Index

C^{1} 15, 134 C^{r} 15, 136 C^{r}(M, N) 265 Cont(n, ℝ) 123 Diff^{r}(M) 265 Diff^{r}(ℝ^{k}) 136 𝔼^{c} 113, 159 𝔼^{s} 113, 159 𝔼_{p}^{s} 269, 391 𝔼^{u} 113, 159 𝔼_{p}^{u} 269, 391 Fix(f) 15 L(ℝ^{k}, ℝ^{n}) 96 Lip(f) 134 m(A) 96, 140, 161 ℕ 30 Per(f) 450 Per(n, f) 15, 450 Per_{h}(f) 450 Per_{h}(k, f) 450 X^{r}(M) 268

α Cantor set 27 α-limit point 23, 148 α-limit set 23, 148 α-recurrent 25, 151 accessible points 308 adapted metric 184 adapted norm 110, 118, 153, 159, 184,

270 adding machine 92 adjacency matrix 276 admissible metric 470 affine conjugacy 42 Andronov-Hopf bifurcation for diffeo-

morphisms 257 Andronov-Hopf bifurcation for flows

251 Anosov Closing Lemma 418 Anosov diffeomorphism 308, 477 asymptotic 16 asymptotically stable 16, 108, 158 attracting 16, 108, 112, 152, 158, 170 attracting fixed point 118 attracting set 326, 404 attractor 326 attractor of Lorenz type 353 Axiom A 419

backward orbit 16 Baker’s map 61 Baker’s map, generalized 400 base space 469 basic sets 423 basin boundary 307 basin of attraction 152, 158, 207, 307 Benedicks and Carleson 343 Birkhoff center 25, 151 Birkhoff Ergodic Theorem 89, 274, 390 Birkhoff normal form 228 Birkhoff Transitivity Theorem 273 Borel measure 273 Borel probability measure 388 Borel sets 388 bounded continuous maps 161 bounded linear map 161 box dimension 394 branched manifold 338, 341, 350 Brusselator 260 bump functions 166 bundle map 469

C^{0}-sup-norm 143 C^{r}-conjugacy 42 C^{r}-diffeomorphism 136 C^{r}-distance between functions 45 C^{r} Section Theorem 472 C^{r} structurally stable 266 C^{r} topology 266 C^{r} topology on vector fields 268 Cantor set 27 capacity dimension 394 center 106, 152 center eigenspace 113, 122 center manifold 201 center-stable manifold 200 center-unstable manifold 200 chain components 26, 151, 404 chain equivalent 26 chain limit set 25, 404 chain recurrent set 26, 151, 404 chain transitive 26 chaos 86, 391 chaotic attractor 327 chaotic map 86 characteristic multipliers 170 closed orbit 148, 168 cobweb plot 18 codimension 453 cones 187, 189, 284, 332 conjugacy 41 conjugate flows 115 conjugate maps 41 Conley’s Fundamental Theorem 404 continuation of a fixed point 160 continuity of solutions with respect to

initial conditions 145 continuous splitting 269 continuously differentiable 15, 134, 136 contracting linear map 118 contraction mapping 141 Contraction Mapping Theorem 141 coordinate chart 263 covering estimate 140 critical point 13 cross section to a flow 168 cycle 356, 425

DA-diffeomorphism 334 Denjoy’s Example 56 Denjoy’s Theorem 56 derivative 134 diffeomorphism 15, 136, 265 differentiable 134, 136 differentiable linearization 156, 160 differentiable map between manifolds

265 differentiably conjugacy 42 differentiably conjugate 368 differential equation on a manifold 268 dimension, box 394 dimension, capacity 394 dimension, Hausdorff 398 dimension, topological 328 disk bundle 470 double of a map 73 doubling map (squaring map) 43, 371 dynamically defined 398

ϵ-chain 25, 151 ϵ-shadow 62, 414 edge subshift 276 eigenvalues of a periodic orbit 170 embedded submanifold 264 embedding 264 entropy, topological 370 equilibrium 148 equivalent flows 115 ergodic 274, 388 eventually periodic 15 eventually positive matrix 76, 125 exact symplectic 226 existence of solutions of differential

equations 143 expanding attractor 328 expanding linear map 118 expansion 112 expansive 85, 418

f-covers 66

family of quadratic maps 20 Feigenbaum constant 82 fiber 469 fiber contraction 469 filtration 441 first return map 168 First Variation Equation 147 fixed point 15, 148, 158 flip bifurcation 246 Floquet theory 203 flow 142 flow box coordinates 206 flow conjugate 205 flow equivalent 115, 205 flow expansiveness 440 flow of a vector field 268 focus 107 forward orbit 16 Frechet derivative 134 full n-shift 275 fundamental domain 45, 46, 47, 49,

