ergodic theorems. De Gruyter Stud. Math. 6, Walter de Gruyter, Berlin,
1985. 45
[61] R. de la Llave, A tutorial on KAM theory. In Smooth ergodic theory and its appli-
cations, ed. by A. Katok et al., Proc. Sympos. Pure Math. 69, Amer. Math. Soc.,
Providence, RI, 2001, 175–292. 238
[62] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist
mappings. In Smooth ergodic theory and its applications, ed. by A. Katok et al., Proc.
Sympos. Pure Math. 69, Amer. Math. Soc., Providence, RI, 2001, 733–746 237
[63] R. Mañé, Ergodic theory and differentiable dynamics. Ergeb. Math. Grenzgeb. 8,
Springer-Verlag, Berlin, 1987. 45
[64] L. Macarini and F. Schlenk, A refinement of the Hofer–Zehnder theorem on the ex-
istence of closed characteristics near a hypersurface. Bull. London Math. Soc. 37 (2)
(2005), 297–300. 314
[65] D. McDuff and D. Salamon, Introduction to symplectic topology. Second edition,
Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York,
NY, 1998. 238, 306
[66] K. R. Meyer and G. R. Hall, Introduction to Hamiltonian dynamical systems and the
N-body problem. Applied Math. Sci. 90, Springer-Verlag, New York, NY, 1992. 238
[67] J. Milnor,
Topology from the differential viewpoint. The University Press of Virginia,
Charlottesville, Va., 1965. 299
[68] J. Milnor, Morse theory. 5th print., Ann. of Math. Stud. 51, Princeton University Press,
Princeton, NJ, 1973. 180
[69] K. Mischaikow and M. Mrozek, The Conley index. In Handbook of dynamical systems,
Vol. 2, ed. by B. Fiedler, North-Holland, Amsterdam, 2002, 393–460. 181
Bibliography 343
[70] J. Moser, On invariant curves of area-preserving mappings of an annulus. Nachr. Akad.
Wiss. Göttingen Math.-Phys. Kl. II 1962 (19562), 1–20. 237
[71] J. Moser, On a theorem of Anosov. J. Differential Equations 5 (1969), 411–440. 126
[72] J. Moser, Dynamical systems – past and present. In Proc. Internat. Congress of Mathe-
maticians (Berlin, 1998), Vol. I, Doc. Math., 1998, 381-402. 235
[73] J. Moser, Stable and random motions in dynamical systems: with special emphasis on
celestial mechanics. Ann. of Math. Stud. 77, Princeton University Press, Princeton,
NJ, 1973. 238
[74] J. Moser and E. Zehnder, Notes on dynamical systems. Courant Lecture Notes in
Mathematics, Courant Institute, New York, NY, 2005. vii, 238, 338
[75] J. C. Oxtoby and S. M. Ulam, Measure preserving homeomorphisms and metrical
transitivity. Ann. of Math. 42 (4) (1941) 874–920.
[76] R. S. Palais, Morse theory on Hilbert manifolds. Topology 2 (1963), 299–340. 181
[77] R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds. Topology 5 (1966),
115–132. 161, 181
[78] R. S. Palais, Critical point theory and the minimax principle. In Global analysis, Proc.
Sympos. Pure Math. 15, Amer. Math. Soc., Providence, RI, 1970, 185–212. 181
[79] R. S. Palais and Chun-lian Terng, Critical point theory and submanifold geometry.
Lecture Notes in Math. 1353, Springer-Verlag, Berlin, 1988. 180
[80] K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal ho-
moclinic points. In Dynamics reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems
Appl. 1, B. G. Teubner and John Wiley and Sons, Stuttgart/Chichester, 1988, 265–306.
126
[81] J. Palis, Jr., and W. de Melo, Geometric theory of dynamical systems: an introduction.
Springer-Verlag, New York, NY, 1982.
