Dynamical systems, ergodic theory and applications:
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| Format: | Book |
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| Language: | English Russian |
| Published: |
Berlin [u.a.]
Springer
2000
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| Edition: | 2., expanded and rev. ed. |
| Series: | Encyclopaedia of mathematical sciences
100 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 1. Aufl. u.d.T.: Dynamical systems. - 2. Ergodic theory with applications to dynamical systems and statistical mechanics. Bildete Vol. 2 der "Encyclopaedia of mathematical sciences" |
| Physical Description: | X, 459 S. |
| ISBN: | 3540663169 |
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| 245 | 1 | 0 | |a Dynamical systems, ergodic theory and applications |c L. A. Bunimovich ... Ed. by Ya. G. Sinai |
| 250 | |a 2., expanded and rev. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2000 | |
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| 650 | 4 | |a Mécanique céleste | |
| 650 | 7 | |a Mécanique céleste |2 ram | |
| 650 | 7 | |a Théorie ergodique |2 ram | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Ergodic theory | |
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Record in the Search Index
| _version_ | 1804127840541605888 |
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| adam_text | CONTENTS I. GENERAL ERGODIC THEORY OF GROUPS OF MEASURE PRESERVING
TRANSFORMATIONS 1 II. ERGODIC THEORY OF SMOOTH DYNAMICAL SYSTEMS 103
III. DYNAMICAL SYSTEMS ON HOMOGENEOUS SPACES 264 IV. THE DYNAMICS OF
BILLIARD FLOWS IN RATIONAL POLYGONS 360 V. DYNAMICAL SYSTEMS OF
STATISTICAL MECHANICS AND KINETIC EQUATIONS 383 SUBJECT INDEX 455 I.
GENERAL ERGODIC THEORY OF GROUPS OF MEASURE PRESERVING TRANSFORMATIONS
CONTENTS CHAPTER 1. BASIC NOTIONS OF ERGODIC THEORY AND EXAMPLES OF
DYNAMICAL SYSTEMS ( I.P. KORNFELD, YA.G. SINAI ) . . . . . . . . . . . .
. . . 2 § 1. DYNAMICAL SYSTEMS WITH INVARIANT MEASURES . . . . . . . . .
. . . . . . . 2 § 2. FIRST COROLLARIES OF THE EXISTENCE OF INVARIANT
MEASURES. ERGODIC THEOREMS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 11 § 3. ERGODICITY. DECOMPOSITION INTO ERGODIC
COMPONENTS. VARIOUS MIXING CONDITIONS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 18 § 4. GENERAL CONSTRUCTIONS . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 23 4.1. DIRECT PRODUCTS
OF DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . 23 4.2. SKEW
PRODUCTS OF DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . 24 4.3.
FACTOR-SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 25 4.4. INTEGRAL AND INDUCED AUTOMORPHISMS . . . . . . . . . .
. . . . . . . 25 4.5. SPECIAL FLOWS AND SPECIAL REPRESENTATIONS OF FLOWS
. . . . . . 26 4.6. NATURAL EXTENSIONS OF ENDOMORPHISMS . . . . . . . .
. . . . . . . . 28 CHAPTER 2. SPECTRAL THEORY OF DYNAMICAL SYSTEMS (
I.P. KORNFELD, YA.G. SINAI ) . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 30 § 1. GROUPS OF UNITARY OPERATORS AND SEMIGROUPS
OF ISOMETRIC OPERATORS ADJOINT TO DYNAMICAL SYSTEMS . . . . . . . . . .
. . . . . . . . 30 § 2. THE STRUCTURE OF THE DYNAMICAL SYSTEMS WITH PURE
POINT AND QUASIDISCRETE SPECTRA . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 33 § 3. EXAMPLES OF SPECTRAL ANALYSIS OF DYNAMICAL
SYSTEMS . . . . . . . . 35 § 4. SPECTRAL ANALYSIS OF GAUSS DYNAMICAL
SYSTEMS . . . . . . . . . . . . . 36 CHAPTER 3. ENTROPY THEORY OF
DYNAMICAL SYSTEMS ( I.P. KORNFELD, YA.G. SINAI ) . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 38 § 1. ENTROPY AND
CONDITIONAL ENTROPY OF A PARTITION . . . . . . . . . . . . . . 39 § 2.
