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Subject Index of Volume 1A
Subject Index of Volume 1A
– metric, 248 – norm, 247 adjoint representation, 817 admissible – lattice, 919 – manifold, 303 – measure, 967 affine – action, 717 – equivalence, 848 algebra – Pinsker, 79, 212 – semisimple Lie, 673 algebraic – entropy, 39 – group, 676 – hull, 692, 728 – linear representation, 823 – Zd -action, 796 almost k-simple, 673 almost complex structure, 457 almost direct product, 819 almost existence, 1132 1133, 1137 almost-conjugacy, 21, 144 almost-isomorphism, 21, 127, 129 amenable, 15, 16, 81–83, 694, 705 – action, 695, 972, 975 – group, 694 analysis, local, 98 Anosov – action, 754, 755 – alternative, 272 – closing lemma, 112, 138, 142, 148, 149, 268 – cocycle, 289, 291 – diffeomorphism, 132, 133, 144, 145, 248, 278, 284 – element, 747 – flow, 252, 850 – – anomalous, 254 – – geodesic, 473 – obstruction, 290
α-invariant function, 690 α-limit set, 24 δ-distance, 209 δ-shadowed, 563 ε-chain, 550 ε-chainable, 552 λ-lemma, 658 µ-harmonic function, 967 µ-stationary measure, 937, 968 π -simple cocycle, 687 ψ -approximable, 905, 906, 910 ψ -multiplicatively approximable, 914 ω-limit set, 24 Ω-stability, 106, 277, 278 1-form, natural, 456 3-sphere, tight, 1165 Abramov formula, 79 absolute continuity, 262, 266 absolutely continuous spectrum, 72 accessibility, 144, 150, 272, 285, 286 action – amenable, 695, 972, 975 – Anosov, 754, 755 – Bernoulli, 717 – functional, 119, 125 – homogeneous, 828 – induced, 13, 23, 688 – integral, 1095, 1100 – isometric, 718 – local, 723 – measurable, 677 – minimal, 954, 992 – projective, 714 – proper, 683 – proximal, 949 – standard, 751 – strongly proximal, 949, 954 action-angle coordinates, 117, 154 Ad-proper Lie group, 845 adapted 1169
1170
Subject Index of Volume 1A
aperiodic matrix, 334 approximable – ψ -, 905, 906, 910 – ψ -multiplicatively, 914 – badly, 906, 907, 910 – – multiplicatively, 914 – very well, 906, 907 – – multiplicatively, 914 – well, 906, 910 approximable set, 883 arithmetic – group, 676 – lattice, 825 Arnold diffusion, 170, 1122 Artin–Mazur zeta function, 411 asymptotic – behavior, 26 – cycle, 27 – density, 49 – distribution, 56 – flag, 1073 – growth, 274 – limit, 1148 – orbit growth, 32 – to a fixed point, 250 asymptotically harmonic manifold, 493 attractor, 143, 371–373, 386, 387, 392, 400, 402, 551 Aubry–Mather set, 157, 164, 165, 174 Auslander subgroup of a Lie group, 835 automorphism, 251 – K-, 73, 212 average – Birkhoff, 14, 49, 50, 87, 180 – ergodic, 14 Axiom A, 248, 278, 411, 635 badly approximable, 906, 907, 910 – multiplicatively, 914 Baire space, 684 Banach contraction principle, 255 barycenter of a measure, 503 basic set, 132, 248, 326–328, 330, 334, 368, 370– 372, 374, 378, 389, 401, 414, 564 basin, 21, 635 – of attraction, 373, 387, 392, 393, 399 behavior – asymptotic, 26 – stable, 151 Bernoulli – action, 717 – measure, 58, 73, 74, 77 – property, 361, 363, 373, 399
– shift, 58, 80, 361, 1179, 1180 billiards, 187–189, 191–193 birecurrent set, 890 Birkhoff – average, 14, 49, 50, 87, 180 – ergodic theorem, 66, 85 – normal form, 166 – periodic orbit, 157 block – isolating, 98, 557 – form, Lyapunov, 300, 302 blow up, 722 Boltzmann ergodic hypothesis, 242 bootstrap, 265, 291 Borel – density theorem, 685, 824 – G-space, standard, 678 – measure, invariant, 56 Borel–Cantelli – family, 885 – lemma, 886 boundary, 713 – entropy, 980 – Furstenberg, 713, 976 – map, 972 – Poisson, 971 bounded – µ-harmonic function, 967 – distortion property, 622 – geometry, 657 Bowen measure, 95 Bowen–Margulis measure, 282, 492 box mapping, 654 brake orbit, 1101 branchwise equivalence, 657 Brjuno condition, 659 Brouwer’s translation theorem, 1158 bubbling off analysis, 1176 bunching, 253, 262, 264 Busemann – density, 489, 490, 518 – functions, 485 C-map, 628 (C, α)-good, 857 canonical – line bundle, 1132 – representation, 586 – transformation, 114 capacity – c0 , 1138, 1139 – symplectic, 116, 1138 capture of celestial bodies, 249
Subject Index of Volume 1A Cartan – decomposition, 820, 827 – involution, 672 – subalgebra, 673 – subgroup of a Lie group, 819 Cauchy–Riemann equation, 1147, 1182 caustic, 166 CE condition, 644 – topological, 645 CE map, 644 CE2 condition, 644 celestial mechanics, 241 center, 25 – manifold, 141 center-stable manifold, 141 center-unstable manifold, 141 central limit theorem, 362–364, 390, 397, 961, 963, 995 chain – heteroclinic, 1113, 1115 – recurrent, 550 – transitive component, 553 character, Q-, 822 characteristic flow, 121 circle, invariant, 164 classical Hamiltonian system, 1135 classification, 105, 130, 145, 187, 278, 291, 292 – Poincaré, 28 closed geodesic – regular, 520, 527 – singular, 522, 536 closing lemma, 112, 268, 270, 277, 278, 306 – Anosov, 112, 138, 142, 148, 149, 268 – Mañé, 112, 146 – – ergodic, 113 – Pugh, 112, 146, 270 cluster property, 347, 361 co-orientation, 1136 coboundary, 11, 142, 171, 172, 187, 188, 363, 686, 789 cocycle, 11, 62, 108, 142, 162, 171, 187, 297, 686 – Anosov, 289–291 – identity, 681 – Lyapunov, 105, 286 – non-compact, 1002 – Radon–Nikodym, 55, 680 – reduction, 691, 730 – rigidity, 188 – stability, 110 – strongly irreducible, 1002 – superrigidity, 736 – tempered, 299 – Zariski dense, 1002 codimension one, 263, 279, 280
1171
coding, 45, 189 cohomological equation, 354 cohomologous, 162, 352, 354, 364, 385, 387, 388, 686 cohomology, 142 Collet–Eckmann map, 636 commensurability, 675 commensurable subgroups, 824 commensurator, 824 compact – (G, µ)-boundary, 969 – (G, µ)-space, 968 complementary series, 700 complete Lyapunov function, 554 completely – integrable system, 117 – positive entropy, 80 complexity, 1080 – function, 45 component – expansive, 772 – invariant, 585 condition – Diophantine, 642 – Furstenberg, 943, 1002 conditional – entropy, 74, 75 – information function, 75 – measures, 54 cone – criterion, 253 – field, 248, 253, 256 – topology, 478 configuration, 335 – space, 118, 122 conjugacy, 753 – smooth, 103, 105 – topological, 18, 103, 768 conjugate points, 462 conjugation-approximation method, 161, 173 Conley index, 560 – homology, 561 Conley–Zehnder index, 1163, 1166, 1167 connecting lemma, 113, 146, 278 constant – cocycle, 687 – expansivity (expansiveness), 30 – type, 602 contact – form, 120, 1136 – – dynamically convex, 1162 – manifold, 120 – structure, 120, 272, 1136
1172
Subject Index of Volume 1A
– – overtwisted, 1151 – – tight, 1151, 1152 contact type hypersurface, 1134, 1135 continuous – representation, 85 – sum, 701 contractible set, 713 contracting, 948 – (semi)group, 948 – sequence, 948, 956 conull set, 678 copying lemma, Ornstein, 218 correlation – coefficient, 68 – function, 362, 363, 390, 391 correspondence principle, Furstenberg, 46, 85 countable spectrum – Lebesgue, 72, 73, 80, 181 – Plancherel, 717 cover, Markov, 48 critical – exponent, 489 – value, 554 cross-ratio, 612 – inequality, 639 cycle, 581 – asymptotic, 27 – heteroclinic, 1180 cylinder, 40, 333, 345, 354, 356, 359, 367 – orbit, 1145, 1148, 1173 ¯ d-distance, 209, 219 DA-map, 251 Dani – correspondence, 907 – subgroup of a Lie group, 835, 837 DE-map, 107, 250 decay of – correlations, 73, 150, 179, 181, 285, 444 – geometry, 656 decomposition – Cartan, 820, 827 – ergodic, 50, 60, 79, 84, 85, 680, 838, 846 – Iwasawa, 819 – Jordan, 822 – Levi, 820 – polar, 827 – root space, 674 – spectral, 43, 132, 142, 143, 147–149, 271, 279 Denjoy theorem, 103, 153 density – asymptotic, 49 – Busemann, 489, 490, 518
– of Axiom A, 636 derived from expanding, 107, 250 descending chain condition, 800 diffeomorphism, Anosov, 132, 133, 144, 145, 248, 278, 284 differentiable stability, 170 differential of the geodesic flow, 456 dimension group, 785 Diophantine, 158, 162, 163, 165–170, 174, 187, 188 – condition, 642 direct integral, 702 discrete – series, 700 – spectrum, 69, 89, 704, 1003 discretization procedure, 976 disjoint transformations, 213 disk, Siegel, 661 distality, 28–30, 176, 181 distortion, 611 distribution – asymptotic, 56 – invariant, 109, 179, 181, 182, 188 divergence type, 489 domino shift, 777 doubling transformation, 649 dual – lattice, 920 – unitary, 699 – variational method, 1098 dynamical system – Lagrangian, 118 – symbolic, 18, 41 dynamical zeta function, 433 dynamics – elliptic, 151–175 – hyperbolic, 127–151 – parabolic, 175–194 – symbolic, 242 element of a Lie group – R-diagonalizable, 818, 822 – partially hyperbolic, 818 – quasi-unipotent, 818 – semisimple, 818, 822 – unipotent, 818, 822 elliptic – dynamics, 151, 175 – fixed point, 166 – solution, 1159 – system, 101 elvel, 587 endomorphism, exact, 80
Subject Index of Volume 1A energy – free, 364, 397 – of interaction, 336 – surface – – stable, 1134 – – star-like, 1135, 1153, 1158 engaging totally, 744 ensemble, Gibbs, 334, 337 entropy, 51, 74, 77, 80, 82, 91, 92, 95, 143, 145, 147, 148, 177, 193, 273, 283, 741, 1077 – algebraic, 39 – as dimension, 35 – boundary, 980 – comparison, 495 – conditional, 74, 75 – expansive, 532 – for random transformations, 995, 996 – formula, 487, 488, 624 – – Pesin, 309 – fundamental-group, 39 – Furstenberg, 716, 980 – Gibbs, 393, 394 – homological, 39 – homotopical, 40 – Kolmogorov–Sinai, 364 – measure-theoretic, 350, 364, 401 – minimal, 502 – of partition, 74 – profile, 983 – random walk, 982 – relative, 79 – relative or fiber, 996 – relative to a partition, 75 – rigidity, 294, 495, 500 – slow, 37, 80, 92 – topological, 34–37, 308, 365, 487, 609 – volume, 487 equation – Jacobi, 252, 253, 459 – Riccati, 252 equilibrium state, 38, 143, 145, 147, 364, 365, 368, 383, 384 equivalence – affine, 848 – finite, 786 – orbit, 8, 18, 59, 67, 686, 697, 739, 740 – shift, 560, 782 – topological, 848 equivalent cocycles, 299 ergodic, 679, 1049 – average, 14 – decomposition, 50, 60, 79, 84, 85, 680, 838, 846 – extension, 743 – measure, 679
