Extremes and Recurrence in Dynamical Systems.

Intro -- Title Page -- COPYRIGHT -- Table of Contents -- DEDICATION -- CHAPTER 1: INTRODUCTION -- 1.1 A TRANSDISCIPLINARY RESEARCH AREA -- 1.2 SOME MATHEMATICAL IDEAS -- 1.3 SOME DIFFICULTIES AND CHALLENGES IN STUDYING EXTREMES -- 1.4 EXTREMES, OBSERVABLES, AND DYNAMICS -- 1.5 THIS BOOK -- ACKNOWLEDGMENTS -- CHAPTER 2: A FRAMEWORK FOR RARE EVENTS IN STOCHASTIC PROCESSES AND DYNAMICAL SYSTEMS -- 2.1 Introducing Rare Events -- 2.2 Extremal Order Statistics -- 2.3 Extremes and Dynamics -- CHAPTER 3: CLASSICAL EXTREME VALUE THEORY -- 3.1 THE i.i.d. SETTING AND THE CLASSICAL RESULTS -- 3.2 STATIONARY SEQUENCES AND DEPENDENCE CONDITIONS -- 3.3 CONVERGENCE OF POINT PROCESSES OF RARE EVENTS -- 3.4 ELEMENTS OF DECLUSTERING -- CHAPTER 4: EMERGENCE OF EXTREME VALUE LAWS FOR DYNAMICAL SYSTEMS -- 4.1 EXTREMES FOR GENERAL STATIONARY PROCESSES-AN UPGRADE MOTIVATED BY DYNAMICS -- 4.2 EXTREME VALUES FOR DYNAMICALLY DEFINED STOCHASTIC PROCESSES -- 4.3 POINT PROCESSES OF RARE EVENTS -- 4.4 CONDITIONS Дq(un), D3(un), Dp(un)* AND DECAY OF CORRELATIONS -- 4.5 SPECIFIC DYNAMICAL SYSTEMS WHERE THE DICHOTOMY APPLIES -- 4.6 EXTREME VALUE LAWS FOR PHYSICAL OBSERVABLES -- CHAPTER 5: HITTING AND RETURN TIME STATISTICS -- 5.1 INTRODUCTION TO HITTING AND RETURN TIME STATISTICS -- 5.2 HTS VERSUS RTS AND POSSIBLE LIMIT LAWS -- 5.3 THE LINK BETWEEN HITTING TIMES AND EXTREME VALUES -- 5.4 UNIFORMLY HYPERBOLIC SYSTEMS -- 5.5 NONUNIFORMLY HYPERBOLIC SYSTEMS -- 5.6 NONEXPONENTIAL LAWS -- CHAPTER 6: EXTREME VALUE THEORY FOR SELECTED DYNAMICAL SYSTEMS -- 6.1 RARE EVENTS AND DYNAMICAL SYSTEMS -- 6.2 INTRODUCTION AND BACKGROUND ON EXTREMES IN DYNAMICAL SYSTEMS -- 6.3 THE BLOCKING ARGUMENT FOR NONUNIFORMLY EXPANDING SYSTEMS -- 6.4 NONUNIFORMLY EXPANDING DYNAMICAL SYSTEMS -- 6.5 NONUNIFORMLY HYPERBOLIC SYSTEMS -- 6.6 HYPERBOLIC DYNAMICAL SYSTEMS.

6.7 SKEW-PRODUCT EXTENSIONS OF DYNAMICAL SYSTEMS -- 6.8 ON THE RATE OF CONVERGENCE TO AN EXTREME VALUE DISTRIBUTION -- 6.9 EXTREME VALUE THEORY FOR DETERMINISTIC FLOWS -- 6.10 PHYSICAL OBSERVABLES AND EXTREME VALUE THEORY -- 6.11 NONUNIFORMLY HYPERBOLIC EXAMPLES: THE HÉNON AND LOZI MAPS -- 6.12 Extreme Value Statistics for the Lorenz '63 Model -- CHAPTER 7: EXTREME VALUE THEORY FOR RANDOMLY PERTURBED DYNAMICAL SYSTEMS -- 7.1 INTRODUCTION -- 7.2 Random Transformations via the Probabilistic Approach: Additive Noise -- 7.3 Random Transformations via the Spectral Approach -- 7.4 RANDOM TRANSFORMATIONS VIA THE PROBABILISTIC APPROACH: RANDOMLY APPLIED STOCHASTIC PERTURBATIONS -- 7.5 OBSERVATIONAL NOISE -- 7.6 NONSTATIONARITY-THE SEQUENTIAL CASE -- CHAPTER 8: A STATISTICAL MECHANICAL POINT OF VIEW -- 8.1 CHOOSING A MATHEMATICAL FRAMEWORK -- 8.2 GENERALIZED PARETO DISTRIBUTIONS FOR OBSERVABLES OF DYNAMICAL SYSTEMS -- 8.3 IMPACTS OF PERTURBATIONS: RESPONSE THEORY FOR EXTREMES -- 8.4 REMARKS ON THE GEOMETRY AND THE SYMMETRIES OF THE PROBLEM -- CHAPTER 9: Extremes as Dynamical and Geometrical Indicators -- 9.1 The Block Maxima Approach -- 9.2 The Peaks Over Threshold Approach -- 9.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure -- 9.4 Chaotic Maps With Singular Invariant Measures -- 9.5 Analysis of the Distance and Physical Observables for the HNON Map -- 9.6 Extremes as Dynamical Indicators -- 9.7 EXTREME VALUE LAWS FOR STOCHASTICALLY PERTURBED SYSTEMS -- CHAPTER 10: EXTREMES AS PHYSICAL PROBES -- 10.1 SURFACE TEMPERATURE EXTREMES -- 10.2 DYNAMICAL PROPERTIES OF PHYSICAL OBSERVABLES: EXTREMES AT TIPPING POINTS -- 10.3 CONCLUDING REMARKS -- CHAPTER 11: CONCLUSIONS -- 11.1 MAIN CONCEPTS OF THIS BOOK -- 11.2 EXTREMES, COARSE GRAINING, AND PARAMETRIZATIONS -- 11.3 EXTREMES OF NONAUTONOMOUS DYNAMICAL SYSTEMS -- 11.4 QUASI-DISCONNECTED ATTRACTORS.

11.5 CLUSTERS AND RECURRENCE OF EXTREMES -- 11.6 TOWARD SPATIAL EXTREMES: COUPLED MAP LATTICE MODELS -- APPENDIX A: CODES -- A.1 Extremal Index -- A.2 Recurrences-Extreme Value Analysis -- A.3 SAMPLE PROGRAM -- REFERENCES -- INDEX -- SERIES PAGE -- End User License Agreement.

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