ENTROPY OF A DYNAMICAL SYSTEM . . . . . . . . . . . . . . . . . . . . .
. . . . 40 § 3. THE STRUCTURE OF DYNAMICAL SYSTEMS OF POSITIVE ENTROPY .
. . . . . 43 CONTENTS 13 § 4. THE ISOMORPHY PROBLEM FOR BERNOULLI
AUTOMORPHISMS AND K -SYSTEMS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 45 § 5. EQUIVALENCE OF DYNAMICAL SYSTEMS
IN THE SENSE OF KAKUTANI . . . 53 § 6. SHIFTS IN THE SPACES OF SEQUENCES
AND GIBBS MEASURES . . . . . . . . 57 CHAPTER 4. PERIODIC APPROXIMATIONS
AND THEIR APPLICATIONS. ERGODIC THEOREMS, SPECTRAL AND ENTROPY THEORY
FOR THE GENERAL GROUP ACTIONS ( I.P. KORNFELD, A.M. VERSHIK ) . . . . .
. . . . . . . . . . . . . . . 61 § 1. APPROXIMATION THEORY OF DYNAMICAL
SYSTEMS BY PERIODIC ONES. FLOWS ON THE TWO-DIMENSIONAL TORUS . . . . . .
. . . . . . . . . . . . . . . 61 § 2. FLOWS ON THE SURFACES OF GENUS P *
1 AND INTERVAL EXCHANGE TRANSFORMATIONS . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 66 § 3. GENERAL GROUP
ACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69 3.1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 69 3.2. GENERAL DEFINITION OF THE ACTIONS OF
LOCALLY COMPACT GROUPS ON LEBESGUE SPACES . . . . . . . . . . . . . . .
. . . . . . . . . 70 3.3. ERGODIC THEOREMS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 71 3.4. SPECTRAL THEORY . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 74 § 4. ENTROPY
THEORY FOR THE ACTIONS OF GENERAL GROUPS . . . . . . . . . . . 76
CHAPTER 5. TRAJECTORY THEORY ( A.M. VERSHIK ) . . . . . . . . . . . . .
. . . . . . . 80 § 1. STATEMENTS OF MAIN RESULTS . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 80 § 2. SKETCH OF THE PROOF. TAME
PARTITIONS . . . . . . . . . . . . . . . . . . . . . . 84 § 3.
TRAJECTORY THEORY FOR AMENABLE GROUPS . . . . . . . . . . . . . . . . .
. . 89 § 4. TRAJECTORY THEORY FOR NON-AMENABLE GROUPS. RIGIDITY . . . .
. . . . 91 § 5. CONCLUDING REMARKS. RELATIONSHIP BETWEEN TRAJECTORY
THEORY AND OPERATOR ALGEBRAS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 94 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 95 ADDITIONAL
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 101 II. ERGODIC THEORY OF SMOOTH DYNAMICAL SYSTEMS CONTENTS
CHAPTER 6. STOCHASTICITY OF SMOOTH DYNAMICAL SYSTEMS. THE ELEMENTS OF
KAM-THEORY ( YA.G. SINAI ) . . . . . . . . . . . . . . . . . . . 106 §
1. INTEGRABLE AND NONINTEGRABLE SMOOTH DYNAMICAL SYSTEMS. THE HIERARCHY
OF STOCHASTIC PROPERTIES OF DETERMINISTIC DYNAMICS 106 § 2. THE
KOLMOGOROV-ARNOLD-MOSER THEORY (KAM-THEORY) . . . . . . . 109 CHAPTER 7.
GENERAL THEORY OF SMOOTH HYPERBOLIC DYNAMICAL SYSTEMS ( YA.B. PESIN ) .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 113 § 1. HYPERBOLICITY OF INDIVIDUAL TRAJECTORIES . . . .
. . . . . . . . . . . . . . . 113 1.1. INTRODUCTORY REMARKS . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 113 1.2. UNIFORM
HYPERBOLICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114 1.3. NONUNIFORM HYPERBOLICITY . . . . . . . . . . . . . . . . . . .