1173
– set, 85 – stable, 150 – strongly, 994, 1003 – theorem, Birkhoff, 66, 85 ergodicity, 50, 60, 67, 69, 71, 84, 88, 90, 135, 144, 147, 150, 156, 160, 170, 172, 175, 178, 181, 182, 185, 186, 191, 192, 266, 678, 704 escapable set, 882, 883 escape rate, 370, 401–404, 947 essential class, 595 Euclidean – Lie group, 818 – manifold, 824 Euler product, 413 Euler–Lagrange equation, 118 even equivalence, 230 EWAS quadratic form, 904 exact – endomorphism, 80 – symplectic form, 122 exit set, 556 expanding map, 621 expansive – component, 772 – subdynamics, 771 expansivity (expansiveness), 30, 94, 95, 128, 140, 147, 149, 266, 273, 274, 282, 564, 768 – constant, 30 exponent – critical, 489 – Hölder, 262, 263, 265, 276, 287 – Lyapunov, 147, 262, 298, 302, 304, 629, 935, 936, 977, 997 exponential – growth rate, 32 – Lie group, 818 – type, 41 extension, 10, 21, 108, 586 – ergodic, 743 – isometric, 22, 62, 108, 182, 226 – Markov, 419, 631 – natural, 10, 22, 61, 80, 107 – of a pattern, 586 extremal process, 220 extreme point, 84 f¯-distance, 229 factor, 10, 21, 52, 60, 72, 79, 144, 208 – orbit, 21 – Radon–Nikodym, 989 – topological, 36, 768 family – Borel–Cantelli, 885
1174
Subject Index of Volume 1A
– of smooth maps, 627 fast stable manifold, 259, 265 Fell topology, 821 field – cone, 248, 253, 256 – Jacobi, 252, 456, 459 filtration, 565 filtration pair, 556, 557 finitary (isomorphism), 225 finite – energy – – cylinder, 1157 – – foliation, 1169 – – – stable, 1171, 1172 – – plane, 1146, 1152 – – sphere, 1169 – – surface, 1146, 1181 – equivalence, 786 – exponential moment, 961 – first moment, 935, 946 – order, 584 finitely determined (process), 220 first moment, finite, 935, 946 first-entry map, 655 first-return map, 19, 59, 107 fixed point – class, 594 – elliptic, 166 – point of the R-action, 1171 – theorem – – hyperbolic, 137, 139, 255, 267, 275 – – Lefschetz, 567 – transverse, 111 flag, 696, 952 – asymptotic, 1073 – variety, 714, 952 flat – strip theorem, 477 – surface, 1022 Floer’s homology theory, 1151 flow, 252–254, 260, 270–272, 274, 280 – Anosov, 252, 850 – geodesic, 120, 123, 135, 150, 242, 252, 253, 265, 272, 280, 441, 831, 833, 852 – Hamiltonian, 115, 252 – homogeneous, 828 – horocycle, 832, 851, 852 – horospherical, 859 – infra-homogeneous, 833 – partially hyperbolic, 880 – rectilinear, 850 – suspension, 382, 388, 391 – unipotent, 854 – Weyl chamber, 850
folding, 587 folklore theorem, 622 Følner set, 14, 83, 234 forces, 581, 582 forcing, 581–583 form – contact, 120, 1136 – index, 463 – normal, 104, 145, 289 – symplectic, 114 formula – entropy, 487, 488, 624 – Pinsker, 211 forward-matching method, 264 frame bundle, 726 Fredholm index, 1171 free – energy, 364, 397 – particle motion, 252 frequency locking, 170 Fuchsian subgroup of a Lie group, 825 full shift, 633 function – µ-harmonic, 967 – bounded µ-harmonic, 967 – complexity, 45 – correlation, 362, 363, 390, 391 – harmonic, 976 – left uniformly continuous, 967 – tempered, 299 – transfer, 11, 142, 287 fundamental cocycle, 790 fundamental-group entropy, 39 Furstenberg – boundary, 713, 976 – condition, 943, 1002 – correspondence principle, 46, 85 – entropy, 716, 980 G– invariant function, 678 – map, 678 – – relative to a measure, 678 – representation – – quasi-regular, 981 – space, 937 – – homogeneous, 712 – – irreducible, 708 (G, µ)-boundary, 969 – compact, 969 (G, µ)-space, 968, 981 – compact, 968 gap, spectral, 946, 965, 995, 1004
Subject Index of Volume 1A Gauss transformation, 624 Gaussian dynamical system, 720 generator, 61, 78, 81, 193, 207 – of cocycle, 297 – one-sided, 78 geodesic – Anosov flow, 473 – flow, 120, 123, 135, 150, 242, 252, 253, 265, 272, 280, 441, 831, 833, 852 – length space, 483 – stretch, 496 geometric decomposition of S 3 , 1174 geometric structure, rigid, 495, 724 Gibbs – ensemble, 334, 337 – entropy, 393, 394 – measure, 334, 349, 352, 359, 361, 364, 369, 375, 377, 381, 385, 389, 390, 393 – state, 334, 339, 342, 345, 347, 348, 351, 352 global – surface of section, 1155, 1162, 1164 – system of transversal sections, 1174 gluing, 587 good periodic approximation, 70, 160, 183 graph – Lagrangian, 461 – shift, 779 – transform, 302 – – Hadamard method, 256 Gromov – representation, 733 – width, 117, 1138 group – algebraic, 676 – amenable, 694 – extension, 22, 182 – semisimple Lie, 673 – stable, 709 – unstable, 709 – Veech, 1059 growth – asymptotic, 274 – volume, 520 H– reduction, 724 – representation, regular, 947 Haar measure, 57, 817, 823 Hadamard – graph transform method, 256 – manifold, 476 Hadamard–Cartan theorem, 476 Hadamard–Perron theorem, 130, 132, 138, 139, 257
1175
half-pinched Anosov diffeomorphism, 751 Hamiltonian – flow, 115, 252 – system, 1131 – vector field, 115 Hamming metric, 81 harmonic function, 493, 976 Hartman–Grobman theorem, 139, 266 Hausdorff topology, 821 Hayashi connecting lemma, 270 Heisenberg group, 707 heteroclinic, 140, 158, 1109 – chain, 1113, 1115 – cycle, 1180 – orbit, 1180 higher rank Abelian action, 897 Hilbert bundle, 701 Hölder – continuity, 40, 41, 73, 95, 133, 142, 143, 246, 261, 262, 265 – exponent, 262, 263, 265, 276, 287 holomorphic dynamics, 125 holonomy, 261, 266, 283 – semigroup, 99 homeomorphism of finite order, 584 homoclinic, 1109 – orbit, 1180 – point, 806 – – transverse, 249 – tangles, 241, 249 homogeneous – G-space, 712 – action, 828 – flow, 828 – measure, 861 – space, 823 – subset, 824 homological entropy, 39 homology – Conley index, 561 – zeta function, 570 homotopical entropy, 40 homotopy rotation class, 48 Hopf argument, 144 horizontal space, 455 horocycle flow, 89, 136, 181, 832, 851, 852 horosphere, 485 horospheric foliation, 265, 486, 491 horospherical – flow, 859 – subgroup of a Lie group, 818, 819, 835 horseshoe, 134, 148, 248, 249, 308 Howe–Moore ergodicity theorem, 708
1176
Subject Index of Volume 1A
hull, 581 – algebraic, 692, 728 hyperbolic, 561 – dynamics, 127, 151 – fixed point theorem, 137, 139, 255, 267, 275 – measure, 147, 304 – point, 303 – set, 131, 142, 144, 248, 257, 263, 561 – – for flow, 252 – solution, 1159 – system, 100, 110 hyperbolicity – normal, 141 – partial, 149 hypersurface, 1134 – contact type, 1134, 1135 – star-like, 1135 iceberg model, 794 immediate basin, 635 in involution, 117, 154 In Phase Theorem, 565 independent partitions, 54 index, 595 – form, 463 – Lefschetz, 566 – lemma, 464 – of periodic solution, 1159 induced – action, 13, 23, 688 – map, 19, 59 – representation, 703 – sequence, 656 inducing domain, 655 inequality – cross-ratio, 639 – Rokhlin, 76 infinitesimal generator, 723 information function, 74 – conditional, 75 infra-homogeneous flow, 833 infranilmanifold, 251, 278 integrable system, 153, 154 – completely, 117 interaction, 335, 336, 343, 348, 351, 352 intermediate value theorem, 47, 607 interval exchange, 90, 176, 177, 183, 186, 187, 191, 192, 1027 invariant – Borel measure, 56 – circle, 164 – component, 585 – curve theorem, 163 – distribution, 109, 179, 181, 182, 188
– manifold, 107 – mean, 994 – measure, 129, 273, 1034 – parabolic, 747 – reduction, 691 – set, 9 – spectral, 64, 698 – tori, 170, 1116 inverse limit, 10, 22, 61, 107 involution, Cartan, 672 irreducible, 698 – G-space, 708 – lattice, 825 – matrix, 333 – representation, 821 – subshift, 633 – totally, 940 isolated invariant set, 551 isolating – block, 98, 557 – neighborhood, 551 isometric – action, 718 – extension, 22, 62, 108, 182, 226 isometry, 88, 152 isomorphism, 7 – spectral, 69, 698 – theorem, Ornstein, 222 isotropic, 461 – subspace, 114 isotropy subgroup of a Lie group, 828 iterated logarithm law, 962 itinerary, 423 Iwasawa decomposition, 674, 819 Jacobi – equation, 252, 253, 459 – field, 252, 456, 459 – tensor, 460, 470 Jacobian, unstable, 307 jet data, 288 joining, 59, 61, 71, 208 joint partition, 54, 75 Jordan decomposition, 822 Jordan curve theorem, 48 Julia set, 427 K– automorphism, 73, 212 – property, 73, 77, 79, 80, 991, 1001 k-prong singularity, 584 K-quasisymmetric, 640 Kakutani equivalence, 60, 63, 79, 80, 175, 228
Subject Index of Volume 1A Kakutani–Markov fixed point property, 83 KAM theory, 155 Kanai connection, 492, 494 Katok entropy rigidity conjecture, 294 Kazhdan property, 706 kernel, tempering, 300 Khintchine–Groshev theorem, 886, 907 Killing field, 728 Klingenberg’s theorem, 475 kneading sequence, 606 knotted periodic orbit, 1159 Kolmogorov theorem, 169 Kolmogorov–Sinai entropy, 364 Kronecker factor, 69 Kryloff–Bogoliouboff theorem, 83, 88, 92 Kupka–Smale theorem, 110–112 L-functions, 438 labeled graph, 787 lag, 560 Lagrange equation, 118 Lagrangian – dynamical system, 118 – graph, 461 – subspace, 461 large deviations, 364, 394, 396, 397 lattice, 675, 823 – admissible, 919 – arithmetic, 825 – dual, 920 – irreducible, 825 – unimodular, 826 law of large numbers, 934 leafwise regularity, 266 least action principle, 124 Lebesgue – point, 53, 69 – space, 53, 296, 298, 301 – spectrum – – countable, 72, 73, 80, 181 – – simple, 73 Lefschetz – fixed point theorem, 567 – index, 566 – number, 566 – zeta function, 412 left uniformly continuous functions, 967 left-invariant mean, 694 Legendre transform, 119 lemma – Borel–Cantelli, 886 – Morse, 123, 480 – Rokhlin, 64, 216 – shadowing, 138, 147–149, 268, 306, 371, 563
1177
length space, geodesic, 483 Levi decomposition, 820 Levi-Civita connection, 455 Lie group – Ad-proper, 845 – Euclidean, 818 – exponential, 818 – nilpotent, 818 – of type (I), 818 – Q-algebraic, 822 – Q-anisotropic, 822 – Q-split, 822 – R-algebraic, 821 – R-split, 819 – reductive, 822 – semisimple, 819 – simple, 819 – solvable, 818 – totally noncompact, 819 – triangular, 818 – unimodular, 817 Lie transformation group, 724 limit – asymptotic, 1148 – quasi-projective, 951, 958 – set, 23 line bundle, canonical, 1132 line elements, projective, 953 linearizable map, 659 linearization, 97 linking arguments, 1105 Liouville measure, 120, 825 Liouville–Arnold theorem, 117, 154, 169 Liouvillian, 158, 160, 164, 166, 168, 171, 173, 174, 178, 187, 192 Lipschitz, 139, 173, 265, 277 Littlewood’s conjecture, 914 Livschitz theorem, 142, 143, 147, 148, 287, 306, 512 local – action, 723 – analysis, 98 – maximality, 21, 130 – product structure, 144, 248, 260 – rigidity, 753 locally – closed, 683 – maximal, 248 – – hyperbolic set, 248, 271 logarithm law, 885 logistic family, 627, 657 loosely Bernoulli, 229 Luzin theorem, 305
