AN INVITATION TO MATHEMATICS
CONTENTS PART 1 PRELIMINARY MATERIAL CHAPTER 1 INTRODUCTION TO CONCEPTS IN FRACTAL GEOMETRY §1 1 MONSTERS 1 1 1 A BRIEF HISTORY OF FRACTAL GEOMETRY 1 1 2 THE COASTLINE OF BRITAIN §1 2 A PRELIMINARY FRACTAL TOOLBOX 1 2 1 MEASURING SETS 1 2 2 LEBESGUE OUTER MEASURE 1 2 3 SELF-SIMILARITY §1 3 NOTIONS OF DIMENSION 1 3 1 TYPICAL PROPERTIES OF A DIMENSION 1 3 2 TOPOLOGICAL DIMENSION 1 3 3 COUNTING BOXES 1 3 4 SIMILARITY DIMENSION 1 3 5 BOX-COUNTING DIMENSION 1 3 6 MINKOWSKI DIMENSION 1 3 7 FRACTAL STRINGS AND ZETA FUNCTIONS §1 4 FRACTAL FUNCTIONS AND MATHEMATICAL MONSTERS 1 4 1 A SURPRISING PAIR OF FUNCTIONS: THE DEVIL’S STAIRCASE 1 4 2 WEIERSTRASS’S NOWHERE DIFFERENTIABLE FUNCTION 1 4 3 PEANO’S SPACE-FILLING CURVE §1 5 RANDOMLY CONSTRUCTED FRACTALS 1 5 1 BROWNIAN MOTION 1 5 2 STATISTICALLY SELF-SIMILAR SETS AND OTHER RANDOM FRACTALS §1 6 COMPLEX DYNAMICS AND FRACTALS: JULIA SETS AND THE MANDELBROT SET 1 6 1 JULIA SETS AND THE MANDELBROT SET §1 7 CHAOTIC DYNAMICS AND FRACTALS: STRANGE ATTRACTORS §1 8 A SAMPLE OF BOOKS ON FRACTAL GEOMETRY EXERCISES CHAPTER 2 METRIC SPACES AND FIXED POINT THEOREM §2 1 DEFINITION AND PROPERTIES §2 2 ANALYSIS AND TOPOLOGY §2 3 CONTRACTION MAPPING PRINCIPLE CHAPTER 3 MEASURE THEORY AND INTEGRALS §3 1 ELEMENTARY FAMILIES AND APPLICATIONS §3 2 MEASURES §3 3 EXTENSION OF A PREMEASURE TO A MEASURE §3 4 LEBESGUE MEASURE §3 5 CONTENT AND THE RIEMANN INTEGRAL §3 6 LEBESGUE INTEGRATION PART 2 DIMENSION THEORY CHAPTER 4 ITERATED FUNCTION SYSTEMS AND SELF-SIMILARITY §4 1 HAUSDORFF METRIC §4 2 ITERATED FUNCTION SYSTEMS §4 3 SELF-SIMILAR SETS §4 4 SELF-AFFINE SETS §4 5 SYMBOLIC DYNAMICS: WORDS AND ADDRESSES §4 6 LATTICE AND NONLATTICE SELF-SIMILAR SYSTEMS §4 7 MORAN’S EQUATION AND SIMILARITY DIMENSION §4 8 SELF-SIMILAR MEASURES §4 9 PERFECT, NOWHERE DENSE SETS §4 10 ADDITIONAL EXAMPLES CHAPTER 5 INTRODUCTION TO HAUSDORFF MEASURE AND DIMENSION §5 1 HAUSDORFF MEASURE §5 2 HAUSDORFF DIMENSION §5 3 HAUSDORFF DIMENSION CALCULATIONS §5 4 MASS DISTRIBUTION PRINCIPLE AND MORAN’S THEOREM §5 5 FURTHER TECHNIQUES FOR CALCULATING HAUSDORFF DIMENSION §5 6 DENSITY AND HAUSDORFF MEASURE OF PRODUCTS CHAPTER 6 ??-APPROXIMATE HAUSDORFF MEASURES, VIA CARATHÉODORY’S THEORY §6 1 A FEW GENERALITIES ABOUT METRIC SPACES §6 2 METRIC OUTER MEASURES AND THEIR ASSOCIATED BOREL MEASURES §6 3 CONSTRUCTION AND PROPERTIES OF ??-APPROXIMATE HAUSDORFF MEASURES 6 3 1 CONSTRUCTION OF ??-APPROXIMATE HAUSDORFF MEASURES FOR VARIOUS FAMILIES 6 3 2 ??-APPROXIMATE HAUSDORFF OUTER MEASURES AND MEASURES 6 3 3 COMPARISON OF ??-APPROXIMATE HAUSDORFF MEASURES FOR VARIOUS FAMILIES 6 3 4 JUNG’S SHARP INEQUALITY AND RELATED RESULTS 6 3 5 TRANSLATION INVARIANCE AND SCALE COVARIANCE CHAPTER 7 CONSTRUCTION AND PROPERTIES OF HAUSDORFF MEASURE §7 1 CONSTRUCTION OF HAUSDORFF MEASURE, VIA ITS ??-APPROXIMATE COUNTERPARTS §7 2 COMPARISON OF HAUSDORFF MEASURES FOR VARIOUS FAMILIES §7 3 KEY GEOMETRIC PROPERTIES OF HAUSDORFF MEASURE 7 3 1 TRANSLATION AND ISOMETRY INVARIANCE, SCALING, AND SIMILARITY COVARIANCE §7 4 LEBESGUE MEASURE VS ??-DIMENSIONAL HAUSDORFF MEASURE ON ??? 