. . . . . . . 115 1.4. LOCAL MANIFOLDS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 116 1.5. GLOBAL MANIFOLDS . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 118 § 2. BASIC CLASSES
OF SMOOTH HYPERBOLIC DYNAMICAL SYSTEMS. DEFINITIONS AND EXAMPLES . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.1. ANOSOV
SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 118 2.2. HYPERBOLIC SETS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 121 2.3. LOCALLY MAXIMAL HYPERBOLIC SETS . . . . .
. . . . . . . . . . . . . . . 124 2.4. AXIOM A -DIFFEOMORPHISMS . . . .
. . . . . . . . . . . . . . . . . . . . . 125 2.5. HYPERBOLIC
ATTRACTORS. REPELLERS . . . . . . . . . . . . . . . . . . . . . 126 2.6.
PARTIALLY HYPERBOLIC DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . 128
2.7. MATHER THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 129 2.8. NONUNIFORMELY HYPERBOLIC DYNAMICAL SYSTEMS.
LYAPUNOV EXPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 131 § 3. ERGODIC PROPERTIES OF SMOOTH HYPERBOLIC DYNAMICAL SYSTEMS
. . 133 3.1. U -GIBBS MEASURES . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 133 3.2. SYMBOLIC DYNAMICS . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 135 3.3. MEASURES OF MAXIMAL ENTROPY
. . . . . . . . . . . . . . . . . . . . . . 137 3.4. CONSTRUCTION OF U
-GIBBS MEASURES . . . . . . . . . . . . . . . . . . . 137 104 CONTENTS
3.5. TOPOLOGICAL PRESSURE AND TOPOLOGICAL ENTROPY . . . . . . . . . .
138 3.6. PROPERTIES OF U -GIBBS MEASURES . . . . . . . . . . . . . . . .
. . . . . 141 3.7. SMALL STOCHASTIC PERTURBATIONS . . . . . . . . . . .
. . . . . . . . . . . 142 3.8. EQUILIBRIUM STATES AND THEIR ERGODIC
PROPERTIES . . . . . . . . . 143 3.9. ERGODIC PROPERTIES OF DYNAMICAL
SYSTEMS WITH NONZERO LYAPUNOV EXPONENTS . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 144 3.10. ERGODIC PROPERTIES OF ANOSOV
SYSTEMS AND OF UPH-SYSTEMS 146 3.11. CONTINUOUS TIME DYNAMICAL SYSTEMS .
. . . . . . . . . . . . . . . . 149 § 4. HYPERBOLIC GEODESIC FLOWS . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.1. MANIFOLDS
WITH NEGATIVE CURVATURE . . . . . . . . . . . . . . . . . . . 149 4.2.
RIEMANNIAN METRICS WITHOUT CONJUGATE (OR FOCAL) POINTS . . 153 4.3.
ENTROPY OF GEODESIC FLOWS . . . . . . . . . . . . . . . . . . . . . . .
. . 156 4.4. RIEMANNIAN METRICS OF NONPOSITIVE CURVATURE . . . . . . . .
. . 157 § 5. GEODESIC FLOWS ON MANIFOLDS WITH CONSTANT NEGATIVE
CURVATURE 158 § 6. DIMENSION-LIKE CHARACTERISTICS OF INVARIANT SETS FOR
DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 161 6.1. INTRODUCTORY REMARKS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2. HAUSDORFF
DIMENSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3. OTHER DIMENSION CHARACTERISTICS . . . . . . . . . . . . . . . . . .
. . . 164 6.4. CARATH´ EODORY DIMENSION STRUCTURE. CARATH´ EODORY
DIMENSION CHARACTERISTICS . . . . . . . . . . . . . . . 167 6.5.
EXAMPLES OF C -STRUCTURES AND CARATH´ EODORY DIMENSION CHARACTERISTICS .
. . . . . . . . . . . . . . . . . . . . . . . . . 169 6.6. MULTIFRACTAL
FORMALISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 §
7. COUPLED MAP LATTICES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 182 ADDITIONAL REFERENCES . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 190 CHAPTER 8. BILLIARDS
AND OTHER HYPERBOLIC SYSTEMS ( L.A. BUNIMOVICH ) . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 § 1.
BILLIARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 192 1.1. THE GENERAL DEFINITION OF A BILLIARD .