1178
Subject Index of Volume 1A
Lyapunov – block form, 300, 302 – cocycle, 105, 286 – exponent, 147, 262, 298, 302, 304, 629, 935, 936, 977, 997 – – upper, 298 – function, complete, 554 – metric, 132, 248, 299, 301 – norm, 247, 301, 302 – scalar product, 301 – spectrum, 935, 936, 942 – – simple, 1003 Mackey range, 13, 62, 689 Mahler – compactness criterion, 826 – measure, 802 – problem, 911, 912 Mañé – closing lemma, 112, 146 – – ergodic, 270 manifold – admissible, 303 – asymptotically harmonic, 493 – contact, 120 – Hadamard, 476 – invariant, 107 – nondegenerate, 912 – rank-1, 282 – slow, 259 – stable, 112, 130, 140, 142, 148, 258, 260, 305, 306 – strong stable, 260 – strong unstable, 260 – symplectic, 114, 1131 – unstable, 112, 140, 258, 260, 306 map – boundary, 972 – G-, 678 – induced, 19, 59 – Markov, 587 – natural, 503 – nondegenerate, 857, 912 – Poincaré, 59 – section, 187 – twist, 156, 164 Margulis – arithmeticity theorem, 825 – measure, 282, 491 marked length spectrum, 510, 515 Markov – chain, topological, 42, 82, 127, 129, 144, 330, 332–334, 338, 340, 368, 381, 632 – cover, 48
– extension, 419, 631 – map, 587 – measure, 58, 73, 78 – operator, 936, 955 – partition, 129, 144, 147, 149, 281, 325, 328, 330, 332, 334, 368, 370, 375, 380, 381, 396, 428 – process, 936, 955 – property, 325, 326 – section, 378, 380, 384, 386, 388, 390, 434 – shift, 773 Mather – set, 1116 – spectrum, 131, 247, 259, 264, 272, 279 matrix – aperiodic, 334 – coefficient, 844 – irreducible, 333 – primitive, 334 – transition, 332, 335, 381 – transitive, 44, 58 Mautner phenomenon, 710, 837, 841, 855 maximal spectral type, 64, 69, 71, 72, 74, 80, 160 mean, invariant, 994 measurable – action, 677 – partition, 53 measurably isometric, 704 measure – µ-stationary, 937, 968 – admissible, 967 – Bernoulli, 58, 73, 74, 77 – class, smooth, 97, 103, 108 – ergodic, 679 – Gibbs, 334, 349–352, 359–361, 364–369, 375, 377, 381–385, 389, 390, 393 – homogeneous, 861 – hyperbolic, 147, 304 – invariant, 129, 273, 1034 – Mahler, 802 – Margulis, 282, 491 – Markov, 58, 73, 78 – of maximal entropy, 82, 94, 282, 366, 487, 528, 532 – proper, 940, 956 – quasi-invariant, 677 – spectral, 68, 69 – transversal, 491 measure-theoretic entropy, 350, 364, 401 measured foliation, 183 measures, conditional, 54 method, variational, 123 metric – adapted, 248
Subject Index of Volume 1A – cylinder, 1024 – isomorphism, 58, 69, 70, 90 – Lyapunov, 132, 248, 299–301 – Rokhlin, 75 mild mixing, 71 Milnor–Thurston zeta function, 423 minimal – action, 954, 992 – entropy, 502 – parabolic subgroup, 714 – set, 19, 88 minimality, 19, 88, 89, 91, 152, 156, 172, 175, 178, 181, 186, 192, 1024 minimizing property of Jacobi fields, 465 mirror equation, 167 mixing, 50, 72, 82, 95, 129, 142–145, 160, 173, 175, 178, 179, 183, 184, 186, 187, 191, 192, 633, 704, 1070 – multiple, 72 – subshift, 633 – topological, 26, 72, 271, 274 – weak, 71, 704 modular surface, 825 moduli space, 1032 momenta, 119 monotone maps, P -, 581 Moore subgroup of a Lie group, 835, 837 Morse – lemma, 123, 480 – sequence, 44, 608 Moser–deLatte normal form, 290 Mostow rigidity, 502 multibump solutions, 1109, 1117, 1121 multiple – mixing, 72 – Poincaré recurrence, 86 – weak mixing, 71 multiplicative ergodic theorem, 298 multiplicity – function, 702 – of exponent, 298 multiply nonwandering, 46 natural – 1-form, 456 – extension, 10, 22, 61, 80, 107 – map, 503 near action, 689 negative puncture, 1147 neighborhood – isolating, 551 – regular, 302 – symmetric, 684 neutral subgroup of a Lie group, 835
1179
Newton method, 159 Nielsen number, 595 Nielsen–Thurston theory, 183 nilmanifold, 824 nilpotent Lie group, 818 nilradical, 818 Noether theorem, 117 non-compact cocycle, 1002 non-squeezing, 1138 nonamenable group, 947, 965 nondegenerate – manifold, 912 – map, 857, 912 nonflat critical point, 618 nonlinearity measure, 611 nonpositive curvature, 476 nonrandom filtration, 942, 998 nonstandard smooth realization, 153, 172 nonuniform hyperbolicity, 132 nonwandering – multiply, 46 – point, 25 – set, 25 norm – adapted, 247 – Lyapunov, 247, 301, 302 normal – form, 104, 145, 289 – – Birkhoff, 166 – hyperbolicity, 141 – subgroup theorem, 757 normalized potential function, 359, 367, 368 NT homeomorphism, 585 nuclear operator, 437 obstruction, Anosov, 290 one-sided generator, 78 open book decomposition, 1157, 1180 operator – Markov, 936, 955 – Riccati, 256 – Ruelle–Perron–Frobenius, 354 – transfer, 57, 108, 426 Oppenheim conjecture, 860, 901 – quantitative versions, 902 orbit – complexity, 128 – cylinder, 1145, 1148, 1173 – equivalence, 8, 18, 59, 67, 686, 697, 739, 740 – factor, 21 – heteroclinic, 1180 – homoclinic, 1180 – periodic, 143, 1052
1180
Subject Index of Volume 1A
– twisted, 413 orbit growth, asymptotic, 32 order of criticality, 618 Ornstein – copying lemma, 218 – isomorphism theorem, 222 Oseledets multiplicative ergodic theorem, 147 overtwisted contact structure, 1152 P– monotone maps, 581 – stationary, 936 p-contracting, 948 – (semi)group, 948 – sequence, 948 p-irreducible, strongly, 940 Palais–Smale condition, 1098 parabolic – dynamics, 175, 194 – invariant, 747 – Levi subgroup of a Lie group, 820 – subgroup, 703, 713 – – minimal, 714 – system, 101 partial hyperbolicity, 101, 133, 149, 284 partially hyperbolic – element of a Lie group, 818 – flow, 880 particle motion, free, 252 partition – function, 337, 343 – Markov, 129, 144, 147–149, 281, 325–328, 330, 332, 334, 368, 370, 375, 380, 381, 396, 428 – measurable, 53 past, 210 pattern, 581, 583 – twist, 585 Patterson–Sullivan measure, 490, 491 period-doubling bifurcations, 648 periodic – data, 105, 287, 288 – orbit, 143, 1052 – – Birkhoff, 157 – point, 7, 31, 142, 145, 411 – – transverse, 111 – solution, 1095 periodic trajectory, stable, 1075 Perron number, 784 Perron–Frobenius operator, 108, 622 Perron–Irwin method, 256 perturbation, 155 Pesin – entropy formula, 309 – set, 147, 305
– tempering kernel, 300 Pestov’s identity, 511, 537, 538 piecewise monotone, 603 pinching, 253, 264, 1101 Pinsker – algebra, 79, 212 – formula, 211 Plancherel – countable spectrum, 717 – formula, 703 Plykin attractor, 251 Poincaré – classification, 28 – map, 59 – recurrence – – multiple, 86 – – theorem, 59, 682 – section map ψ , 1162 – series, 489 Poincaré–Bendixson theory, 48 point – homoclinic, 806 – hyperbolic, 303 – Lebesgue, 53, 69 – nonwandering, 25 – periodic, 7, 31, 142, 145, 411 – regular, 302, 629 pointed space map, 560 Poisson – boundary, 971 – bracket, 117 – transform, 968 polar decomposition, 827 polygonal billiard, 1017 polynomial-like extension, 653 positive puncture, 1147 potential – function, 349–352, 354, 364, 368, 369, 384 – – normalized, 359, 367, 368 – singular, 1106 pressure, 38, 92, 94, 95 – topological, 343, 349, 365, 384, 402 primary pattern, 592 prime number theorem, 275 primitive matrix, 334 principal series, 700 principle, variational, 93, 94, 125, 148, 374, 393, 402, 487, 770, 1141 process, Markov, 936, 955 product, 149 – relative, 56 – structure, local, 144, 248, 260 profile, entropy, 983
Subject Index of Volume 1A profinitely dense, 744 projective – action, 714 – line elements, 953 prongs, 584 proper – action, 683 – measure, 940, 956 – rectangle, 324, 325, 328, 332, 333, 379 property – Bernoulli, 361–363, 373, 399 – K-, 73, 77, 79, 80, 991, 1001 – Markov, 325, 326 – specification, 269 – T , 281, 705 property-(D), 855 proximal – action, 949 – strongly, 713 proximality, 28, 713, 949 pseudo-Anosov, 186, 584, 585 – single-fixed point, 593 pseudo-orbit, 138, 268 pseudoholomorphic curve, 1141, 1153 Pugh closing lemma, 112, 146, 270 puncture – negative, 1147 – positive, 1147 – removable, 1147 pure point spectrum, 70, 71, 89 Q– character, 822 – rank, 822 Q-algebraic – Lie group, 822 – representation, 823 Q-anisotropic Lie group, 822 Q-split Lie group, 822 QNS, 617 quadratic differential, 191, 1022 quadrature, 154 quasi-geodesic, 479 quasi-invariant, 52 – measure, 677 quasi-isometry, 479 quasi-lattice, 824 quasi-negative Schwarzian, 617 quasi-projective – limit, 951, 958 – transformation, 950 quasi-regular – G-representation, 981
– representation, 945, 946, 981, 1004 quasi-unipotent – element of a Lie group, 818 – subgroup of a Lie group, 818 quasiminimality, 184–186 quasisymmetric, 620 R-algebraic Lie group, 821 R-diagonalizable – element of a Lie group, 818, 822 – subgroup of a Lie group, 822 R-rank, 819 R-split Lie group, 819 R-property, 861 radical, 818 Radon–Nikodym – cocycle, 55, 680 – factor, 989 Raghunathan’s conjecture, 860 random ergodic theorem, 991, 993, 995 random walk entropy, 982 rank – of a nonpositively curved manifold, 515 – Q-, 822 – real, 673 – rigidity, 515 rank-1 – manifold, 282 – space, 515 rate of convergence, 992 rational – polygon, 1020 – zeta function, 414 Ratner’s theorem, 742, 861–863 Rauch’s comparison estimates, 465 real Fatou conjecture, 657 real rank, 673 rectangle, 324, 328, 370, 375, 379 – proper, 324–328, 332, 333, 379 rectifiable set, 892 rectilinear flow, 850 recurrence, 24, 129, 682 – uniform, 24 reducible, 585 reducing curves, 584 reduction theory, 826 reductive – group, 672 – Lie group, 822 Reeb vector field, 1136, 1160, 1173 refinement, 54 region of instability, 165 regional recurrence, 25, 271, 272
1181
