## Mathematical Problems of Statistical Mechanics and Dyanamics: A Collection of SurveysApproach your problems from the It isn't that they can't see the solution. right end and begin with the answers. It is that they can't see the problem. Then one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father Brown 'The point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. |

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### Contents

Phase Diagrams for ContinuousSpin Models An Extension of the PirogovSinai Theory | 1 |

Formulation of the Main Result | 4 |

13 HAMILTONIANS | 5 |

15 ASSUMPTIONS OF THE MAIN THEOREM | 6 |

16 THE MAIN THEOREM | 10 |

17 STRATEGY OF THE PROOF | 11 |

THE CORRELATION DECAY | 17 |

CLUSTER EXPANSIONS OF PERTURBED GAUSSIAN FIELDS | 30 |

Small QuasiStochastic Perturbations | 180 |

Ergodic Properties of Dynamical System Discretizations | 184 |

References | 196 |

Statistical Properties of Smooth Smale Horseshoes | 199 |

General Background | 203 |

12 UNIFORMLY HYPERBOLIC TRANSFORMATIONS OF Z X x Y | 204 |

13 A SUFFICIENT CONDITION FOR UNIFORM HYPERBOLICITY IN Z X x Y | 206 |

14 LEAVES AND RECTANGLES | 207 |

The Main Lemma | 62 |

32 REDUCTION TO THE MAIN LEMMA | 64 |

Proof of The Main Lemma | 76 |

42 REDUCTION TO A CONTOUR MODEL | 79 |

43 ESTIMATES OF THE MAIN TERM G₂𝛤 DECOMPOSITION OF THE CONTOUR ENERGY | 88 |

44 BOUNDARY TERMS OF PARTITION FUNCTIONS OF CONTOUR MODELS | 95 |

45 CONCLUSION OF THE PROOF OF THE MAIN LEMMA | 106 |

References | 122 |

SpaceTime Entropy of Infinite Classical Systems | 125 |

Statistical Estimates of the Gibbs Distribution | 126 |

Reduction to Partial Flows | 129 |

Estimate of SpaceTime Entropy | 132 |

References | 136 |

Spectrum Analysis and Scattering Theory for a ThreeParticle Cluster Operator | 139 |

ThreeParticle Cluster Operators | 140 |

Equations for the Resolvent of a SelfAdjoint ThreeParticle Cluster Operator | 143 |

Study of Equations 3436 | 146 |

The Main Result | 155 |

Proof of Theorem 511 Scattering Theory | 157 |

References | 160 |

Stochastic Attractors and their Small Perturbations | 161 |

Dynamical Systems with Stochastic Attractors | 164 |

Stochastic Perturbations Regular Case | 170 |

The Law of Exponential Decay and Small Stochastic Perturbations | 176 |

Stochastic Perturbations Singular Case | 179 |

15 THE SMALE HORSESHOE | 209 |

Expanding and Contracting Fibrations of a Smale Horseshoe | 211 |

22 EXPANDING AND CONTRACTING FIBRATIONS ARE HÖLDERIAN | 213 |

24 THE HÖLDER PROPERTY OF THE CANONICAL ISOMORPHISM DEFINED BY A FIBRATION | 215 |

Smooth Invariant Conditional Probability Distributions on Fibrations | 216 |

32 THE EXISTENCE OF A TINVARIANT SMOOTH FAMILY OF PROBABILITY DISTRIBUTIONS ON FIBRES AT | 217 |

33 COMPARISON OF DENSITIES OF CONDITIONAL PROBABILITY DISTRIBITIONS ON DIFFERENT FIBRES | 218 |

34 THE DEPENDENCE OF TINVARIANT CONDITIONAL DENSITIES ON THE NUMBER OF THE FIBRE | 219 |

Smooth NonSingular Probability Distributions on a Smale Horseshoe | 220 |

42 AN AVERAGE DESCRIPTION OF THE EVOLUTION OF MEASURES FROM THE CLASS | 222 |

43 THE CONSTRUCTION OF AN EIGENMEASURE FOR A SMALE HORSESHOE | 224 |

A Natural Invariant Probability Distribution on the Hyperbolic Set of a Smale Horseshoe | 226 |

52 THE COMPUTATION OF THE ASYMPTOTICS OF VIA THE MATRIX TECHNIQUE | 228 |

53 THE WEAK LIMIT 𝜇₀ OF THE SEQUENCE OF MEASURES 𝜇ₘ | 233 |

Some Properties of the Constructed Limit Probability Distributions on a Smale Horseshoe | 235 |

62 WEAK BERNOULLI PARTITION FOR THE TINVARIANT MEASURE 𝜇₀ | 236 |

63 THE EIGENFUNCTION el OF A SMALE HORSESHOE | 239 |

Evolution of Probability Distributions on a Smale Horseshoe | 240 |

72 THE ASYMPTOTICS OF INTEGRALS | 244 |

73 MAPPINGS WHICH POSSESS A SMALE HORSESHOE | 250 |

Ergodic Properties of Positive Matrices with Bounded Ratio of Rows | 253 |

256 | |

257 | |

259 | |

### Common terms and phrases

analogous analytical assume assumptions Borel Borel sets bounded cluster expansion cluster operators components conditional probability distributions configuration consider constant construction continuous function contour ensemble contour models contour weight convergence COROLLARY corresponding cycle defined definition Denote density depending diagram diffeomorphism Dobrushin domain dynamical system eigenvalue entropy equation exists fibration finite fixed formula frame Gaussian fields Gibbs Gibbsian given graph Hamiltonian holds hyperbolic hyperbolic set implies inequality integral interactions intrinsically stable invariant measures investigate K-functional kernel Lebesgue measure limit Lipschitzizable Main Lemma Main Theorem mathematical matrix Minlos normed notation NOTE obtain particles partition functions perturbed systems piecewise expanding Proposition prove rectangle relation respect satisfy the condition semi-invariants sequence Sinai Smale horseshoe small neighbourhood space statement stochastic attractor stochastic perturbations subset sufficiently small supp Suppose T-invariant Theorem theory trajectory translation-invariant vacuum diagrams values variables write Y-leaves

### Popular passages

Page 256 - On infinite processes leading to differentiability in the complement of a point, Differential and combinatorial topology (A symposium in honor of Marston Morse), Princeton University Press, Princeton, NJ, 1965, pp.