. . . . . . . . . . . . . . . . . 192 1.2. BILLIARDS IN POLYGONS AND
POLYHEDRONS . . . . . . . . . . . . . . . . 194 1.3. BILLIARDS IN
DOMAINS WITH SMOOTH CONVEX BOUNDARY . . . . . 196 1.4. DISPERSING OR
SINAI BILLIARDS . . . . . . . . . . . . . . . . . . . . . . . . 198 1.5.
THE LORENTZ GAS AND HARD SPHERES GAS . . . . . . . . . . . . . . . 206
1.6. SEMI-DISPERSING BILLIARDS AND BOLTZMANN HYPOTHESES . . . . . 206
1.7. BILLIARDS IN DOMAINS WITH BOUNDARY POSSESSING FOCUSING COMPONENTS .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209 1.8. HYPERBOLIC DYNAMICAL SYSTEMS WITH SINGULARITIES (A GENERAL
APPROACH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
1.9. MARKOV APPROXIMATIONS AND SYMBOLIC DYNAMICS FOR HYPERBOLIC
BILLIARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
1.10. STATISTICAL PROPERTIES OF DISPERSING BILLIARDS AND OF THE LORENTZ
GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 1.11.
TRANSPORT COEFFICIENTS FOR THE SIMPLEST MECHANICAL MODELS 222 CONTENTS
105 § 2. STRANGE ATTRACTORS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 224 2.1. DEFINITION OF A STRANGE ATTRACTOR .
. . . . . . . . . . . . . . . . . . . . 224 2.2. THE LORENZ ATTRACTOR .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2.3. SOME
OTHER EXAMPLES OF HYPERBOLIC STRANGE ATTRACTORS . . . 230 ADDITIONAL
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 231 CHAPTER 9. ERGODIC THEORY OF ONE-DIMENSIONAL MAPPINGS (
M.V. JAKOBSON ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 234 § 1. EXPANDING MAPS . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 234 1.1.
DEFINITIONS, EXAMPLES, THE ENTROPY FORMULA . . . . . . . . . . . . 234
1.2. WALTERS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 237 § 2. ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR
NONEXPANDING MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 239 2.1. SOME EXAMPLES . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 239 2.2. INTERMITTENCY OF STOCHASTIC AND
STABLE SYSTEMS . . . . . . . . . . 241 2.3. ERGODIC PROPERTIES OF
ABSOLUTELY CONTINUOUS INVARIANT MEASURES . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 243 § 3. FEIGENBAUM UNIVERSALITY LAW . . .
. . . . . . . . . . . . . . . . . . . . . . . . 245 3.1. THE PHENOMENON
OF UNIVERSALITY . . . . . . . . . . . . . . . . . . . . 245 3.2.
DOUBLING TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . . .
. . . 247 3.3. NEIGHBORHOOD OF THE FIXED POINT . . . . . . . . . . . . .
. . . . . . . 249 3.4. PROPERTIES OF MAPS BELONGING TO THE STABLE
MANIFOLD OF * 251 § 4. RATIONAL ENDOMORPHISMS OF THE RIEMANN SPHERE . .
. . . . . . . . . . 252 4.1. THE JULIA SET AND ITS COMPLEMENT . . . . .
. . . . . . . . . . . . . . . 252 4.2. THE STABILITY PROPERTIES OF
RATIONAL ENDOMORPHISMS . . . . . . 254 4.3. ERGODIC AND DIMENSIONAL
PROPERTIES OF JULIA SETS . . . . . . . . 255 BIBLIOGRAPHY . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 256 III. DYNAMICAL SYSTEMS ON HOMOGENEOUS SPACES CONTENTS CHAPTER 10.
DYNAMICAL SYSTEMS ON HOMOGENEOUS SPACES ( S.G. DANI ) . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
266 § 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 266 1.1. MEASURES ON HOMOGENEOUS SPACES
. . . . . . . . . . . . . . . . . . . . 266 1.2. EXAMPLES OF LATTICES .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 1.3.
ERGODICITY AND ITS CONSEQUENCES . . . . . . . . . . . . . . . . . . . .
. 271 1.4. ISOMORPHISMS AND FACTORS OF AFFINE AUTOMORPHISMS . . . . . .
. 272 § 2. AFFINE AUTOMORPHISMS OF TORI AND NILMANIFOLDS . . . . . . . .