1182 regular – closed geodesic, 520, 527 – H -representation, 947 – neighborhood, 302 – point, 302, 629 – representation, 699, 820, 946, 947 regularity, 261 – of the horospherical foliation, 492 – of topological entropy, 500 relative – entropy, 79 – product, 56 relatively independent joinings, 208 relaxation oscillations, 241, 249 removable puncture, 1147 renormalization, 610, 654 renormalized functional, 1114 repeller, 21, 251, 402, 403, 552 representation – adjoint, 817 – algebraic linear, 823 – canonical, 586 – continuous, 85 – Gromov, 733 – induced, 703 – irreducible, 821 – Q-algebraic, 823 – quasi-regular, 945, 946, 981, 1004 – regular, 699, 820, 946, 947 – unipotent, 824 – unitary, 697, 698, 820 resonance, 158, 289 restricted root, 674 restrictive interval I , 631 return – map, 183, 1156 – probability, 965, 966 Riccati – equation, 252 – operator, 256 Riemannian metric, 455 rigid – geometric structure, 495, 724 – surface, 1173 rigidity, 70, 71, 160, 180 – cocycle, 188 – entropy, 294, 495, 500 – local, 753 – rank, 515 – smooth, 286, 292 – spectral, 509 Rokhlin – inequality, 76 – lemma, 64, 216
Subject Index of Volume 1A – metric, 75 root space, 674 – decomposition, 674 rotation number, 27, 28, 602 – of constant type, 602 Ruelle zeta function, 424 Ruelle–Perron–Frobenius – operator, 354 – theorem, 354 saddle connection, 184, 1024 Sasaki metric, 457 Schwarzian derivative, 614 section, 62, 107, 182, 183, 189 – map, 187 – Markov, 378, 380, 384–386, 388, 390, 434 sectional curvature, 252 self-joining, 59, 60 self-linking number, 1164, 1174 semisimple – element of a Lie group, 818, 822 – Lie – – algebra, 673 – – group, 673, 819 sensitive dependence, 127, 149 separated sets, 34 sequence – contracting, 948, 956 – induced, 656 – Morse, 44, 608 – p-contracting, 948 – totally contracting, 948 series – discrete, 700 – Poincaré, 489 set – Aubry–Mather, 157, 164, 165, 174 – basic, 132, 248, 326–328, 330, 334, 368, 370– 372, 374, 378, 389, 401, 414, 564 – ergodic, 85 – hyperbolic, 131, 142–144, 248, 257, 263, 561 – invariant, 9 – Mather, 1116 – minimal, 19, 88 – nonwandering, 25 – Pesin, 147, 305 – symmetric, 684 shadowing, 94, 267 – lemma, 138, 147, 149, 268, 306, 371, 563 – theorem, 269, 275 Shannon–McMillan–Breiman theorem, 76, 214 Sharkovsky – ordering, 582
Subject Index of Volume 1A – theorem, 601, 610 sharp determinant, 430 shear, 176 shift, 18, 41–47, 73, 129 – Bernoulli, 58, 80, 361, 1179, 1180 – equivalence, 560, 782 – homeomorphism, 333, 354 – Markov, 773 Siegel – disk, 661 – summation formula, 826 simple – Lebesgue spectrum, 73 – Lie group, 819 – Lyapunov – – exponent, 948 – – spectrum, 948, 949 – spectrum, 70, 160 Sinai–Ruelle–Bowen (SRB) measure, 143, 148, 283, 309, 369, 373–375, 385–387, 391, 392, 395–399, 402, 404 singular – closed geodesic, 522, 536 – potential, 1106 – spectrum, 72 skew product, 12, 79, 970, 979, 991, 1004 sliding block code, 780 slow – entropy, 37, 80, 92 – manifold, 259 – subbundle, 259 Smale attractor, 107, 134, 250 small denominator, 105, 162 smooth – conjugacy, 103, 105 – measure class, 97, 103, 108 – rigidity, 286, 292 – stability, 170 sofic, 45, 94, 774, 787 solenoid, 107, 134, 250 solution – elliptic, 1159 – hyperbolic, 1159 – periodic, 1095 solvable Lie group, 818 solvmanifold, 824 space – average, 50 – homogeneous, 823 – Lebesgue, 53, 296, 298, 301 – of partitions, 76 – rank-1, 515 – Teichmüller, 1031 spanning set, 35
1183
special flow, 13, 63, 172, 186, 187, 191, 229, 254 specification, 82, 94, 95, 143, 269, 273, 274, 282 – for flows, 270 – property, 269 – – for flows, 270 – weak, 769 spectral – decomposition, 43, 132, 142, 143, 147, 149, 271, 279 – gap, 946, 965, 995, 1004 – invariant, 64, 698 – isomorphism, 69, 698 – measure, 68, 69 – rigidity, 509 – theorem, 702 spectrum – discrete, 69, 89, 704, 1003 – Lyapunov, 935, 936, 942 – Mather, 131, 247, 259, 264, 272, 279 – simple, 70, 160 – singular, 72 sphere – at infinity, 478 – topology, 478 spherical finite energy foliation, 1171 Sprindžuk’s conjectures, 912, 915 stability – cocycle, 110 – of the solar system, 169, 170 – smooth, 170 – theorem, 146, 276 – topological, 106, 275 stable – and unstable – – foliations, 486 – – manifolds, 563 – – spaces, 472 – behavior, 151 – energy surface, 1134 – ergodicity, 286 – finite energy foliation, 1171, 1172 – group, 709 – manifold, 112, 130, 140, 142, 148, 258, 260, 305, 306 – – at periodic point, 112 – – strong, 260 – – theorem, 255, 563 – periodic trajectory, 1075 standard – action, 751 – Borel G-space, 678 star-like – energy surface, 1135, 1153, 1158
1184
Subject Index of Volume 1A
– hypersurface, 1135 state, Gibbs, 334, 339, 342, 345, 347, 348, 351, 352 stationary measure, 715 stochastic, 58, 127 strange attractor, 127 stratum, 1032 stretch, geodesic, 496 strict G-map, 678 strictly convex energy surface, 1158, 1159 strong – force condition, 1107 – shift equivalence, 781 – specification, 769 – stable manifold, 260 – unstable manifold, 260 strongly – ergodic, 994, 1003 – irreducible – – cocycle, 1002 – – group, 939, 940 – – measure, 939, 940 – – semigroup, 939, 940 – p-irreducible, 940 – proximal, 713 – – action, 949, 954 structural stability, 106, 127, 129, 139, 145, 275, 276, 484, 634 structure, contact, 120, 272, 1136 structures of finite type, 724 subalgebra, Cartan, 673 subbundles, 261 subgroup – of a Lie group – – Auslander, 835 – – Cartan, 819 – – Dani, 835, 837 – – epimorphic, 895 – – Fuchsian, 825 – – horospherical, 818, 819, 835 – – isotropy, 828 – – Moore, 835, 837 – – neutral, 835 – – parabolic Levi, 820 – – R-diagonalizable, 822 – – quasi-unipotent, 818 – – uniform, 823 – – unipotent, 818, 822 – parabolic, 703, 713 subshift, 633, 773 – irreducible, 633 – mixing, 633 – of finite type, 42 subspace – isotropic, 114
– Lagrangian, 461 Sullivan conjecture, 295 sum, continuous, 701 support, 49 surface, 193 – flat, 1022 – of section, global, 1155, 1162, 1164 – rigid, 1173 – Veech, 1059 suspension, 11, 23, 62, 108, 253, 272, 688 – flow, 382, 388, 391 symbolic – dynamical system, 18, 41 – dynamics, 242 symmetric – neighborhood, 684 – set, 684 symplectic – capacity, 116, 1138 – form, 114 – – exact, 122 – manifold, 114, 1131 – structure on T M, 457 syndetic, 24 system – elliptic, 101 – Hamiltonian, 1131 – hyperbolic, 100, 110 – of transversal sections, global, 1174 – parabolic, 101 Szemerédi theorem, 85 (T , T −1 )-transformation, 228 tame, 683 tangles, homoclinic, 241, 249 Teichmüller – space, 1031 – theory, 183 telescope construction, 646 tempered – cocycle, 299 – function, 299 tempering, 299 – kernel, 300 – – Pesin, 300 tensor, Jacobi, 460, 470 tent maps, 607, 609 theorem – Ruelle–Perron–Frobenius, 354 – shadowing, 269, 275 – Sharkovsky, 601, 610 – spectral, 702 – stability, 146, 276
Subject Index of Volume 1A – transversality, 111 thermodynamic limit, 337–339 thick set, 882 tight – 3-sphere, 1165 – contact structure, 1152 tightening, 587 time average, 14, 49 time change, 8, 188, 254 Toeplitz shift, 44 topological – CE condition, 645 – conjugacy, 18, 103, 768 – entropy, 34–37, 308, 365, 487, 609 – – for noncompact spaces, 36 – – original definition, 36 – equivalence, 848 – factor, 36, 768 – Markov chain, 42, 82, 127, 129, 144, 330, 332, 334, 338, 340, 368, 381, 632 – mixing, 26, 72, 271, 274 – pressure, 343, 349, 365, 384, 402 – stability, 106, 275 – transitivity, 19, 26, 88, 89, 271, 272, 1026 – weak mixing, 433 topologically engaging, 744 topology – cone, 478 – sphere, 478 – Zariski, 821 tori, invariant, 170, 1116 total Conley–Zehnder index, 1165 totally – contracting, 948 – – (semi)group, 948 – – sequence, 948 – engaging, 744 – irreducible, 940 – noncompact Lie group, 819 transfer – function, 11, 142, 287 – operator, 57, 108, 426 transform – graph, 302 – Legendre, 119 – Poisson, 968 transformation – canonical, 114 – quasi-projective, 950 transition – matrix, 332, 335, 381 – probabilities, 936 transitive, 142, 144, 145 – matrix, 44, 58
1185
transitivity, 150, 152, 156, 178, 182, 184 – topological, 19, 26, 88, 89, 271, 272, 1026 transversal, 107, 566 – measure, 491 transversality, 110, 139, 276 – theorem, 111 transverse – fixed point, 111 – homoclinic point, 249 – periodic point, 111 triangular Lie group, 818 twist, 585 – interval, 157 – map, 156, 164 – pattern, 585 twisted – orbit, 413 – product, 13, 23, 688 typical point, 66, 85 u-Gibbs measure, 369 UHC, 645 uniform – hyperbolicity on cycles, 645 – recurrence, 24 – subgroup of a Lie group, 823 unimodal pattern, 594 unimodular – group, 675 – lattice, 826 – Lie group, 817 unipotent – element of a Lie group, 818, 822 – flow, 854 – radical, 822 – representation, 824 – subgroup of a Lie group, 818, 822 unique ergodicity, 87–91, 1034 unitary – dual, 699 – representation, 697, 698, 820 universality, 654 unstable – group, 709 – manifold, 112, 140, 258, 260, 306 – – at periodic point, 112 – – strong, 260 upper Lyapunov exponent, 629 value, critical, 554 variational – method, 123 – principle, 93, 94, 125, 148, 374, 393, 402, 487, 770, 1141
1186 Veech – group, 1059 – surface, 1059 vertex shift, 779 vertical space, 455 very weak Bernoulli process, 220 volume – density, 732 – entropy, 487 – growth, 520 – lemma – – first, 370 – – second, 370 von Neumann mean ergodic theorem, 65 wandering interval, 603, 638 Wang tiles, 777 weak – mixing, 71, 704 – – multiple, 71 – – topological, 433 – specification, 769 weakly – hyperbolic action, 753 – hyperbolic space, 515
Subject Index of Volume 1A – orbit equivalent, 740 Weinstein conjecture, 1136, 1141, 1150 Weyl chamber flow, 850 width, Gromov, 117, 1138 Wronskian, 460 Zd -action, algebraic, 796 Zariski – dense, 821, 824 – – cocycle, 1002 – topology, 821 zeta function, 32, 33, 38, 274, 366, 569, 784 – Artin–Mazur, 411 – for Axiom A diffeomorphisms, 411 – for Axiom A flows, 431 – for subshifts, 415 – homology, 570 – Lefschetz, 412 – Milnor–Thurston, 423 – rational, 414 – Ruelle, 424 Zimmer’s program, 735 Zygmund, 265 – smoothness, 614