[82] O. Perron, Über die Stabilität und asymptotisches Verhalten der Integrale von Differ-
entialgleichungssystemen. Math. Z. 29 (1928), 129–160. 80
[83] Ya. B. Pesin, General theory of smooth hyperbolic dynamical systems. Chapter 7 in
Dynamical systems II, Encyclopedia Math. Sci. 2, Springer-Verlag, Berlin, 1989. 126
[84] Ya. B. Pesin, Lectures on partial hyperbolicity and stable ergodicity. Zürich Lect.Adv.
Math., European Math. Soc. Publishing House, Zürich, 2004. 126
[85] S. Y. Pilyugin, Introduction to structurally stable systems of differential equations.
Birkhäuser, Basel, 1992. 126
[86] J. Pöschel, A lecture on the classical KAM theorem. In Smooth ergodic theory and its
applications, Proc. Sympos. Pure Math. 69, Amer. Math. Soc., Providence, RI, 2001,
707–732. 238
[87] H. Poincaré, Sur le problème des trois corps et les équations de la dynamique. Mémoire
couronné du prix de S. M. le roi Oscar III de Suède, Acta Math. 13 (1890), 1–270. 80
[88] M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds.
London Math. Soc. Lecture Note Ser. 180, Cambridge University Press, Cambridge,
1993. 45
344 Bibliography
[89] P. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math.
31 (1978), 157–184. 270
[90] P. Rabinowitz, Periodic solutions of Hamiltonian systems on a prescribed energy
surface. J. Differential Equations 33 (1979), 336–352. 270, 320
[91] C. Robinson, Dynamical systems: stability, symbolic dynamics, and chaos. 2nd edi-
tion, Stud. Adv. Math., CRC Press, Boca Raton, 1999. 45, 114, 123
[92] H. L. Royden, Real analysis. 2nd edition, Macmillan Publishing, New York, NY,
1968.
[93] D. Ruelle, Elements of differentiable dynamics and bifurcation theory. Academic
Press, Boston, Mass., 1989.
[94] K. P. Rybakowski, The homotopy index and partial differential equations. Universitext,
Springer-Verlag, New York, NY, 1987. 181
[95] D. G. Saari and J. Xia, Off to infinity in finite time. Notices Amer. Math. Soc. 42 (5)
(1995), 538–546. 3
[96] D. Salamon, Connected simple systems and the Conley index of isolated invariant
sets. Trans. Amer. Math. Soc. 291 (1985), 1–41. 181
[97] D. Salamon, The Kolmogorov–Arnold–Moser theorem. Math. Phys. Electron. J. 10
(2004), Paper 3, 37 pp. 238
[98] D. Salamon and E. Zehnder, KAM theory in configuration space. Comment. Math.
Helv. 64 (1989), 84–132. 238
[99] F. Schlenk, Embedding problems in symplectic geometry. De Gruyter Exp. Math. 40,
Walter de Gruyter, Berlin, 2005. 306
[100] M. Schwarz, Morse homology. Progr. Math. 111, Birkhäuser Verlag, Basel, 1993. 181
[101] J. T. Schwartz, Nonlinear functional analysis. Notes on mathematics and its applica-
tions, Gordon and Breach, New York, NY, 1969. 160, 161, 180
[102] L. P. Shil’nikov, Homoclinic trajectories: From Poincaré to the present. In Mathemat-
ical events of the twentieth century, ed. by A. A. Bolibruch, Yu. S. Osipov and Ya. G.
Sinai, Springer-Verlag, Berlin, 2006, 347–370. 80
[103] M. Shub, Global stability of dynamical systems. Springer-Verlag, New York, 1987.
123
[104] K. F. Siburg, Symplectic capacities in two dimensions. Manuscripta Math. 78 (1993),
149–163. 262
[105] C. L. Siegel, On the integrals of canonical systems. Ann. of Math. 42 (3) (1941),
806–822.
[106] C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Diffe-
rentialgleichungen in der Nähe einer Gleichgewichtslösung. Math. Ann. 128 (1954),
144–170.