. . . . . . 273 2.1. ERGODIC PROPERTIES; THE CASE OF TORI . . . . . . .
. . . . . . . . . . . . 273 2.2. ERGODIC PROPERTIES ON NILMANIFOLDS . .
. . . . . . . . . . . . . . . . . 275 2.3. UNIPOTENT AFFINE
AUTOMORPHISMS . . . . . . . . . . . . . . . . . . . . . 278 2.4.
QUASI-UNIPOTENT AFFINE AUTOMORPHISMS . . . . . . . . . . . . . . . . .
280 2.5. CLOSED INVARIANT SETS OF AUTOMORPHISMS . . . . . . . . . . . .
. . . 281 2.6. DYNAMICS OF HYPERBOLIC AUTOMORPHISMS . . . . . . . . . .
. . . . . 281 2.7. MORE ON INVARIANT SETS OF HYPERBOLIC TORAL
AUTOMORPHISMS . 283 2.8. DISTRIBUTION OF ORBITS OF HYPERBOLIC
AUTOMORPHISMS . . . . . . . 285 2.9. DYNAMICS OF ERGODIC TORAL
AUTOMORPHISMS . . . . . . . . . . . . . . 286 2.10. ACTIONS OF GROUPS OF
AFFINE AUTOMORPHISMS . . . . . . . . . . . . . 287 § 3. GROUP-INDUCED
TRANSLATION FLOWS; SPECIAL CASES . . . . . . . . . . . . . . 289 3.1.
FLOWS ON SOLVMANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . .
. . 289 3.2. HOMOGENEOUS SPACES OF SEMISIMPLE GROUPS . . . . . . . . . .
. . 292 3.3. FLOWS ON LOW-DIMENSIONAL HOMOGENEOUS SPACES . . . . . . . .
. 295 § 4. ERGODIC PROPERTIES OF FLOWS ON GENERAL HOMOGENEOUS SPACES . .
. . 297 4.1. HOROSPHERICAL SUBGROUPS AND MAUTNER PHENOMENON . . . . . .
298 4.2. ERGODICITY OF ONE-PARAMETER FLOWS . . . . . . . . . . . . . . .
. . . . . 300 4.3. INVARIANT FUNCTIONS AND ERGODIC DECOMPOSITION . . . .
. . . . . . 301 4.4. ACTIONS OF SUBGROUPS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 303 4.5. DUALITY . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.6. SPECTRUM
AND MIXING OF GROUP-INDUCED FLOWS . . . . . . . . . . . 305 CONTENTS 265
4.7. MIXING OF HIGHER ORDERS . . . . . . . . . . . . . . . . . . . . . .
. . . . . 306 4.8. ENTROPY . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 307 4.9. K-MIXING, BERNOULLICITY . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 308 § 5. GROUP-INDUCED
FLOWS WITH HYPERBOLIC STRUCTURE . . . . . . . . . . . . . . 309 5.1.
ANOSOV AUTOMORPHISMS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 309 5.2. AFFINE AUTOMORPHISMS WITH A HYPERBOLIC FIXED POINT . . . .
. . 311 5.3. ANOSOV FLOWS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 312 § 6. INVARIANT MEASURES OF GROUP-INDUCED
FLOWS . . . . . . . . . . . . . . . . . 313 6.1. INVARIANT MEASURES OF
AD-UNIPOTENT FLOWS . . . . . . . . . . . . . . 313 6.2. INVARIANT
MEASURES AND EPIMORPHIC SUBGROUPS . . . . . . . . . . . 316 6.3.
INVARIANT MEASURES OF ACTIONS OF DIAGONALISABLE GROUPS . . . . 318 6.4.
A WEAK RECURRENCE PROPERTY AND INFINITE INVARIANT MEASURES 318 6.5.
DISTRIBUTION OF ORBITS AND POLYNOMIAL TRAJECTORIES . . . . . . . . 320
6.6. A UNIFORM VERSION OF UNIFORM DISTRIBUTION . . . . . . . . . . . . .
321 6.7. DISTRIBUTION OF TRANSLATES OF CLOSED ORBITS . . . . . . . . . .
. . . . 323 § 7. ORBIT CLOSURES OF GROUP-INDUCED FLOWS . . . . . . . . .