– metric, 248 – norm, 247 adjoint representation, 817 admissible – lattice, 919 – manifold, 303 – measure, 967 affine – action, 717 – equivalence, 848 algebra – Pinsker, 79, 212 – semisimple Lie, 673 algebraic – entropy, 39 – group, 676 – hull, 692, 728 – linear representation, 823 – Zd -action, 796 almost k-simple, 673 almost complex structure, 457 almost direct product, 819 almost existence, 1132 1133, 1137 almost-conjugacy, 21, 144 almost-isomorphism, 21, 127, 129 amenable, 15, 16, 81–83, 694, 705 – action, 695, 972, 975 – group, 694 analysis, local, 98 Anosov – action, 754, 755 – alternative, 272 – closing lemma, 112, 138, 142, 148, 149, 268 – cocycle, 289, 291 – diffeomorphism, 132, 133, 144, 145, 248, 278, 284 – element, 747 – flow, 252, 850 – – anomalous, 254 – – geodesic, 473 – obstruction, 290
α-invariant function, 690 α-limit set, 24 δ-distance, 209 δ-shadowed, 563 ε-chain, 550 ε-chainable, 552 λ-lemma, 658 µ-harmonic function, 967 µ-stationary measure, 937, 968 π -simple cocycle, 687 ψ -approximable, 905, 906, 910 ψ -multiplicatively approximable, 914 ω-limit set, 24 Ω-stability, 106, 277, 278 1-form, natural, 456 3-sphere, tight, 1165 Abramov formula, 79 absolute continuity, 262, 266 absolutely continuous spectrum, 72 accessibility, 144, 150, 272, 285, 286 action – amenable, 695, 972, 975 – Anosov, 754, 755 – Bernoulli, 717 – functional, 119, 125 – homogeneous, 828 – induced, 13, 23, 688 – integral, 1095, 1100 – isometric, 718 – local, 723 – measurable, 677 – minimal, 954, 992 – projective, 714 – proper, 683 – proximal, 949 – standard, 751 – strongly proximal, 949, 954 action-angle coordinates, 117, 154 Ad-proper Lie group, 845 adapted 1169
1170
Subject Index of Volume 1A
aperiodic matrix, 334 approximable – ψ -, 905, 906, 910 – ψ -multiplicatively, 914 – badly, 906, 907, 910 – – multiplicatively, 914 – very well, 906, 907 – – multiplicatively, 914 – well, 906, 910 approximable set, 883 arithmetic – group, 676 – lattice, 825 Arnold diffusion, 170, 1122 Artin–Mazur zeta function, 411 asymptotic – behavior, 26 – cycle, 27 – density, 49 – distribution, 56 – flag, 1073 – growth, 274 – limit, 1148 – orbit growth, 32 – to a fixed point, 250 asymptotically harmonic manifold, 493 attractor, 143, 371–373, 386, 387, 392, 400, 402, 551 Aubry–Mather set, 157, 164, 165, 174 Auslander subgroup of a Lie group, 835 automorphism, 251 – K-, 73, 212 average – Birkhoff, 14, 49, 50, 87, 180 – ergodic, 14 Axiom A, 248, 278, 411, 635 badly approximable, 906, 907, 910 – multiplicatively, 914 Baire space, 684 Banach contraction principle, 255 barycenter of a measure, 503 basic set, 132, 248, 326–328, 330, 334, 368, 370– 372, 374, 378, 389, 401, 414, 564 basin, 21, 635 – of attraction, 373, 387, 392, 393, 399 behavior – asymptotic, 26 – stable, 151 Bernoulli – action, 717 – measure, 58, 73, 74, 77 – property, 361, 363, 373, 399
– shift, 58, 80, 361, 1179, 1180 billiards, 187–189, 191–193 birecurrent set, 890 Birkhoff – average, 14, 49, 50, 87, 180 – ergodic theorem, 66, 85 – normal form, 166 – periodic orbit, 157 block – isolating, 98, 557 – form, Lyapunov, 300, 302 blow up, 722 Boltzmann ergodic hypothesis, 242 bootstrap, 265, 291 Borel – density theorem, 685, 824 – G-space, standard, 678 – measure, invariant, 56 Borel–Cantelli – family, 885 – lemma, 886 boundary, 713 – entropy, 980 – Furstenberg, 713, 976 – map, 972 – Poisson, 971 bounded – µ-harmonic function, 967 – distortion property, 622 – geometry, 657 Bowen measure, 95 Bowen–Margulis measure, 282, 492 box mapping, 654 brake orbit, 1101 branchwise equivalence, 657 Brjuno condition, 659 Brouwer’s translation theorem, 1158 bubbling off analysis, 1176 bunching, 253, 262, 264 Busemann – density, 489, 490, 518 – functions, 485 C-map, 628 (C, α)-good, 857 canonical – line bundle, 1132 – representation, 586 – transformation, 114 capacity – c0 , 1138, 1139 – symplectic, 116, 1138 capture of celestial bodies, 249
Subject Index of Volume 1A Cartan – decomposition, 820, 827 – involution, 672 – subalgebra, 673 – subgroup of a Lie group, 819 Cauchy–Riemann equation, 1147, 1182 caustic, 166 CE condition, 644 – topological, 645 CE map, 644 CE2 condition, 644 celestial mechanics, 241 center, 25 – manifold, 141 center-stable manifold, 141 center-unstable manifold, 141 central limit theorem, 362–364, 390, 397, 961, 963, 995 chain – heteroclinic, 1113, 1115 – recurrent, 550 – transitive component, 553 character, Q-, 822 characteristic flow, 121 circle, invariant, 164 classical Hamiltonian system, 1135 classification, 105, 130, 145, 187, 278, 291, 292 – Poincaré, 28 closed geodesic – regular, 520, 527 – singular, 522, 536 closing lemma, 112, 268, 270, 277, 278, 306 – Anosov, 112, 138, 142, 148, 149, 268 – Mañé, 112, 146 – – ergodic, 113 – Pugh, 112, 146, 270 cluster property, 347, 361 co-orientation, 1136 coboundary, 11, 142, 171, 172, 187, 188, 363, 686, 789 cocycle, 11, 62, 108, 142, 162, 171, 187, 297, 686 – Anosov, 289–291 – identity, 681 – Lyapunov, 105, 286 – non-compact, 1002 – Radon–Nikodym, 55, 680 – reduction, 691, 730 – rigidity, 188 – stability, 110 – strongly irreducible, 1002 – superrigidity, 736 – tempered, 299 – Zariski dense, 1002 codimension one, 263, 279, 280
1171
coding, 45, 189 cohomological equation, 354 cohomologous, 162, 352, 354, 364, 385, 387, 388, 686 cohomology, 142 Collet–Eckmann map, 636 commensurability, 675 commensurable subgroups, 824 commensurator, 824 compact – (G, µ)-boundary, 969 – (G, µ)-space, 968 complementary series, 700 complete Lyapunov function, 554 completely – integrable system, 117 – positive entropy, 80 complexity, 1080 – function, 45 component – expansive, 772 – invariant, 585 condition – Diophantine, 642 – Furstenberg, 943, 1002 conditional – entropy, 74, 75 – information function, 75 – measures, 54 cone – criterion, 253 – field, 248, 253, 256 – topology, 478 configuration, 335 – space, 118, 122 conjugacy, 753 – smooth, 103, 105 – topological, 18, 103, 768 conjugate points, 462 conjugation-approximation method, 161, 173 Conley index, 560 – homology, 561 Conley–Zehnder index, 1163, 1166, 1167 connecting lemma, 113, 146, 278 constant – cocycle, 687 – expansivity (expansiveness), 30 – type, 602 contact – form, 120, 1136 – – dynamically convex, 1162 – manifold, 120 – structure, 120, 272, 1136
1172
Subject Index of Volume 1A
– – overtwisted, 1151 – – tight, 1151, 1152 contact type hypersurface, 1134, 1135 continuous – representation, 85 – sum, 701 contractible set, 713 contracting, 948 – (semi)group, 948 – sequence, 948, 956 conull set, 678 copying lemma, Ornstein, 218 correlation – coefficient, 68 – function, 362, 363, 390, 391 correspondence principle, Furstenberg, 46, 85 countable spectrum – Lebesgue, 72, 73, 80, 181 – Plancherel, 717 cover, Markov, 48 critical – exponent, 489 – value, 554 cross-ratio, 612 – inequality, 639 cycle, 581 – asymptotic, 27 – heteroclinic, 1180 cylinder, 40, 333, 345, 354, 356, 359, 367 – orbit, 1145, 1148, 1173 ¯ d-distance, 209, 219 DA-map, 251 Dani – correspondence, 907 – subgroup of a Lie group, 835, 837 DE-map, 107, 250 decay of – correlations, 73, 150, 179, 181, 285, 444 – geometry, 656 decomposition – Cartan, 820, 827 – ergodic, 50, 60, 79, 84, 85, 680, 838, 846 – Iwasawa, 819 – Jordan, 822 – Levi, 820 – polar, 827 – root space, 674 – spectral, 43, 132, 142, 143, 147–149, 271, 279 Denjoy theorem, 103, 153 density – asymptotic, 49 – Busemann, 489, 490, 518
– of Axiom A, 636 derived from expanding, 107, 250 descending chain condition, 800 diffeomorphism, Anosov, 132, 133, 144, 145, 248, 278, 284 differentiable stability, 170 differential of the geodesic flow, 456 dimension group, 785 Diophantine, 158, 162, 163, 165–170, 174, 187, 188 – condition, 642 direct integral, 702 discrete – series, 700 – spectrum, 69, 89, 704, 1003 discretization procedure, 976 disjoint transformations, 213 disk, Siegel, 661 distality, 28–30, 176, 181 distortion, 611 distribution – asymptotic, 56 – invariant, 109, 179, 181, 182, 188 divergence type, 489 domino shift, 777 doubling transformation, 649 dual – lattice, 920 – unitary, 699 – variational method, 1098 dynamical system – Lagrangian, 118 – symbolic, 18, 41 dynamical zeta function, 433 dynamics – elliptic, 151–175 – hyperbolic, 127–151 – parabolic, 175–194 – symbolic, 242 element of a Lie group – R-diagonalizable, 818, 822 – partially hyperbolic, 818 – quasi-unipotent, 818 – semisimple, 818, 822 – unipotent, 818, 822 elliptic – dynamics, 151, 175 – fixed point, 166 – solution, 1159 – system, 101 elvel, 587 endomorphism, exact, 80
Subject Index of Volume 1A energy – free, 364, 397 – of interaction, 336 – surface – – stable, 1134 – – star-like, 1135, 1153, 1158 engaging totally, 744 ensemble, Gibbs, 334, 337 entropy, 51, 74, 77, 80, 82, 91, 92, 95, 143, 145, 147, 148, 177, 193, 273, 283, 741, 1077 – algebraic, 39 – as dimension, 35 – boundary, 980 – comparison, 495 – conditional, 74, 75 – expansive, 532 – for random transformations, 995, 996 – formula, 487, 488, 624 – – Pesin, 309 – fundamental-group, 39 – Furstenberg, 716, 980 – Gibbs, 393, 394 – homological, 39 – homotopical, 40 – Kolmogorov–Sinai, 364 – measure-theoretic, 350, 364, 401 – minimal, 502 – of partition, 74 – profile, 983 – random walk, 982 – relative, 79 – relative or fiber, 996 – relative to a partition, 75 – rigidity, 294, 495, 500 – slow, 37, 80, 92 – topological, 34–37, 308, 365, 487, 609 – volume, 487 equation – Jacobi, 252, 253, 459 – Riccati, 252 equilibrium state, 38, 143, 145, 147, 364, 365, 368, 383, 384 equivalence – affine, 848 – finite, 786 – orbit, 8, 18, 59, 67, 686, 697, 739, 740 – shift, 560, 782 – topological, 848 equivalent cocycles, 299 ergodic, 679, 1049 – average, 14 – decomposition, 50, 60, 79, 84, 85, 680, 838, 846 – extension, 743 – measure, 679
1173