[107] C. L. Siegel and J. Moser, Lectures on celestial mechanics. Classics Math., Springer-
Verlag, Berlin, 1995. 238
Bibliography 345
[108] Ya. G. Sinai, Introduction to ergodic theory. Math. Notes 18, Princeton University
Press, Princeton, NJ, 1977. 45
[109] Ya. G. Sinai, Topics in ergodic theory. Princeton Math. Ser. 44, Princeton University
Press, Princeton, NJ, 1994 45
[110] S. Smale, Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967),
747–817.
[111] K. R. Stromberg, An introduction to classical real analysis. Wadsworth and Brooks
Cole, Pacific Grove, Cal., 1981. 35
[112] M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every
energy surface. Bol. Soc. Bras. Mat. 20 (1990), 49–58. 317
[113] W. Szlenk, An introduction to the theory of smooth dynamical systems. PWN - Polish
Scientific Publishers/Wiley, Warszawa/Chichester, 1984. 23, 45, 126
[114] C. Viterbo, A proof of Weinstein’s conjecture in R
2n
. Ann. Inst. H. Poincaré Anal.
Non Linéaire 4 (4) (1987), 337–356. 320
[115] P. Walters, An introduction to ergodic theory. Grad. Texts in Math. 79, Springer-Verlag,
New York, NY, 2000. 45
[116] A. Weinstein, Periodic orbits for convex Hamiltonian systems. Ann. of Math. 108
(1978), 507–518. 320
[117] A. Weinstein, On the hypothesis of Rabinowitz’s periodic orbit theorems. J. Differen-
tial Equations 33 (1979), 353–358. 318
[118] J. Xia. Homoclinic points for area preserving surface diffeomorphisms. Preprint 2006,
arXiv:math/0606291. 80
[119] J. C. Yoccoz, An Introduction to hyperbolic dynamics in real and complex dynamical
systems. Nato ASI Series Ser. C 464 (1995), 265–291. 80, 126
[120] E. Zehnder, Homoclinic points near elliptic fixed points. Comm. Pure Appl. Math. 26
(1973), 131–182. 80, 238
List of Symbols
A, 26
A./, 248
A.M /, 140
a.x/, 278
A
˛, 191
˛ ^ ˇ, 190, 195
b.x/, 283
O
b.x/, 283
ˇ
j
, 165
c.f;F/, 272
c.M;!/, 253
c
0
.M; !/, 261
cat
M
, 158
.M /, 167
d , 166, 195, 205
E, 277
E
C
;E
, 47
E
C
x
, E
x
, 81
E
?
, 183
N
f , 8
f
.z/, 8
fF;Gg, 213
G.M; !/, 258
H
j
.M /, 165
H
s
.S
1
; R
2n
/, 277
H.M; !/, 260
H
a
.M; !/, 261
I.M/, 142
i
X
˛, 198
J , 183
j , 281
j
, 281
K, 152
K
a
, 153
L, L.X; A;m/, 26
L
X
˛, 198
L
X
Y , 201
L.X; Y /, L.X/, 46
L
S
, 246
, 247, 323
M
a
, 169
M
a
b
, 169
m
j
, 165
m.H /, 260
m.t; V /, 165
.x
k
/, 163
O.x/, O
C
.x/, O
.x/, 5
k
, 194, 205
!, 206
!.x/, 137
!
.x/, 137
P.t;M/, 166
P
˙
p
, 82
ˆ.x/, 269
Q, 58
.R
2n
;!