. . . . . . . . . . . . 323 7.1. HOMOGENEITY OF ORBIT CLOSURES . . . . .
. . . . . . . . . . . . . . . . . 323 7.2. ORBIT CLOSURES OF
HOROSPHERICAL SUBGROUPS . . . . . . . . . . . . . . 325 7.3. ORBITS OF
REDUCTIVE SUBGROUPS . . . . . . . . . . . . . . . . . . . . . . . 327
7.4. ORBIT CLOSURES OF ONE-PARAMETER FLOWS . . . . . . . . . . . . . . .
. . 328 7.5. DENSE ORBITS AND MINIMAL SETS OF FLOWS . . . . . . . . . .
. . . . . . 330 7.6. DIVERGENT TRAJECTORIES OF FLOWS . . . . . . . . . .
. . . . . . . . . . . . . 332 7.7. BOUNDED ORBITS AND ESCAPABLE SETS . .
. . . . . . . . . . . . . . . . . 333 § 8. DUALITY AND LATTICE-ACTIONS
ON VECTOR SPACES . . . . . . . . . . . . . . . . 335 8.1. DUALITY
BETWEEN ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
335 8.2. DUALITY OF INVARIANT MEASURES . . . . . . . . . . . . . . . . .
. . . . . . 336 § 9. APPLICATIONS TO DIOPHANTINE APPROXIMATION . . . . .
. . . . . . . . . . . . 338 9.1. POLYNOMIALS IN ONE VARIABLE . . . . . .
. . . . . . . . . . . . . . . . . . 338 9.2. VALUES OF LINEAR FORMS . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 338 9.3.
DIOPHANTINE APPROXIMATION WITH DEPENDENT QUANTITIES . . . . . 339 9.4.
VALUES OF QUADRATIC FORMS . . . . . . . . . . . . . . . . . . . . . . .
. . . 340 9.5. FORMS OF HIGHER DEGREE . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 343 9.6. INTEGRAL POINTS ON ALGEBRAIC VARIETIES .
. . . . . . . . . . . . . . . . . 343 § 10. CLASSIFICATION AND RELATED
QUESTIONS . . . . . . . . . . . . . . . . . . . . . . . 344 10.1. METRIC
ISOMORPHISMS AND FACTORS . . . . . . . . . . . . . . . . . . . . . 345
10.2. METRIC RIGIDITY . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 346 10.3. TOPOLOGICAL CONJUGACY . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 347 10.4. TOPOLOGICAL EQUIVALENCE
. . . . . . . . . . . . . . . . . . . . . . . . . . . 349 BIBLIOGRAPHY .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 350 IV. THE DYNAMICS OF BILLIARD FLOWS IN RATIONAL
POLYGONS CONTENTS CHAPTER 11. THE DYNAMICS OF BILLIARD FLOWS IN RATIONAL
POLYGONS OF DYNAMICAL SYSTEMS ( J. SMILLIE ) . . . . . . . . . . . . . .
. . . . . . . . . . . . . 360 § 1. TWO EXAMPLES . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 362 § 2. FORMAL
PROPERTIES OF THE BILLIARD FLOW . . . . . . . . . . . . . . . . . . . .
. 364 § 3. THE FLOW IN A FIXED DIRECTION . . . . . . . . . . . . . . . .
. . . . . . . . . . . 367 § 4. BILLIARD TECHNIQUES: MINIMALITY AND
CLOSED ORBITS . . . . . . . . . . . 369 § 5. BILLIARD TECHNIQUES: UNIQUE
ERGODICITY . . . . . . . . . . . . . . . . . . . . 372 § 6. DYNAMICS ON
MODULI SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
374 § 7. THE LATTICE EXAMPLES OF VEECH . . . . . . . . . . . . . . . . .
. . . . . . . . . 377 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 380 V. DYNAMICAL
SYSTEMS OF STATISTICAL MECHANICS AND KINETIC EQUATIONS CONTENTS CHAPTER
12. DYNAMICAL SYSTEMS OF STATISTICAL MECHANICS ( R.L. DOBRUSHIN, YA.G.
SINAI, YU.M. SUKHOV ) . . . . . . . . . . . . . . . . . . . . 384 § 1.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 384 § 2. PHASE SPACE OF SYSTEMS OF STATISTICAL
MECHANICS AND GIBBS MEASURES . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 386 2.1. THE CONFIGURATION SPACE . . . . . . . .