– set, 85 – stable, 150 – strongly, 994, 1003 – theorem, Birkhoff, 66, 85 ergodicity, 50, 60, 67, 69, 71, 84, 88, 90, 135, 144, 147, 150, 156, 160, 170, 172, 175, 178, 181, 182, 185, 186, 191, 192, 266, 678, 704 escapable set, 882, 883 escape rate, 370, 401–404, 947 essential class, 595 Euclidean – Lie group, 818 – manifold, 824 Euler product, 413 Euler–Lagrange equation, 118 even equivalence, 230 EWAS quadratic form, 904 exact – endomorphism, 80 – symplectic form, 122 exit set, 556 expanding map, 621 expansive – component, 772 – subdynamics, 771 expansivity (expansiveness), 30, 94, 95, 128, 140, 147, 149, 266, 273, 274, 282, 564, 768 – constant, 30 exponent – critical, 489 – Hölder, 262, 263, 265, 276, 287 – Lyapunov, 147, 262, 298, 302, 304, 629, 935, 936, 977, 997 exponential – growth rate, 32 – Lie group, 818 – type, 41 extension, 10, 21, 108, 586 – ergodic, 743 – isometric, 22, 62, 108, 182, 226 – Markov, 419, 631 – natural, 10, 22, 61, 80, 107 – of a pattern, 586 extremal process, 220 extreme point, 84 f¯-distance, 229 factor, 10, 21, 52, 60, 72, 79, 144, 208 – orbit, 21 – Radon–Nikodym, 989 – topological, 36, 768 family – Borel–Cantelli, 885
1174
Subject Index of Volume 1A
– of smooth maps, 627 fast stable manifold, 259, 265 Fell topology, 821 field – cone, 248, 253, 256 – Jacobi, 252, 456, 459 filtration, 565 filtration pair, 556, 557 finitary (isomorphism), 225 finite – energy – – cylinder, 1157 – – foliation, 1169 – – – stable, 1171, 1172 – – plane, 1146, 1152 – – sphere, 1169 – – surface, 1146, 1181 – equivalence, 786 – exponential moment, 961 – first moment, 935, 946 – order, 584 finitely determined (process), 220 first moment, finite, 935, 946 first-entry map, 655 first-return map, 19, 59, 107 fixed point – class, 594 – elliptic, 166 – point of the R-action, 1171 – theorem – – hyperbolic, 137, 139, 255, 267, 275 – – Lefschetz, 567 – transverse, 111 flag, 696, 952 – asymptotic, 1073 – variety, 714, 952 flat – strip theorem, 477 – surface, 1022 Floer’s homology theory, 1151 flow, 252–254, 260, 270–272, 274, 280 – Anosov, 252, 850 – geodesic, 120, 123, 135, 150, 242, 252, 253, 265, 272, 280, 441, 831, 833, 852 – Hamiltonian, 115, 252 – homogeneous, 828 – horocycle, 832, 851, 852 – horospherical, 859 – infra-homogeneous, 833 – partially hyperbolic, 880 – rectilinear, 850 – suspension, 382, 388, 391 – unipotent, 854 – Weyl chamber, 850
folding, 587 folklore theorem, 622 Følner set, 14, 83, 234 forces, 581, 582 forcing, 581–583 form – contact, 120, 1136 – index, 463 – normal, 104, 145, 289 – symplectic, 114 formula – entropy, 487, 488, 624 – Pinsker, 211 forward-matching method, 264 frame bundle, 726 Fredholm index, 1171 free – energy, 364, 397 – particle motion, 252 frequency locking, 170 Fuchsian subgroup of a Lie group, 825 full shift, 633 function – µ-harmonic, 967 – bounded µ-harmonic, 967 – complexity, 45 – correlation, 362, 363, 390, 391 – harmonic, 976 – left uniformly continuous, 967 – tempered, 299 – transfer, 11, 142, 287 fundamental cocycle, 790 fundamental-group entropy, 39 Furstenberg – boundary, 713, 976 – condition, 943, 1002 – correspondence principle, 46, 85 – entropy, 716, 980 G– invariant function, 678 – map, 678 – – relative to a measure, 678 – representation – – quasi-regular, 981 – space, 937 – – homogeneous, 712 – – irreducible, 708 (G, µ)-boundary, 969 – compact, 969 (G, µ)-space, 968, 981 – compact, 968 gap, spectral, 946, 965, 995, 1004
Subject Index of Volume 1A Gauss transformation, 624 Gaussian dynamical system, 720 generator, 61, 78, 81, 193, 207 – of cocycle, 297 – one-sided, 78 geodesic – Anosov flow, 473 – flow, 120, 123, 135, 150, 242, 252, 253, 265, 272, 280, 441, 831, 833, 852 – length space, 483 – stretch, 496 geometric decomposition of S 3 , 1174 geometric structure, rigid, 495, 724 Gibbs – ensemble, 334, 337 – entropy, 393, 394 – measure, 334, 349, 352, 359, 361, 364, 369, 375, 377, 381, 385, 389, 390, 393 – state, 334, 339, 342, 345, 347, 348, 351, 352 global – surface of section, 1155, 1162, 1164 – system of transversal sections, 1174 gluing, 587 good periodic approximation, 70, 160, 183 graph – Lagrangian, 461 – shift, 779 – transform, 302 – – Hadamard method, 256 Gromov – representation, 733 – width, 117, 1138 group – algebraic, 676 – amenable, 694 – extension, 22, 182 – semisimple Lie, 673 – stable, 709 – unstable, 709 – Veech, 1059 growth – asymptotic, 274 – volume, 520 H– reduction, 724 – representation, regular, 947 Haar measure, 57, 817, 823 Hadamard – graph transform method, 256 – manifold, 476 Hadamard–Cartan theorem, 476 Hadamard–Perron theorem, 130, 132, 138, 139, 257
1175
half-pinched Anosov diffeomorphism, 751 Hamiltonian – flow, 115, 252 – system, 1131 – vector field, 115 Hamming metric, 81 harmonic function, 493, 976 Hartman–Grobman theorem, 139, 266 Hausdorff topology, 821 Hayashi connecting lemma, 270 Heisenberg group, 707 heteroclinic, 140, 158, 1109 – chain, 1113, 1115 – cycle, 1180 – orbit, 1180 higher rank Abelian action, 897 Hilbert bundle, 701 Hölder – continuity, 40, 41, 73, 95, 133, 142, 143, 246, 261, 262, 265 – exponent, 262, 263, 265, 276, 287 holomorphic dynamics, 125 holonomy, 261, 266, 283 – semigroup, 99 homeomorphism of finite order, 584 homoclinic, 1109 – orbit, 1180 – point, 806 – – transverse, 249 – tangles, 241, 249 homogeneous – G-space, 712 – action, 828 – flow, 828 – measure, 861 – space, 823 – subset, 824 homological entropy, 39 homology – Conley index, 561 – zeta function, 570 homotopical entropy, 40 homotopy rotation class, 48 Hopf argument, 144 horizontal space, 455 horocycle flow, 89, 136, 181, 832, 851, 852 horosphere, 485 horospheric foliation, 265, 486, 491 horospherical – flow, 859 – subgroup of a Lie group, 818, 819, 835 horseshoe, 134, 148, 248, 249, 308 Howe–Moore ergodicity theorem, 708
1176
Subject Index of Volume 1A
hull, 581 – algebraic, 692, 728 hyperbolic, 561 – dynamics, 127, 151 – fixed point theorem, 137, 139, 255, 267, 275 – measure, 147, 304 – point, 303 – set, 131, 142, 144, 248, 257, 263, 561 – – for flow, 252 – solution, 1159 – system, 100, 110 hyperbolicity – normal, 141 – partial, 149 hypersurface, 1134 – contact type, 1134, 1135 – star-like, 1135 iceberg model, 794 immediate basin, 635 in involution, 117, 154 In Phase Theorem, 565 independent partitions, 54 index, 595 – form, 463 – Lefschetz, 566 – lemma, 464 – of periodic solution, 1159 induced – action, 13, 23, 688 – map, 19, 59 – representation, 703 – sequence, 656 inducing domain, 655 inequality – cross-ratio, 639 – Rokhlin, 76 infinitesimal generator, 723 information function, 74 – conditional, 75 infra-homogeneous flow, 833 infranilmanifold, 251, 278 integrable system, 153, 154 – completely, 117 interaction, 335, 336, 343, 348, 351, 352 intermediate value theorem, 47, 607 interval exchange, 90, 176, 177, 183, 186, 187, 191, 192, 1027 invariant – Borel measure, 56 – circle, 164 – component, 585 – curve theorem, 163 – distribution, 109, 179, 181, 182, 188
– manifold, 107 – mean, 994 – measure, 129, 273, 1034 – parabolic, 747 – reduction, 691 – set, 9 – spectral, 64, 698 – tori, 170, 1116 inverse limit, 10, 22, 61, 107 involution, Cartan, 672 irreducible, 698 – G-space, 708 – lattice, 825 – matrix, 333 – representation, 821 – subshift, 633 – totally, 940 isolated invariant set, 551 isolating – block, 98, 557 – neighborhood, 551 isometric – action, 718 – extension, 22, 62, 108, 182, 226 isometry, 88, 152 isomorphism, 7 – spectral, 69, 698 – theorem, Ornstein, 222 isotropic, 461 – subspace, 114 isotropy subgroup of a Lie group, 828 iterated logarithm law, 962 itinerary, 423 Iwasawa decomposition, 674, 819 Jacobi – equation, 252, 253, 459 – field, 252, 456, 459 – tensor, 460, 470 Jacobian, unstable, 307 jet data, 288 joining, 59, 61, 71, 208 joint partition, 54, 75 Jordan decomposition, 822 Jordan curve theorem, 48 Julia set, 427 K– automorphism, 73, 212 – property, 73, 77, 79, 80, 991, 1001 k-prong singularity, 584 K-quasisymmetric, 640 Kakutani equivalence, 60, 63, 79, 80, 175, 228
Subject Index of Volume 1A Kakutani–Markov fixed point property, 83 KAM theory, 155 Kanai connection, 492, 494 Katok entropy rigidity conjecture, 294 Kazhdan property, 706 kernel, tempering, 300 Khintchine–Groshev theorem, 886, 907 Killing field, 728 Klingenberg’s theorem, 475 kneading sequence, 606 knotted periodic orbit, 1159 Kolmogorov theorem, 169 Kolmogorov–Sinai entropy, 364 Kronecker factor, 69 Kryloff–Bogoliouboff theorem, 83, 88, 92 Kupka–Smale theorem, 110–112 L-functions, 438 labeled graph, 787 lag, 560 Lagrange equation, 118 Lagrangian – dynamical system, 118 – graph, 461 – subspace, 461 large deviations, 364, 394, 396, 397 lattice, 675, 823 – admissible, 919 – arithmetic, 825 – dual, 920 – irreducible, 825 – unimodular, 826 law of large numbers, 934 leafwise regularity, 266 least action principle, 124 Lebesgue – point, 53, 69 – space, 53, 296, 298, 301 – spectrum – – countable, 72, 73, 80, 181 – – simple, 73 Lefschetz – fixed point theorem, 567 – index, 566 – number, 566 – zeta function, 412 left uniformly continuous functions, 967 left-invariant mean, 694 Legendre transform, 119 lemma – Borel–Cantelli, 886 – Morse, 123, 480 – Rokhlin, 64, 216 – shadowing, 138, 147–149, 268, 306, 371, 563
1177
length space, geodesic, 483 Levi decomposition, 820 Levi-Civita connection, 455 Lie group – Ad-proper, 845 – Euclidean, 818 – exponential, 818 – nilpotent, 818 – of type (I), 818 – Q-algebraic, 822 – Q-anisotropic, 822 – Q-split, 822 – R-algebraic, 821 – R-split, 819 – reductive, 822 – semisimple, 819 – simple, 819 – solvable, 818 – totally noncompact, 819 – triangular, 818 – unimodular, 817 Lie transformation group, 724 limit – asymptotic, 1148 – quasi-projective, 951, 958 – set, 23 line bundle, canonical, 1132 line elements, projective, 953 linearizable map, 659 linearization, 97 linking arguments, 1105 Liouville measure, 120, 825 Liouville–Arnold theorem, 117, 154, 169 Liouvillian, 158, 160, 164, 166, 168, 171, 173, 174, 178, 187, 192 Lipschitz, 139, 173, 265, 277 Littlewood’s conjecture, 914 Livschitz theorem, 142, 143, 147, 148, 287, 306, 512 local – action, 723 – analysis, 98 – maximality, 21, 130 – product structure, 144, 248, 260 – rigidity, 753 locally – closed, 683 – maximal, 248 – – hyperbolic set, 248, 271 logarithm law, 885 logistic family, 627, 657 loosely Bernoulli, 229 Luzin theorem, 305