116, 119, 444 fundamental matrix solution 102 generalized Baker’s map 400 generalized eigenvector 96 generalized tent map 59 generating function 226 geometric horseshoe 277 geometric model of Lorenz equations

347 global stable manifold 185 global stable manifolds of points 271 global unstable manifold 185 global unstable manifolds of points 271 gradient flow 358 gradient-like 359 gradient-like vector fields 404 gradient vector field 358, 404 graph transform 185, 470 Gronwall’s inequality 144

Hamiltonian differential equations 216 Hamiltonian matrix 220 Hamiltonian vector field 216 Hartman-Grobman Theorem 154, 155,

159, 173 Hausdorff dimension 398 Hausdorff metric 452 Hénon attractor 341 Hénon map 6, 259, 283, 341 heteroclinically related 420 higher derivatives 136 Hölder 135 homeomorphism 15 homoclinic 218 homoclinic point 287 homoclinically related 420 homology zeta function 324 Hopf bifurcation for diffeomorphisms

257 Hopf bifurcation for flows 251 horseshoe 6, 277, 287 h-related 420 hyperbolic 113, 122, 152, 159, 170, 183 hyperbolic estimates 190 hyperbolic fixed point 152 hyperbolic invariant set 269, 272 hyperbolic linear differential equation

113 hyperbolic linear map 122, 183 hyperbolic periodic point 159 hyperbolic splitting 113 hyperbolic structure 269, 272 hyperbolic toral automorphisms 309 immersed submanifold 264 immersion 264 Implicit Function Theorem 137, 138 In Phase Theorem 417 Inclination Lemma 203 index of fixed point 360 infinitesimally symplectic 220 Intermediate Value Theorem 14 invariant measure 84, 388 invariant section 469 Invariant Section Theorem 470 invariant set 23 invariant splitting, differentiability 477 Inverse Function Theorem 140 inverse limit 334 irreducible 275 irreducible matrix 76, 125 isoclines 175, 208 isolated invariant set 290, 414 isolating neighborhood 404, 414 itinerary map 39, 281, 315

Jakobson 84, 344

Jordan canonical form 96

k-cycle 356, 425

knots 353

Kupka-Smale Theorem 450

Lambda Lemma 203

least period 15

Lefschetz Fixed Point Theorem 323 Lefschetz index of a fixed point 323 Lefschetz number of a diffeomorphism

323 Liapunov exponents 86, 88, 388 Liapunov function 157, 359, 404 Liapunov stable 16, 108, 158 Lienard equations 174 lift 50 limit cycle 149, 181 limit set 25, 151, 420 linear conjugacy 42 linear contraction 118 linear expansion 118 linear sink 118 linear source 118 Liouville’s formula 80, 103, 180, 210 Lipschitz 134 local product structure 316, 422, 426 local stable manifold 184 local unstable manifold 184 locally eventually onto 351 Lorenz attractor 347 Lorenz equations 8, 344 L-stable 16

manifold 263 Markov chain, topological 75 Markov partition 316, 426 Markov property for ambient boxes

281, 293 maximal interval of definition of the

solution 145 maximal invariant set 288, 414 Mean Value Theorem 14 measure 388 Melnikov function 304 Melnikov method 7, 304 middle-α Cantor set 27 middle-third Cantor 28 minimal set 25 minimum norm 96, 140, 161, 190 mixing 40 monotonically decreasing 58 monotonically increasing 58 Morse inequalities 360 Morse-Smale diffeomorphism or flow

354 Morse-Smale flow 357 Multiplicative Ergodic Theorem 388 multiplicatively independent 231

(n, ϵ)-separated 370 (n, ϵ)-spanning set 379 negatively invariant set 23 no cycle property 425 node 107 nondegenerate critical point 13, 358 nonhomogeneous linear equations 117 nonnegative matrix 125 nonuniformly hyperbolic structure 387,

391 nonwandering 151 nonwandering point 25 nonwandering set 25 normal bundle 481 normal bundle of a closed orbit 173 normally contracting 481 normally contracting manifold, persis-

tence 481 normally hyperbolic 481 nowhere dense 26 n-shift 38, 275 n-sphere 264

ω-limit point 23, 148 ω-limit set 23, 148 ω-recurrent 25, 151 Ω-Stability Theorem 441 Ω-stable 48, 441 one-sided shift 38 openness of Anosov diffeomorphisms