. . . . . . . . . . . . . . . . . . . 386 2.2. POISSON MEASURES . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 2.3. THE
GIBBS CONFIGURATION PROBABILITY DISTRIBUTION . . . . . . . . 388 2.4.
POTENTIAL OF THE PAIR INTERACTION. EXISTENCE AND UNIQUENESS OF A GIBBS
CONFIGURATION PROBABILITY DISTRIBUTION . . . . . . . . 390 2.5. THE
PHASE SPACE. THE GIBBS PROBABILITY DISTRIBUTION . . . . . 393 2.6. GIBBS
MEASURES WITH A GENERAL POTENTIAL . . . . . . . . . . . . . . 395 2.7.
THE MOMENT MEASURE AND MOMENT FUNCTION . . . . . . . . . . . 396 § 3.
DYNAMICS OF A SYSTEM OF INTERACTING PARTICLES . . . . . . . . . . . . .
. 398 3.1. STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . .
. . . . . . . . 398 3.2. CONSTRUCTION OF THE DYNAMICS AND TIME EVOLUTION
. . . . . . . 400 3.3. HIERARCHY OF THE BOGOLYUBOV EQUATIONS . . . . . .
. . . . . . . . . 402 § 4. EQUILIBRIUM DYNAMICS . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 403 4.1. DEFINITION AND
CONSTRUCTION OF EQUILIBRIUM DYNAMICS . . . . 403 4.2. THE GIBBS
POSTULATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405 4.3. DEGENERATE MODELS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 407 4.4. ASYMPTOTIC PROPERTIES OF THE MEASURES PT . . .
. . . . . . . . . . . 408 § 5. IDEAL GAS AND RELATED SYSTEMS . . . . . .
. . . . . . . . . . . . . . . . . . . . 408 5.1. THE POISSON
SUPERSTRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . 408
5.2. ASYMPTOTIC BEHAVIOUR OF THE PROBABILITY DISTRIBUTION PT AS T ** . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
410 5.3. THE DYNAMICAL SYSTEM OF ONE-DIMENSIONAL HARD RODS . . . 411 §
6. KINETIC EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 412 6.1. STATEMENT OF THE PROBLEM . . . . . . . . .
. . . . . . . . . . . . . . . . . 412 6.2. THE BOLTZMANN EQUATION . . .
. . . . . . . . . . . . . . . . . . . . . . . . 415 6.3. THE VLASOV
EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
6.4. THE LANDAU EQUATION . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 420 6.5. HYDRODYNAMIC EQUATIONS . . . . . . . . . . . . . . .
. . . . . . . . . . . 421 BIBLIOGRAPHY . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 423 CHAPTER 13.
EXISTENCE AND UNIQUENESS THEOREMS FOR THE BOLTZMANN EQUATION ( N.B.
MASLOVA ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 430 § 1. FORMULATION OF BOUNDARY PROBLEMS. PROPERTIES OF
INTEGRAL OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 430 1.1. THE BOLTZMANN EQUATION . . . . .
. . . . . . . . . . . . . . . . . . . . . . 430 1.2. FORMULATION OF
BOUNDARY PROBLEMS . . . . . . . . . . . . . . . . . . . 434 1.3.
PROPERTIES OF THE COLLISION INTEGRAL . . . . . . . . . . . . . . . . . .
. 435 § 2. LINEAR STATIONARY PROBLEMS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 437 2.1. ASYMPTOTICS . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 437 2.2. INTERNAL PROBLEMS .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 2.3.
EXTERNAL PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 439 2.4. KRAMERS* PROBLEM . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 441 § 3. NONLINEAR STATIONARY PROBLEMS . . .
. . . . . . . . . . . . . . . . . . . . . . . 441 § 4. NON-STATIONARY
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
443 4.1. RELAXATION IN A HOMOGENEOUS GAS . . . . . . . . . . . . . . . .
. . . 443 4.2. THE CAUCHY PROBLEM . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 444 4.3. BOUNDARY PROBLEMS . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 445 § 5. ON A CONNECTION OF THE
BOLTZMANN EQUATION WITH HYDRODYNAMIC EQUATIONS . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 5.1.
STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . .
. . 446 5.2. LOCAL SOLUTIONS. REDUCTION TO EULER EQUATIONS . . . . . . .
. . . 448 5.3. A GLOBAL THEOREM. REDUCTION TO NAVIER-STOKES EQUATIONS .
450 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 452
|
| any_adam_object | 1 |
| author_GND | (DE-588)124663079 (DE-588)104216557 |
| building | Verbundindex |
| bvnumber | BV013141627 |
| callnumber-first | Q - Science |
| callnumber-label | QA313 |
| callnumber-raw | QA313 QC19.2 |
| callnumber-search | QA313 QC19.2 |
| callnumber-sort | QA 3313 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 810 SK 880 |
| classification_tum | MAT 344f |
| ctrlnum | (OCoLC)44054681 (DE-599)BVBBV013141627 |
| dewey-full | 515/.42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.42 |
| dewey-search | 515/.42 |
| dewey-sort | 3515 242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2., expanded and rev. ed. |
| format | Book |
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| id | DE-604.BV013141627 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:39:45Z |
| institution | BVB |
| isbn | 3540663169 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008953207 |
| oclc_num | 44054681 |
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| owner_facet | DE-210 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-703 DE-29T DE-384 DE-20 DE-188 |
| physical | X, 459 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Encyclopaedia of mathematical sciences |
| series2 | Encyclopaedia of mathematical sciences [Encyclopaedia of mathematical sciences / Mathematical physics] |
| spelling | Dynamical systems, ergodic theory and applications L. A. Bunimovich ... Ed. by Ya. G. Sinai 2., expanded and rev. ed. Berlin [u.a.] Springer 2000 X, 459 S. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 100 [Encyclopaedia of mathematical sciences / Mathematical physics] 1 1. Aufl. u.d.T.: Dynamical systems. - 2. Ergodic theory with applications to dynamical systems and statistical mechanics. Bildete Vol. 2 der "Encyclopaedia of mathematical sciences" Dynamische systemen gtt Ergodiciteit gtt Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 s DE-604 Dynamisches System (DE-588)4013396-5 s Statistische Mechanik (DE-588)4056999-8 s Sinaj, Jakov G. 1935- Sonstige (DE-588)124663079 oth Bunimovich, Leonid A. 1947- Sonstige (DE-588)104216557 oth Mathematical physics] [Encyclopaedia of mathematical sciences 1 (DE-604)BV013141609 1 Encyclopaedia of mathematical sciences 100 (DE-604)BV024126459 100 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008953207&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dynamical systems, ergodic theory and applications Encyclopaedia of mathematical sciences Dynamische systemen gtt Ergodiciteit gtt Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Statistische Mechanik (DE-588)4056999-8 gnd Ergodentheorie (DE-588)4015246-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4056999-8 (DE-588)4015246-7 (DE-588)4013396-5 |
| title | Dynamical systems, ergodic theory and applications |
| title_auth | Dynamical systems, ergodic theory and applications |
| title_exact_search | Dynamical systems, ergodic theory and applications |
| title_full | Dynamical systems, ergodic theory and applications L. A. Bunimovich ... Ed. by Ya. G. Sinai |
| title_fullStr | Dynamical systems, ergodic theory and applications L. A. Bunimovich ... Ed. by Ya. G. Sinai |
| title_full_unstemmed | Dynamical systems, ergodic theory and applications L. A. Bunimovich ... Ed. by Ya. G. Sinai |
| title_short | Dynamical systems, ergodic theory and applications |
| title_sort | dynamical systems ergodic theory and applications |
| topic | Dynamische systemen gtt Ergodiciteit gtt Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Statistische Mechanik (DE-588)4056999-8 gnd Ergodentheorie (DE-588)4015246-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Dynamische systemen Ergodiciteit Mécanique analytique Mécanique céleste Théorie ergodique Differentiable dynamical systems Ergodic theory Statistische Mechanik Ergodentheorie Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008953207&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013141609 (DE-604)BV024126459 |
| work_keys_str_mv | AT sinajjakovg dynamicalsystemsergodictheoryandapplications AT bunimovichleonida dynamicalsystemsergodictheoryandapplications |