1 Introduction 1 1.1 A Transdisciplinary Research Area 1 1.2 Some Mathematical Ideas 4 1.3 Some Difficulties and Challenges in Studying Extremes 6 1.3.1 Finiteness of Data 6 1.3.2 Correlation and Clustering 8 1.3.3 Time Modulations and Noise 9 1.4 Extremes, Observables, and Dynamics 10 1.5 This Book 12 Acknowledgments 14 2 A Framework for Rare Events in Stochastic Processes and Dynamical Systems 17 2.1 Introducing Rare Events 17 2.2 Extremal Order Statistics 19 2.3 Extremes and Dynamics 20 3 Classical Extreme Value Theory 23 3.1 The i.i.d. Setting and the Classical Results 24 3.1.1 Block Maxima and the Generalized Extreme Value Distribution 24 3.1.2 Examples 26 3.1.3 Peaks Over Threshold and the Generalized Pareto Distribution 28 3.2 Stationary Sequences and Dependence Conditions 29 3.2.1 The Blocking Argument 30 3.2.2 The Appearance of Clusters of Exceedances 31 3.3 Convergence of Point Processes of Rare Events 32 3.3.1 Definitions and Notation 33 3.3.2 Absence of Clusters 35 3.3.3 Presence of Clusters 35 3.4 Elements of Declustering 37 4 Emergence of Extreme Value Laws for Dynamical Systems 39 4.1 Extremes for General Stationary Processes an Upgrade Motivated by Dynamics 40 4.1.1 Notation 41 4.1.2 The New Conditions 42 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 44 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 46 4.2 Extreme Values for Dynamically Defined Stochastic Processes 51 4.2.1 Observables and Corresponding Extreme Value Laws 53 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 57 4.2.3 Example Revisited 59 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 61 4.3 Point Processes of Rare Events 62 4.3.1 Absence of Clustering 62 4.3.2 Presence of Clustering 63 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 65 4.4 Conditions q(un), D3(un), Dp(un) and Decay of Correlations 66 4.5 Specific Dynamical Systems Where the Dichotomy Applies 70 4.5.1 Rychlik Systems 70 4.5.2 Piecewise Expanding Maps in Higher Dimensions 71 4.6 Extreme Value Laws for Physical Observables 72 5 Hitting and Return Time Statistics 75 5.1 Introduction to Hitting and Return Time Statistics 75 5.1.1 Definition of Hitting and Return Time Statistics 76 5.2 HTS Versus RTS and Possible Limit Laws 77 5.3 The Link Between Hitting Times and Extreme Values 78 5.4 Uniformly Hyperbolic Systems 84 5.4.1 Gibbs Measures 85 5.4.2 First HTS Theorem 86 5.4.3 Markov Partitions 86 5.4.4 Two-Sided Shifts 88 5.4.5 Hyperbolic Diffeomorphisms 89 5.4.6 Additional Uniformly Hyperbolic Examples 90 5.5 Nonuniformly Hyperbolic Systems 91 5.5.1 Induced System 91 5.5.2 Intermittent Maps 92 5.5.3 Interval Maps with Critical Points 93 5.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems 94 5.6 Nonexponential Laws 95 6 Extreme Value Theory for Selected Dynamical Systems 97 6.1 Rare Events and Dynamical Systems 97 6.2 Introduction and Background on Extremes in Dynamical Systems 98 6.3 The Blocking Argument for Nonuniformly Expanding Systems 99 6.3.1 Assumptions on the Invariant Measure 99 6.3.2 Dynamical Assumptions on (f , , ) 99 6.3.3 Assumption on the Observable Type 100 6.3.4 Statement or Results 101 6.3.5 The Blocking Argument in One Dimension 102 6.3.6 Quantification of the Error Rates 102 6.3.7 Proof of Theorem 6.3.1 107 6.4 Nonuniformly Expanding Dynamical Systems 108 6.4.1 Uniformly Expanding Maps 108 6.4.2 Nonuniformly Expanding Quadratic Maps 109 6.4.3 One-Dimensional Lorenz Maps 110 6.4.4 Nonuniformly Expanding Intermittency Maps 110 6.5 Nonuniformly Hyperbolic Systems 113 6.5.1 Proof of Theorem 6.5.1 115 6.6 Hyperbolic Dynamical Systems 116 6.6.1 Arnold Cat Map 116 6.6.2 Lozi-Like Maps 118 6.6.3 Sinai Dispersing Billiards 119 6.6.4 Henon Maps 119 6.7 