Multifractals: theory and applications

SECTION I: INTRODUCTION AND PRELIMINARIES MOTIVATION AND BACKGROUND Fractal Sets and Multifractal Measures Dynamical Systems Turbulence Rainfall Fields Earthquake Modelling Other Applications Concept of Multifractals Overview of Book THE MULTIFRACTAL FORMALISM Historical Development of Generalised Renyi Dimensions Generlised Renyi Lattice Dimensions Generalised Renyi Point Centred Dimensions Multifractal Spectrum and Formalism Review of Related Lattice Based Results Review of Related Point Centred Results THE MULTINOMIAL MEASURE Local Behaviour Global Averaging and Legendre Transforms Fractal Dimensions Point Centred Construction SECTION II: MULTIFRACTAL FORMALISM USING LARGE DEVIATIONS LATTICE BASED MULTIFRACTALS Large Deviation Formalism Uniform Spatial Sampling Measure A Family of Sampling Measures Hausdorff Dimensions POINT CENTERED MULTIFRACTALS Large Deviation Formalism A Family of Sampling Measures Hausdorff Dimensions Relationship Between Lattice and Point Centred Constructions MULTIPLICATIVE CASCADE PROCESSES Moran Cascades Processes Random Cascades Other Cascade Processes SECTION III: ESTIMATION OF THE RENYI DIMENSIONS INTERPOINT DISTANCES OF ORDER q AND INTRINSIC BIAS Boundary Effect Multiplicity of Boundaries Decomposition of FY(y) Differentiable Distribution ESTIMATION OF POINT CENTRED RENYI DIMENSIONS WITH q=2 Generalised Grassberger-Procaccia Algorithm Takens Estimator Hill Estimator Bootstrap Estimation Procedure Discussion and Examples EXTRINSIC SOURCES OF BIAS Imposed Boundary Effect Rounding Effect Effect of Noise APPLICATIONS OF DIMENSION ESTIMATION More on Estimation and Interpretation Spatial and Temporal Point Patterns Dynamical Systems Is a Process Stochastic or Deterministic? Stochastic Processes with Powerlaw Properties EARTHQUAKE ANALYSES Sources of Data Effects Causing Bias Results Comparison of Results and Conclusions APPENDICES Properties and Dimensions of Sets Large Deviations REFERENCES