1178
Subject Index of Volume 1A
Lyapunov – block form, 300, 302 – cocycle, 105, 286 – exponent, 147, 262, 298, 302, 304, 629, 935, 936, 977, 997 – – upper, 298 – function, complete, 554 – metric, 132, 248, 299, 301 – norm, 247, 301, 302 – scalar product, 301 – spectrum, 935, 936, 942 – – simple, 1003 Mackey range, 13, 62, 689 Mahler – compactness criterion, 826 – measure, 802 – problem, 911, 912 Mañé – closing lemma, 112, 146 – – ergodic, 270 manifold – admissible, 303 – asymptotically harmonic, 493 – contact, 120 – Hadamard, 476 – invariant, 107 – nondegenerate, 912 – rank-1, 282 – slow, 259 – stable, 112, 130, 140, 142, 148, 258, 260, 305, 306 – strong stable, 260 – strong unstable, 260 – symplectic, 114, 1131 – unstable, 112, 140, 258, 260, 306 map – boundary, 972 – G-, 678 – induced, 19, 59 – Markov, 587 – natural, 503 – nondegenerate, 857, 912 – Poincaré, 59 – section, 187 – twist, 156, 164 Margulis – arithmeticity theorem, 825 – measure, 282, 491 marked length spectrum, 510, 515 Markov – chain, topological, 42, 82, 127, 129, 144, 330, 332–334, 338, 340, 368, 381, 632 – cover, 48
– extension, 419, 631 – map, 587 – measure, 58, 73, 78 – operator, 936, 955 – partition, 129, 144, 147, 149, 281, 325, 328, 330, 332, 334, 368, 370, 375, 380, 381, 396, 428 – process, 936, 955 – property, 325, 326 – section, 378, 380, 384, 386, 388, 390, 434 – shift, 773 Mather – set, 1116 – spectrum, 131, 247, 259, 264, 272, 279 matrix – aperiodic, 334 – coefficient, 844 – irreducible, 333 – primitive, 334 – transition, 332, 335, 381 – transitive, 44, 58 Mautner phenomenon, 710, 837, 841, 855 maximal spectral type, 64, 69, 71, 72, 74, 80, 160 mean, invariant, 994 measurable – action, 677 – partition, 53 measurably isometric, 704 measure – µ-stationary, 937, 968 – admissible, 967 – Bernoulli, 58, 73, 74, 77 – class, smooth, 97, 103, 108 – ergodic, 679 – Gibbs, 334, 349–352, 359–361, 364–369, 375, 377, 381–385, 389, 390, 393 – homogeneous, 861 – hyperbolic, 147, 304 – invariant, 129, 273, 1034 – Mahler, 802 – Margulis, 282, 491 – Markov, 58, 73, 78 – of maximal entropy, 82, 94, 282, 366, 487, 528, 532 – proper, 940, 956 – quasi-invariant, 677 – spectral, 68, 69 – transversal, 491 measure-theoretic entropy, 350, 364, 401 measured foliation, 183 measures, conditional, 54 method, variational, 123 metric – adapted, 248
Subject Index of Volume 1A – cylinder, 1024 – isomorphism, 58, 69, 70, 90 – Lyapunov, 132, 248, 299–301 – Rokhlin, 75 mild mixing, 71 Milnor–Thurston zeta function, 423 minimal – action, 954, 992 – entropy, 502 – parabolic subgroup, 714 – set, 19, 88 minimality, 19, 88, 89, 91, 152, 156, 172, 175, 178, 181, 186, 192, 1024 minimizing property of Jacobi fields, 465 mirror equation, 167 mixing, 50, 72, 82, 95, 129, 142–145, 160, 173, 175, 178, 179, 183, 184, 186, 187, 191, 192, 633, 704, 1070 – multiple, 72 – subshift, 633 – topological, 26, 72, 271, 274 – weak, 71, 704 modular surface, 825 moduli space, 1032 momenta, 119 monotone maps, P -, 581 Moore subgroup of a Lie group, 835, 837 Morse – lemma, 123, 480 – sequence, 44, 608 Moser–deLatte normal form, 290 Mostow rigidity, 502 multibump solutions, 1109, 1117, 1121 multiple – mixing, 72 – Poincaré recurrence, 86 – weak mixing, 71 multiplicative ergodic theorem, 298 multiplicity – function, 702 – of exponent, 298 multiply nonwandering, 46 natural – 1-form, 456 – extension, 10, 22, 61, 80, 107 – map, 503 near action, 689 negative puncture, 1147 neighborhood – isolating, 551 – regular, 302 – symmetric, 684 neutral subgroup of a Lie group, 835
1179
Newton method, 159 Nielsen number, 595 Nielsen–Thurston theory, 183 nilmanifold, 824 nilpotent Lie group, 818 nilradical, 818 Noether theorem, 117 non-compact cocycle, 1002 non-squeezing, 1138 nonamenable group, 947, 965 nondegenerate – manifold, 912 – map, 857, 912 nonflat critical point, 618 nonlinearity measure, 611 nonpositive curvature, 476 nonrandom filtration, 942, 998 nonstandard smooth realization, 153, 172 nonuniform hyperbolicity, 132 nonwandering – multiply, 46 – point, 25 – set, 25 norm – adapted, 247 – Lyapunov, 247, 301, 302 normal – form, 104, 145, 289 – – Birkhoff, 166 – hyperbolicity, 141 – subgroup theorem, 757 normalized potential function, 359, 367, 368 NT homeomorphism, 585 nuclear operator, 437 obstruction, Anosov, 290 one-sided generator, 78 open book decomposition, 1157, 1180 operator – Markov, 936, 955 – Riccati, 256 – Ruelle–Perron–Frobenius, 354 – transfer, 57, 108, 426 Oppenheim conjecture, 860, 901 – quantitative versions, 902 orbit – complexity, 128 – cylinder, 1145, 1148, 1173 – equivalence, 8, 18, 59, 67, 686, 697, 739, 740 – factor, 21 – heteroclinic, 1180 – homoclinic, 1180 – periodic, 143, 1052
1180
Subject Index of Volume 1A
– twisted, 413 orbit growth, asymptotic, 32 order of criticality, 618 Ornstein – copying lemma, 218 – isomorphism theorem, 222 Oseledets multiplicative ergodic theorem, 147 overtwisted contact structure, 1152 P– monotone maps, 581 – stationary, 936 p-contracting, 948 – (semi)group, 948 – sequence, 948 p-irreducible, strongly, 940 Palais–Smale condition, 1098 parabolic – dynamics, 175, 194 – invariant, 747 – Levi subgroup of a Lie group, 820 – subgroup, 703, 713 – – minimal, 714 – system, 101 partial hyperbolicity, 101, 133, 149, 284 partially hyperbolic – element of a Lie group, 818 – flow, 880 particle motion, free, 252 partition – function, 337, 343 – Markov, 129, 144, 147–149, 281, 325–328, 330, 332, 334, 368, 370, 375, 380, 381, 396, 428 – measurable, 53 past, 210 pattern, 581, 583 – twist, 585 Patterson–Sullivan measure, 490, 491 period-doubling bifurcations, 648 periodic – data, 105, 287, 288 – orbit, 143, 1052 – – Birkhoff, 157 – point, 7, 31, 142, 145, 411 – – transverse, 111 – solution, 1095 periodic trajectory, stable, 1075 Perron number, 784 Perron–Frobenius operator, 108, 622 Perron–Irwin method, 256 perturbation, 155 Pesin – entropy formula, 309 – set, 147, 305
– tempering kernel, 300 Pestov’s identity, 511, 537, 538 piecewise monotone, 603 pinching, 253, 264, 1101 Pinsker – algebra, 79, 212 – formula, 211 Plancherel – countable spectrum, 717 – formula, 703 Plykin attractor, 251 Poincaré – classification, 28 – map, 59 – recurrence – – multiple, 86 – – theorem, 59, 682 – section map ψ , 1162 – series, 489 Poincaré–Bendixson theory, 48 point – homoclinic, 806 – hyperbolic, 303 – Lebesgue, 53, 69 – nonwandering, 25 – periodic, 7, 31, 142, 145, 411 – regular, 302, 629 pointed space map, 560 Poisson – boundary, 971 – bracket, 117 – transform, 968 polar decomposition, 827 polygonal billiard, 1017 polynomial-like extension, 653 positive puncture, 1147 potential – function, 349–352, 354, 364, 368, 369, 384 – – normalized, 359, 367, 368 – singular, 1106 pressure, 38, 92, 94, 95 – topological, 343, 349, 365, 384, 402 primary pattern, 592 prime number theorem, 275 primitive matrix, 334 principal series, 700 principle, variational, 93, 94, 125, 148, 374, 393, 402, 487, 770, 1141 process, Markov, 936, 955 product, 149 – relative, 56 – structure, local, 144, 248, 260 profile, entropy, 983
Subject Index of Volume 1A profinitely dense, 744 projective – action, 714 – line elements, 953 prongs, 584 proper – action, 683 – measure, 940, 956 – rectangle, 324, 325, 328, 332, 333, 379 property – Bernoulli, 361–363, 373, 399 – K-, 73, 77, 79, 80, 991, 1001 – Markov, 325, 326 – specification, 269 – T , 281, 705 property-(D), 855 proximal – action, 949 – strongly, 713 proximality, 28, 713, 949 pseudo-Anosov, 186, 584, 585 – single-fixed point, 593 pseudo-orbit, 138, 268 pseudoholomorphic curve, 1141, 1153 Pugh closing lemma, 112, 146, 270 puncture – negative, 1147 – positive, 1147 – removable, 1147 pure point spectrum, 70, 71, 89 Q– character, 822 – rank, 822 Q-algebraic – Lie group, 822 – representation, 823 Q-anisotropic Lie group, 822 Q-split Lie group, 822 QNS, 617 quadratic differential, 191, 1022 quadrature, 154 quasi-geodesic, 479 quasi-invariant, 52 – measure, 677 quasi-isometry, 479 quasi-lattice, 824 quasi-negative Schwarzian, 617 quasi-projective – limit, 951, 958 – transformation, 950 quasi-regular – G-representation, 981
– representation, 945, 946, 981, 1004 quasi-unipotent – element of a Lie group, 818 – subgroup of a Lie group, 818 quasiminimality, 184–186 quasisymmetric, 620 R-algebraic Lie group, 821 R-diagonalizable – element of a Lie group, 818, 822 – subgroup of a Lie group, 822 R-rank, 819 R-split Lie group, 819 R-property, 861 radical, 818 Radon–Nikodym – cocycle, 55, 680 – factor, 989 Raghunathan’s conjecture, 860 random ergodic theorem, 991, 993, 995 random walk entropy, 982 rank – of a nonpositively curved manifold, 515 – Q-, 822 – real, 673 – rigidity, 515 rank-1 – manifold, 282 – space, 515 rate of convergence, 992 rational – polygon, 1020 – zeta function, 414 Ratner’s theorem, 742, 861–863 Rauch’s comparison estimates, 465 real Fatou conjecture, 657 real rank, 673 rectangle, 324, 328, 370, 375, 379 – proper, 324–328, 332, 333, 379 rectifiable set, 892 rectilinear flow, 850 recurrence, 24, 129, 682 – uniform, 24 reducible, 585 reducing curves, 584 reduction theory, 826 reductive – group, 672 – Lie group, 822 Reeb vector field, 1136, 1160, 1173 refinement, 54 region of instability, 165 regional recurrence, 25, 271, 272
1181