7 4 1 TESSELLATIONS BY DYADIC CUBES AND THEIR APPLICATIONS 7 4 2 EQUALITY OF ???AND ??? ??????, VIA TRANSFINITE INDUCTION 7 4 3 TRANSFINITE INDUCTION AND BOREL HIERARCHY 7 4 4 UNIQUENESS OF TRANSLATION-INVARIANT BOREL MEASURES 7 4 5 BOREL REGULARITY OF THE ??-DIMENSIONAL HAUSDORFF AND LEBESGUE MEASURES ???AND ??? 7 4 6 ISODIAMETRIC INEQUALITY AND EXACT VALUE OF THE PROPORTIONALITY CONSTANT BETWEEN ???AND ??? §7 5 HAUSDORFF DIMENSION IN GENERAL METRIC SPACES CHAPTER 8 MINKOWSKI CONTENT AND MINKOWSKI DIMENSION §8 1 MINKOWSKI CONTENT AND DIMENSION §8 2 EXAMPLES: THE ??-STRING AND THE CANTOR STRING (OR SET) §8 3 BOX-COUNTING DIMENSION AND CONTENT §8 4 PROPERTIES OF MINKOWSKI AND BOX-COUNTING DIMENSIONS §8 5 HAUSDORFF VERSUS MINKOWSKI §8 6 TOPOLOGICAL VERSUS HAUSDORFF AND MINKOWSKI DIMENSIONS 8 6 1 PROPERTIES OF THE TOPOLOGICAL DIMENSION 8 6 2 COMPARISON BETWEEN TOPOLOGICAL, HAUSDORFF, AND MINKOWSKI DIMENSIONS §8 7 DIMENSIONS OF ATTRACTORS §8 8 DISTINCT UPPER AND LOWER BOX-COUNTING (OR MINKOWSKI) DIMENSIONS §8 9 GENERALIZED HAUSDORFF, PACKING, AND MINKOWSKI DIMENSIONS 8 9 1 GAUGE FUNCTIONS (OR SCALING LAWS) 8 9 2 GENERALIZED HAUSDORFF MEASURES AND DIMENSION FUNCTIONS 8 9 3 GENERALIZED PACKING MEASURES AND DIMENSION FUNCTIONS 8 9 4 PACKING AND MODIFIED BOX-COUNTING DIMENSIONS 8 9 5 GENERALIZED MINKOWSKI CONTENTS AND DIMENSION FUNCTIONS PART 3 FRACTAL CURVES AND THEIR COMPLEX DIMENSIONS CHAPTER 9 EPILOGUE: A PRIMER OF FRACTAL CURVES AND THEIR COMPLEX DIMENSIONS §9 1 CONTINUOUS CURVES: THEIR LENGTHS AND POLYGONAL APPROXIMATIONS §9 2 INTERCHANGE OF LENGTHS AND UNIFORM OR HAUSDORFF LIMITS §9 3 INTERLUDE: AN INTRODUCTION TO COMPLEX DIMENSIONS AND FRACTAL ZETA FUNCTIONS 9 3 1 LIGHTNING SPEED REVIEW OF COMPLEX ANALYSIS 9 3 2 DISTANCE ZETA FUNCTION 9 3 3 TUBE ZETA FUNCTION AND FUNCTIONAL EQUATION 9 3 4 COMPLEX DIMENSIONS 9 3 5 DEFINITION OF FRACTALITY AND GEOMETRIC INTERPRETATION 9 3 6 RESIDUES AND ORDINARY OR AVERAGE MINKOWSKI CONTENT 9 3 7 FRACTAL TUBE FORMULAS 9 3 8 RELATIVE FRACTAL DRUMS AND SYMMETRIES §9 4 DEVIL’S STAIRCASE (OR CANTOR GRAPH): A SELF-AFFINE AND RECTIFIABLE FRACTAL CURVE 9 4 1 FIRST CONSTRUCTION OF THE DEVIL’S STAIRCASE: VIA MONOTONIC OR CONTINUOUS EXTENSION 9 4 2 SECOND CONSTRUCTION OF THE DEVIL’S STAIRCASE: UNIFORM APPROXIMATION BY PIECEWISE LINEAR FUNCTIONS 9 4 3 RECTIFIABILITY AND EXACT LENGTH 9 4 4 THIRD CONSTRUCTION: VIA EXPLICIT TERNARY EXPANSIONS 9 4 5 SINGULARITY OF THE CANTOR–LEBESGUE FUNCTION 9 4 6 SELF-AFFINITY, SYMMETRIES, AND FUNCTIONAL EQUATIONS 9 4 7 COMPLEX DIMENSIONS AND FRACTALITY OF THE DEVIL’S STAIRCASE §9 5 KOCH CURVE: A SELF-SIMILAR AND UNRECTIFIABLE CURVE 9 5 1 POLYGONAL APPROXIMATIONS, UNRECTIFIABILITY, AND NOWHERE DIFFERENTIABILITY 9 5 2 SELF-SIMILARITY AND FINITE GRAPH APPROXIMATIONS 9 5 3 FRACTAL TUBE FORMULAS AND COMPLEX DIMENSIONS §9 6 SIERPINSKI GASKET AND CARPET: UNIVERSALITY AND BEYOND 9 6 1 FINITE VS INFINITE RAMIFICATION 9 6 2 SIERPINSKI CARPET AND MENGER SPONGE: UNIVERSALITY THEOREMS 9 6 3 COMPLEX DIMENSIONS §9 7 WEIERSTRASS CURVE: A CONTINUOUS BUT NOWHERE DIFFERENTIABLE CURVE 9 7 1 INTRODUCTION: HISTORY AND COMPLEXITY 9 7 2 GEOMETRIC PROPERTIES AND NONLINEAR IFS 9 7 3 COMPLEX DIMENSIONS, FRACTAL TUBE FORMULA, AND OPTIMAL HÖLDER CONTINUITY