436 operator norm 96, 160 orbit 16, 146 orbitally Liapunov stable 171 orientation preserving, reversing 119 overflowing 469

partial ordering of basic sets 425 past history 184 perfect 27 period 15, 148, 158 period doubling bifurcation 246 period doubling route to chaos 82 periodic attractor 170 periodic orbit 15, 148, 168 periodic point 15, 148, 158, 168 periodic repeller 170 periodic sink 170 periodic source 170 Perron method for stable manifolds 186 Perron-Frobenius Theorem 125, 377 persistence of a fixed point 160 phase portrait 2, 104, 150 phase space 104 pitchfork bifurcation 259 Plykin attractor 339 Poincaré map 168 Poincaré metric 36 Poincaré norm 36 Poincaré Recurrence Theorem 361 Poincaré-Bendixson Theorem 181 positive matrix 76, 125 positively invariant set 23 predator-prey 180 proper 166

quadratic map 20 quasi-periodic 233 R-stable 441 r-continuously differentiable 136 ρ-distance 34 rectangle (Markov partition) 316, 426 recurrent 25, 151 repelling 16, 112, 152, 159, 170 repelling fixed point 118 repelling set 404 residual subset 273 Riemannian metric 268 Riemannian norm 268 rigid rotation of S^{1} 51 rooftop map 43 rotation number 50 rotation of S^{1} 51

saddle 113, 152, 159 saddle linear differential equation 113 saddle-node bifurcation 238 Schwarz Lemma 35 second derivative 135 sectors 187 semi-conjugacy 41 semi-stability of Anosov Diffeomor-

phisms 435 semi-stable 435 sensitive dependence on initial condi-

tions 85, 418 separated set 370 shadow 62, 414 Sharkovskii ordering 67 Sharkovskii’s Theorem 68 shift space 38, 275, 317 Sinai-Ruelle-Bowen measure 393 sink 16, 112, 152, 159, 170 Smale horseshoe 277, 287 solenoid 328 source 16, 112, 152, 159, 170 spanning set 379 Spectral Decomposition Theorem 422 spectrum of a linear map 183 sphere 264 spiral 107 squaring map (doubling map) 371 SRB measure 393 stability of an hyperbolic invariant set

437 stable bundle of a periodic orbit 202 stable cones 189 stable disk 188 stable eigenspace 113, 122 stable focus 107 stable manifold 185 stable manifold of an invariant set 417 Stable Manifold Theorem 184, 200 Stable Manifold Theorem for a hyper-

bolic set 271, 411 stable node 107 stable set 16 stable spiral 107 stable subbundle 269 stable subspaces 184 stair step method 18 standard map 227 Stefan cycle 69 Stefan transition graph 69 Sternberg linearization 156, 160 strongly gradient-like 404 structural stability 45, 266 structural stability, necssary conditions

461 structural stability, nondensity 464 structural stability of Anosov diffeo-

morphisms 436 structural stability of Anosov flows 438 structural stability of hyperbolic toral

automorphisms 309 structural stability of Morse-Smale dif-

feomorphisms 356 Structural Stability Theorem 443 structurally stable 9, 45, 266 subadditive 63 subshift 75 subshift of finite type 75, 275, 276, 282 sup-norm 96, 143, 160 support 457 surface 264 suspension of a map 173 symbol space 38 symbolic dynamics 4, 38, 275, 281, 317 symplectic diffeomorphism 223 symplectic form 216 symplectic matrix 220

tangent bundle 266 tangent space 266 tangent vector 266 Taylor’s expansion 136 template 353 tent map 43, 58, 59, 61 Thom Transversality Theorem 454 topological conjugacy 39, 41 topological dimension 328 topological entropy 87, 370 topological Markov chain 75 topologically conjugate flows 115 topologically equivalent flows 115 topologically mixing 40, 294 topologically transitive 6, 40, 273 topologically transverse 297 toral Anosov diffeomorphism 308 torus 264 totally disconnected 26 transition graph 68 transition matrix 75, 276, 317 transitive 40, 273 transversal to a flow 168 transversality condition 443 Transversality Theorem, Parametric

454 Transversality Theorem, Thom 454 transverse 203, 354, 453 transverse homoclinic point 287 trapping region 326, 404 tubular neighborhood 483 two-shift 38 two-sided shift space 275

uniform hyperbolic structure 269 uniqueness of solutions of differential

equations 143 unstable cones 189 unstable disk 188 unstable eigenspace 113, 122 unstable focus 107 unstable manifold 185 unstable manifold of an invariant set

417 unstable node 107 unstable set 16 unstable spiral 107 unstable subbundle 269 unstable subspaces 184

Van der Pol equation 5, 174 variation of parameters 117 vector bundle 469 vector field 142, 268 vertex subshift 276 Volterra-Lotka equations 180

weak Liapunov function 157, 359 weak trapping region 405

zeta function 80, 322