Skew-Product Extensions of Dynamical Systems 120 6.8 On the Rate of Convergence to an Extreme Value Distribution 121 6.8.1 Error Rates for Specific Dynamical Systems 123 6.9 Extreme Value Theory for Deterministic Flows 126 6.9.1 Lifting to Xh 129 6.9.2 The Normalization Constants 129 6.9.3 The Lap Number 130 6.9.4 Proof of Theorem 6.9.1 131 6.10 Physical Observables and Extreme Value Theory 133 6.10.1 Arnold Cat Map 133 6.10.2 Uniformly Hyperbolic Attractors: The Solenoid Map 137 6.11 Nonuniformly Hyperbolic Examples: the HENON and LOZI Maps 140 6.12 Extreme Value Statistics for the Lorenz 63 Model 141 7 Extreme Value Theory for Randomly Perturbed Dynamical Systems 145 7.1 Introduction 145 7.2 Random Transformations via the Probabilistic Approach: Additive Noise 146 7.2.1 Main Results 149 7.3 Random Transformations via the Spectral Approach 155 7.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations 159 7.5 Observational Noise 163 7.6 Nonstationarity the Sequential Case 165 8 A Statistical Mechanical Point of View 167 8.1 Choosing a Mathematical Framework 167 8.2 Generalized Pareto Distributions for Observables of Dynamical Systems 168 8.2.1 Distance Observables 169 8.2.2 Physical Observables 172 8.2.3 Derivation of the Generalized Pareto Distribution Parameters for the Extremes of a Physical Observable 174 8.2.4 Comments 176 8.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow 177 8.2.6 Expressing the Shape Parameter in Terms of the GPD Moments and of the Invariant Measure of the System 178 8.3 Impacts of Perturbations: Response Theory for Extremes 180 8.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments 182 8.3.2 Sensitivity of the Shape Parameter as Determined by the Modification of the Geometry 185 8.4 Remarks on the Geometry and the Symmetries of the Problem 188 9 Extremes as Dynamical and Geometrical Indicators 189 9.1 The Block Maxima Approach 190 9.1.1 Extreme Value Laws and the Geometry of the Attractor 191 9.1.2 Computation of the Normalizing Sequences 192 9.1.3 Inference Procedures for the Block Maxima Approach 194 9.2 The Peaks Over Threshold Approach 196 9.2.1 Inference Procedures for the Peaks Over Threshold Approach 196 9.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure 197 9.3.1 Maximum Likelihood versus L-Moment Estimators 203 9.3.2 Block Maxima versus Peaks Over Threshold Methods 204 9.4 Chaotic Maps With Singular Invariant Measures 204 9.4.1 Normalizing Sequences 205 9.4.2 Numerical Experiments 208 9.5 Analysis of the Distance and Physical Observables for the HNON Map 212 9.5.1 Remarks 218 9.6 Extremes as Dynamical Indicators 218 9.6.1 The Standard Map: Peaks Over Threshold Analysis 219 9.6.2 The Standard Map: Block Maxima Analysis 220 9.7 Extreme Value Laws for Stochastically Perturbed Systems 223 9.7.1 Additive Noise 225 9.7.2 Observational Noise 229 10 Extremes as Physical Probes 233 10.1 Surface Temperature Extremes 233 10.1.1 Normal, Rare and Extreme Recurrences 235 10.1.2 Analysis of the Temperature Records 235 10.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points 238 10.2.1 Extremes of Energy for the Plane Couette Flow 239 10.2.2 Extremes for a Toy Model of Turbulence 245 10.3 Concluding Remarks 247 11 Conclusions 249 11.1 Main Concepts of This Book 249 11.2 Extremes, Coarse Graining, and Parametrizations 253 11.3 Extremes of Nonautonomous Dynamical Systems 255 11.3.1 A Note on Randomly Perturbed Dynamical Systems 258 11.4 Quasi-Disconnected Attractors 260 11.5 Clusters and Recurrence of Extremes 261 11.6 Toward Spatial Extremes: Coupled Map Lattice Models 262 Appendix A Codes 265 A.1 Extremal Index 266 A.2 Recurrences Extreme Value Analysis 267 A.3 Sample Program 271 References 273 Index 293