Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities
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Index 351
L-stable, 143
La Salle’s principle, 141
Lagrangian, 187
lattice, 229
Lebesgue, proposition by, 314
Leray–Schauder degree, 301
Lie bracket, 201
Lie derivative
of a form, 198
of a vector field, 201
line bundle, canonical, 246
linear bounded vector field, 130
Liouville (proposition), 216
Liouville (theorem), 223
Liouville coordinates, 230
Liouville form, 323
local invariant manifolds, 58
local stable manifold
of the orbit, 121
local straightening out theorem, 133
local unstable manifolds
of the orbit, 121
Lusternik–Schnirelman (theorem),
155, 160
Lusternik–Schnirelman category, 158
Lyapunov (theorem), 143
Lyapunov function, 140
Lyapunov-stable, 143
Mañé, .R., 45
Macarini, L., 314
Macarini–Schlenk (theorem), 314
manifold, 203
maximal ergodic lemma, 40
maximal invariant set, 142
McGehee, R., 3
mean value in time, 8
mean value of f over the space, 8
mean value of the function f over
the orbit, 8
measurable, 26
measure preserving, 27
measure space, 26
Melnikov formula, 113
minimal, 14
minimax, 272
minimax principle, 272
Morse (theorem), 167
Morse decomposition, 153
Morse function, 163
Morse index, 163
Morse lemma, 172
Morse polynomial, 165
Moser (theorem), 240
Moser, J., 235
Mountain pass lemma, 274
mountain range, 274
N -body problem, 1
non-squeezing theorem, 256
nondegenerate form, 207
!-limit set, 137
!
-limit set, 137
orbit, 4, 89
oscillation, 260
outer regularity, 155
Painlevé, P., 3
Palais, R. S., 271
parameterized orbit, 4
pendulum
mathematical, 107
periodically excited, 109
with friction, 144
period
minimal, 4
periodic orbit, 94
periodic point, 4, 95, 99
Plancherel identity, 36
planetary system, 235
Poincaré (theorem), 97, 328
Poincaré lemma, 200
Poincaré map, 329
Poincaré polynomial, 166
352 Index
Poisson bracket, 213
positively invariant, 142
potential energy, 323
P.S.-condition, 272
pseudo orbit, 89
pullback, 191, 195
R-integrable, 8
rational approximation, 14
recurrence, 28, 29
recurrence theorem of Poincaré, 28,
29, 245
regularity theorem, 286
reparametrization of time, 135
resonance condition, 236
rest point, 152
rest points, 155
Riemann metric, 161
Riemann–Lebesgue lemma, 35
rotation
irrational, 6
rigid, 37
rigidity, 5–14
rotation number, 97
Saari, D., 3
scalar product, 161
Schlenk, F., 314
semilocally contractible, 159
shadowing lemma, 89
shadowing orbit, 89
shift map, 19, 101
singular point of a vector field, 135
Smale (theorem), 70, 102
Smale, S., 271
spectral proposition for Sp.n/, 191
spectral radius, 48
spectrum, 47
stable manifold, 50
starlike hypersurface, 318
straightening out theorem, local,
see local straightening
out theorem
strictly convex hypersurface, 319
structural stability, 18, 23, 124
of Anosov automorphism, 126
of torus automorphism, 126
structurally stable, 22
subadditivity, 155
sublevel set, 169
symbol, sequence of symbols, 100
symplectic
basis, 185
capacity, 253, 262
coordinates, 185, 207
cylinder, see cylinder
elementary transformation,
see elementary symplectic
transformation
embedding, 241
form, 183
group, 188
linear map, 187, 193
manifold, 206, 210
map, 209
matrix, 188
orthogonal complement, 183
plane, 242
standardspace, 183
structure, 183, 206, 210
subspace, 184
vectorspace, 182
symplectic capacity, 261
theorem of Riesz–Fischer, 36
time T map, 76
topological linking (lemma), 295
topologically conjugated, 22
torus, 11
torus automorphism, 116
torusautomorphismus
hyperbolic, 120
total volume, 239
transitive, 14, 33
transitivity of ergodic systems, 33
Index 353
transitivity theorem by G. Birkhoff, 15
transversal map, 329
transversal section, 329
transversal section map, 333
unparametrized orbit, 4
unstable manifold, 50
Viterbo (corollary), 320
Viterbo, C., 320
Weyl, H., 8
Xia, J., 3