1182 regular – closed geodesic, 520, 527 – H -representation, 947 – neighborhood, 302 – point, 302, 629 – representation, 699, 820, 946, 947 regularity, 261 – of the horospherical foliation, 492 – of topological entropy, 500 relative – entropy, 79 – product, 56 relatively independent joinings, 208 relaxation oscillations, 241, 249 removable puncture, 1147 renormalization, 610, 654 renormalized functional, 1114 repeller, 21, 251, 402, 403, 552 representation – adjoint, 817 – algebraic linear, 823 – canonical, 586 – continuous, 85 – Gromov, 733 – induced, 703 – irreducible, 821 – Q-algebraic, 823 – quasi-regular, 945, 946, 981, 1004 – regular, 699, 820, 946, 947 – unipotent, 824 – unitary, 697, 698, 820 resonance, 158, 289 restricted root, 674 restrictive interval I , 631 return – map, 183, 1156 – probability, 965, 966 Riccati – equation, 252 – operator, 256 Riemannian metric, 455 rigid – geometric structure, 495, 724 – surface, 1173 rigidity, 70, 71, 160, 180 – cocycle, 188 – entropy, 294, 495, 500 – local, 753 – rank, 515 – smooth, 286, 292 – spectral, 509 Rokhlin – inequality, 76 – lemma, 64, 216
Subject Index of Volume 1A – metric, 75 root space, 674 – decomposition, 674 rotation number, 27, 28, 602 – of constant type, 602 Ruelle zeta function, 424 Ruelle–Perron–Frobenius – operator, 354 – theorem, 354 saddle connection, 184, 1024 Sasaki metric, 457 Schwarzian derivative, 614 section, 62, 107, 182, 183, 189 – map, 187 – Markov, 378, 380, 384–386, 388, 390, 434 sectional curvature, 252 self-joining, 59, 60 self-linking number, 1164, 1174 semisimple – element of a Lie group, 818, 822 – Lie – – algebra, 673 – – group, 673, 819 sensitive dependence, 127, 149 separated sets, 34 sequence – contracting, 948, 956 – induced, 656 – Morse, 44, 608 – p-contracting, 948 – totally contracting, 948 series – discrete, 700 – Poincaré, 489 set – Aubry–Mather, 157, 164, 165, 174 – basic, 132, 248, 326–328, 330, 334, 368, 370– 372, 374, 378, 389, 401, 414, 564 – ergodic, 85 – hyperbolic, 131, 142–144, 248, 257, 263, 561 – invariant, 9 – Mather, 1116 – minimal, 19, 88 – nonwandering, 25 – Pesin, 147, 305 – symmetric, 684 shadowing, 94, 267 – lemma, 138, 147, 149, 268, 306, 371, 563 – theorem, 269, 275 Shannon–McMillan–Breiman theorem, 76, 214 Sharkovsky – ordering, 582
Subject Index of Volume 1A – theorem, 601, 610 sharp determinant, 430 shear, 176 shift, 18, 41–47, 73, 129 – Bernoulli, 58, 80, 361, 1179, 1180 – equivalence, 560, 782 – homeomorphism, 333, 354 – Markov, 773 Siegel – disk, 661 – summation formula, 826 simple – Lebesgue spectrum, 73 – Lie group, 819 – Lyapunov – – exponent, 948 – – spectrum, 948, 949 – spectrum, 70, 160 Sinai–Ruelle–Bowen (SRB) measure, 143, 148, 283, 309, 369, 373–375, 385–387, 391, 392, 395–399, 402, 404 singular – closed geodesic, 522, 536 – potential, 1106 – spectrum, 72 skew product, 12, 79, 970, 979, 991, 1004 sliding block code, 780 slow – entropy, 37, 80, 92 – manifold, 259 – subbundle, 259 Smale attractor, 107, 134, 250 small denominator, 105, 162 smooth – conjugacy, 103, 105 – measure class, 97, 103, 108 – rigidity, 286, 292 – stability, 170 sofic, 45, 94, 774, 787 solenoid, 107, 134, 250 solution – elliptic, 1159 – hyperbolic, 1159 – periodic, 1095 solvable Lie group, 818 solvmanifold, 824 space – average, 50 – homogeneous, 823 – Lebesgue, 53, 296, 298, 301 – of partitions, 76 – rank-1, 515 – Teichmüller, 1031 spanning set, 35
1183
special flow, 13, 63, 172, 186, 187, 191, 229, 254 specification, 82, 94, 95, 143, 269, 273, 274, 282 – for flows, 270 – property, 269 – – for flows, 270 – weak, 769 spectral – decomposition, 43, 132, 142, 143, 147, 149, 271, 279 – gap, 946, 965, 995, 1004 – invariant, 64, 698 – isomorphism, 69, 698 – measure, 68, 69 – rigidity, 509 – theorem, 702 spectrum – discrete, 69, 89, 704, 1003 – Lyapunov, 935, 936, 942 – Mather, 131, 247, 259, 264, 272, 279 – simple, 70, 160 – singular, 72 sphere – at infinity, 478 – topology, 478 spherical finite energy foliation, 1171 Sprindžuk’s conjectures, 912, 915 stability – cocycle, 110 – of the solar system, 169, 170 – smooth, 170 – theorem, 146, 276 – topological, 106, 275 stable – and unstable – – foliations, 486 – – manifolds, 563 – – spaces, 472 – behavior, 151 – energy surface, 1134 – ergodicity, 286 – finite energy foliation, 1171, 1172 – group, 709 – manifold, 112, 130, 140, 142, 148, 258, 260, 305, 306 – – at periodic point, 112 – – strong, 260 – – theorem, 255, 563 – periodic trajectory, 1075 standard – action, 751 – Borel G-space, 678 star-like – energy surface, 1135, 1153, 1158
1184
Subject Index of Volume 1A
– hypersurface, 1135 state, Gibbs, 334, 339, 342, 345, 347, 348, 351, 352 stationary measure, 715 stochastic, 58, 127 strange attractor, 127 stratum, 1032 stretch, geodesic, 496 strict G-map, 678 strictly convex energy surface, 1158, 1159 strong – force condition, 1107 – shift equivalence, 781 – specification, 769 – stable manifold, 260 – unstable manifold, 260 strongly – ergodic, 994, 1003 – irreducible – – cocycle, 1002 – – group, 939, 940 – – measure, 939, 940 – – semigroup, 939, 940 – p-irreducible, 940 – proximal, 713 – – action, 949, 954 structural stability, 106, 127, 129, 139, 145, 275, 276, 484, 634 structure, contact, 120, 272, 1136 structures of finite type, 724 subalgebra, Cartan, 673 subbundles, 261 subgroup – of a Lie group – – Auslander, 835 – – Cartan, 819 – – Dani, 835, 837 – – epimorphic, 895 – – Fuchsian, 825 – – horospherical, 818, 819, 835 – – isotropy, 828 – – Moore, 835, 837 – – neutral, 835 – – parabolic Levi, 820 – – R-diagonalizable, 822 – – quasi-unipotent, 818 – – uniform, 823 – – unipotent, 818, 822 – parabolic, 703, 713 subshift, 633, 773 – irreducible, 633 – mixing, 633 – of finite type, 42 subspace – isotropic, 114
– Lagrangian, 461 Sullivan conjecture, 295 sum, continuous, 701 support, 49 surface, 193 – flat, 1022 – of section, global, 1155, 1162, 1164 – rigid, 1173 – Veech, 1059 suspension, 11, 23, 62, 108, 253, 272, 688 – flow, 382, 388, 391 symbolic – dynamical system, 18, 41 – dynamics, 242 symmetric – neighborhood, 684 – set, 684 symplectic – capacity, 116, 1138 – form, 114 – – exact, 122 – manifold, 114, 1131 – structure on T M, 457 syndetic, 24 system – elliptic, 101 – Hamiltonian, 1131 – hyperbolic, 100, 110 – of transversal sections, global, 1174 – parabolic, 101 Szemerédi theorem, 85 (T , T −1 )-transformation, 228 tame, 683 tangles, homoclinic, 241, 249 Teichmüller – space, 1031 – theory, 183 telescope construction, 646 tempered – cocycle, 299 – function, 299 tempering, 299 – kernel, 300 – – Pesin, 300 tensor, Jacobi, 460, 470 tent maps, 607, 609 theorem – Ruelle–Perron–Frobenius, 354 – shadowing, 269, 275 – Sharkovsky, 601, 610 – spectral, 702 – stability, 146, 276
Subject Index of Volume 1A – transversality, 111 thermodynamic limit, 337–339 thick set, 882 tight – 3-sphere, 1165 – contact structure, 1152 tightening, 587 time average, 14, 49 time change, 8, 188, 254 Toeplitz shift, 44 topological – CE condition, 645 – conjugacy, 18, 103, 768 – entropy, 34–37, 308, 365, 487, 609 – – for noncompact spaces, 36 – – original definition, 36 – equivalence, 848 – factor, 36, 768 – Markov chain, 42, 82, 127, 129, 144, 330, 332, 334, 338, 340, 368, 381, 632 – mixing, 26, 72, 271, 274 – pressure, 343, 349, 365, 384, 402 – stability, 106, 275 – transitivity, 19, 26, 88, 89, 271, 272, 1026 – weak mixing, 433 topologically engaging, 744 topology – cone, 478 – sphere, 478 – Zariski, 821 tori, invariant, 170, 1116 total Conley–Zehnder index, 1165 totally – contracting, 948 – – (semi)group, 948 – – sequence, 948 – engaging, 744 – irreducible, 940 – noncompact Lie group, 819 transfer – function, 11, 142, 287 – operator, 57, 108, 426 transform – graph, 302 – Legendre, 119 – Poisson, 968 transformation – canonical, 114 – quasi-projective, 950 transition – matrix, 332, 335, 381 – probabilities, 936 transitive, 142, 144, 145 – matrix, 44, 58
1185
transitivity, 150, 152, 156, 178, 182, 184 – topological, 19, 26, 88, 89, 271, 272, 1026 transversal, 107, 566 – measure, 491 transversality, 110, 139, 276 – theorem, 111 transverse – fixed point, 111 – homoclinic point, 249 – periodic point, 111 triangular Lie group, 818 twist, 585 – interval, 157 – map, 156, 164 – pattern, 585 twisted – orbit, 413 – product, 13, 23, 688 typical point, 66, 85 u-Gibbs measure, 369 UHC, 645 uniform – hyperbolicity on cycles, 645 – recurrence, 24 – subgroup of a Lie group, 823 unimodal pattern, 594 unimodular – group, 675 – lattice, 826 – Lie group, 817 unipotent – element of a Lie group, 818, 822 – flow, 854 – radical, 822 – representation, 824 – subgroup of a Lie group, 818, 822 unique ergodicity, 87–91, 1034 unitary – dual, 699 – representation, 697, 698, 820 universality, 654 unstable – group, 709 – manifold, 112, 140, 258, 260, 306 – – at periodic point, 112 – – strong, 260 upper Lyapunov exponent, 629 value, critical, 554 variational – method, 123 – principle, 93, 94, 125, 148, 374, 393, 402, 487, 770, 1141
1186 Veech – group, 1059 – surface, 1059 vertex shift, 779 vertical space, 455 very weak Bernoulli process, 220 volume – density, 732 – entropy, 487 – growth, 520 – lemma – – first, 370 – – second, 370 von Neumann mean ergodic theorem, 65 wandering interval, 603, 638 Wang tiles, 777 weak – mixing, 71, 704 – – multiple, 71 – – topological, 433 – specification, 769 weakly – hyperbolic action, 753 – hyperbolic space, 515
Subject Index of Volume 1A – orbit equivalent, 740 Weinstein conjecture, 1136, 1141, 1150 Weyl chamber flow, 850 width, Gromov, 117, 1138 Wronskian, 460 Zd -action, algebraic, 796 Zariski – dense, 821, 824 – – cocycle, 1002 – topology, 821 zeta function, 32, 33, 38, 274, 366, 569, 784 – Artin–Mazur, 411 – for Axiom A diffeomorphisms, 411 – for Axiom A flows, 431 – for subshifts, 415 – homology, 570 – Lefschetz, 412 – Milnor–Thurston, 423 – rational, 414 – Ruelle, 424 Zimmer’s program, 735 Zygmund, 265 – smoothness, 614