0
/, 183
SB.M /, 168
SM.2n/, 252
Sp.n/, 188
†
A
, 100
, 101
.A/, 47
T
p
M , 204
t
.x/, t
C
.x/, 127
V
a
, 153
P
V , 140
W
C
, W
, 50
z
W
C
, 71
W
C
loc
, W
loc
, 58
V
k
.R
d
/, 190
X
H
, 211
ŒX; Y , 201
x t , 136
Z.r/, 242
Index
action, 226
action functional of classical
mechanics, 269
action of a closed curve, 248
action variable, 226
action–angle variables, 226
adapted norm, 49, 88
admissible function, 260
alphabet, finite, 99, 102
angle variable, 226
Anosov diffeomorphism, 116
Anosov, D., 89
Arnold, V. I., 227
Arnold–Jost (theorem), 227
asymptotically stable, 143
atlas, 203
attracted, 140
attractor, 140
Baire category theorem, 16
Bernoulli system, 99, 101
Betti numbers, 165
Birkhoff (theorem), 96
Birkhoff, G., 30
Bolotin, S. V., 327
bounded vector field, 129
Brouwer mapping degree, 298
canonical line bundle, see line bundle,
canonical, 246
capacity, see symplectic
capacity, dynamic, see dynamic capacity
Carleson, L., 36
Cartan, 199
Cartan’s formula, 199
characteristic, 247
characteristic function , 8
characteristic, closed,
see closed characteristic
chart, 203
chart transformation, 204
closed characteristic, 247
closed form, 207
closing lemma by Anosov, 94
coercive, 142
coisotropic, 187
commuting vector fields, 203
compact map, 279
constant orbit, 152
contact type, 318–320, 323
contractible in M , 157
contraction, 4
contraction principle, 4
critical point, 152, 162
nondegenerate, 163
critical value of the function V , 153
cylinder
isotropic, 241
symplectic, 242
Dacorogna–Moser (theorem), 239
Darboux (theorem), 207
Darboux chart, 207
Darboux coordinates, 207
de Rham, 165
de Rham cohomology algebra, 166
deformation, 154
deformation lemma, 153, 170
deformation monotonicity, 155
deformation trick by R. Thom and
J. Moser, 172, 207
ı-shadowing orbit, see shadowing orbit
derivative of a Lyapunov function
along the flow, 140
differential forms, 194
Diophantine, 236
Dirichlet, pigeon hole principle, 7
350 Index
domain of attraction, 140
dual basis, 189
Ekeland, I., 244
Ekeland, I. and Hofer, H., vi
elementary symplectic
transformations, 220
Eliashberg–Gromov–Ekeland–Hofer
(theorem), 244
ellipsoid, 255
energy conservation, 215
energy surface, 216
"-pseudo orbit, see pseudo orbit
equidistribution (corollary), 13, 32
equidistribution theorem, 8, 13
equilibrium point of a vector field, 135
ergodic, 30, 32, 33, 237
ergodic theorem, 30
ergodic theorem criterion, 121
ergodicity criterion, 31
ergodicity, criterion for, 30
Euler characteristic, 167
expansion, 23, 38
expansive, 101
exterior derivative, 166
exterior derivative, 195, 205
exterior product, 190
Fermi, E., 237
Floquet multipliers, 328
flow on a compact metric space M , 151
Fourier coefficients, 35
Fourier series, 36
Garsia, A. M., 40
generating function, 220
global flow (or a complete flow), 128
gradientlike flow, 152, 160
Gromov width, 258
Gromov, M., 255
Gronwall (lemma), 130
Hadamard–Perron (theorem), 59
Hamiltonian function, 211
Hamiltonian vector field, 211
Hartman–Grobman (theorem),
53
Hesse matrix, 163
heteroclinic point, 79, 99
transversal, 79, 111
Hofer, H., 244
Hofer–Zehnder (theorem), 310
Hofer–Zehnder capacity, 261
Hofer-Zehnder capacity, vi
homoclinic orbit, 74
homoclinic point, 74, 95, 97, 99
transversal, 74, 95, 97
Hunt, R.A., 37
hyperbolic
fixed point, 47, 335
isomorphism, 48
hyperbolic set, 81
immersed manifolds, 71
immersion, 70
index function, 155
integrable, 26
integrable system, 225
integrable vector field, 225
integral, 225, 337
interior product, 198
invariant, 17
invariant splitting, 47
invariant torus, 236
isotropic, 187
isotropic cylinder, see cylinder
Jost, R., 227
KAM theory, 237
k-form, 190, 194, 205
kinetic energy, 323
Kolmogorov, A. N., 36
Kronecker, density (corollary), 13
Kronecker, rational approximation
(corollary), 14