Geometry and Analysis of Fractals: Hong Kong, December 2012 (Springer Proceedings in Mathematics & Statistics Book 88)

This volume collects thirteen expository or survey articles on topics including Fractal Geometry, Analysis of Fractals, Multifractal Analysis, Ergodic Theory and Dynamical Systems, Probability and Stochastic Analysis, written by the leading experts in their respective fields. The articles are based on papers presented at the International Conference on Advances on Fractals and Related Topics, held on December 10-14, 2012 at the Chinese University of Hong Kong. The volume offers insights into a number of exciting, cutting-edge developments in the area of fractals, which has close ties to and applications in other areas such as analysis, geometry, number theory, probability and mathematical physics. Foreword......Page 6 Contents......Page 8 1 Introduction......Page 10 2 Mandelbrot Cascades......Page 11 3.1 Fixed Points of the Smoothing Transformation......Page 15 4 Directed Polymers on mathscrA*: Partition Functions, Free Energies and Gibbs Measures......Page 19 4.2 Second Order Phase Transition: βcin(0,infty) and Wβc'(1-)='(βc-)βc-(βc)=0......Page 21 4.3 First Order Phase Transition: βcin(0,infty) and Wβc'(1-)='(βc-)βc-(βc)>0......Page 24 5 Fine Geometric Properties of Statistically Self-similar Measures......Page 25 5.1 Dimension, Modulus of Continuity, and Multifractal Analysis of Mandelbrot and Critical Mandelbrot Measures......Page 26 5.2 Multifractal Analysis of Lévy-Mandelbrot and Critical Lévy-Mandelbrot Measures......Page 36 5.3 KPZ Formula......Page 39 6 On Signed and Complex Multiplicative Cascades......Page 40 6.1 Some Convergence Theorems......Page 41 6.2 Multifractal Analysis of Roughness in the Graph of F......Page 43 7.1 A Dynamical System......Page 45 7.2 The Limit Process X as the Limit of an Additive Cascade......Page 48 7.3 Fine Properties of X......Page 49 References......Page 50 2 Law of Pure Types and Some Exotic Spectra of Fractal Spectral Measures......Page 55 1 Introduction to the General Spectral Measures......Page 56 2 Law of Pure Types......Page 58 3 Spectral Properties of Cantor Measures on mathbbR......Page 62 References......Page 71 3 The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative......Page 73 2 Analysis of Infinite Products......Page 74 2.1 What Measures on BmathbbN have a Transfer Operator?......Page 81 2.2 Subalgebras in Linfty(Ω,Σ) and a Conditional Expectation......Page 83 2.3 A Stochastic Process Indexed by mathbbN......Page 86 2.4 Application to Random Walks......Page 88 2.5 An Application to Integral Operators......Page 90 3.1 Preliminaries About r:BrightarrowB......Page 91 3.2 Compact Groups......Page 98 4 Isometries......Page 99 References......Page 102 2 Lq-Dimensions and Images of Measures......Page 104 3 The Main Inequality......Page 106 4 Images of Measures Under Gaussian Processes......Page 110 5 Measures on Almost Self-affine Sets......Page 112 6 Random Multiplicative Cascade Measures......Page 117 References......Page 119 1 Introduction......Page 121 2.1 Multifractal Spectrum of a Sequence of Walsh Functions......Page 125 2.2 Box Dimension of Some Multiplicatively Invariant Set......Page 126 3.1 Potential Theory......Page 128 3.2 Hausdorff Dimensions of a Measures......Page 129 3.3 Sums, Products, Convolutions, Projections of Measures......Page 130 3.4 Ergodicity and Dimension......Page 132 4.1 Oriented Walks......Page 133 4.2 Riesz Products......Page 135 4.3 Evolution Measures......Page 136 5 Multiple Birkhoff Averages......Page 137 5.1 Thermodynamic Formalism......Page 138 5.2 Telescopic Product Measures......Page 139 5.4 Multiplicatively Invariant Sets......Page 141 6.2 Invariant Spectrum and Mixing Spectrum......Page 144 6.4 Subshifts of Finite Type......Page 145 6.6 Mutual Absolute Continuity of Two Riesz Products......Page 146 6.7 Doubling and Tripling......Page 147 References......Page 148 6 Heat Kernels on Metric Measure Spaces......Page 152 1.1 Examples of Heat Kernels......Page 153 1.2 Abstract Heat Kernels......Page 159 1.3 Heat Semigroups......Page 160 1.4 Dirichlet Forms......Page 161 1.5 More Examples of Heat Kernels......Page 165 2.1 Identifying Φ in the Non-local Case......Page 166 2.2 Volume of Balls......Page 167 2.3 Besov Spaces......Page 168 2.4 Subordinated Semigroups......Page 169 2.5 The Walk Dimension......Page 170 2.6 Inequalities for the Walk Dimension......Page 174 2.7 Identifying Φ in the Local Case......Page 176 3.1 Ultracontractive Semigroups......Page 177 3.2 Restriction of the Dirichlet Form......Page 178 3.3 Faber-Krahn and Nash Inequalities......Page 179 3.4 Off-diagonal Upper Bounds......Page 182 4.1 Using Elliptic Harnack Inequality......Page 190 4.2 Matching Upper and Lower Bounds......Page 196 4.3 Further Results......Page 200 5.1 Upper Bounds for Non-local Dirichlet Forms......Page 202 5.2 Upper Bounds Using Effective Resistance......Page 208 References......Page 210 1 Stochastic Completeness of a Diffusion......Page 213 2 Jump Processes......Page 217 3 Random Walks on Graphs......Page 222 References......Page 227 1.1 Self-similar Sets......Page 229 1.2 Dimension of Self-similar Sets......Page 230 1.3 Progress Towards the Conjecture......Page 232 2.2 Minimal Growth......Page 234 2.4 Power Growth, the ``Fractal'' Regime......Page 235 2.5 Trees and Tree-Measures......Page 237 2.6 Inverse Theorems in the Power-Growth Regime......Page 239 3 A Conceptual Proof of Theorem 1.2......Page 241 3.1 Sumset Structure of Self-similar Sets......Page 242 3.2 From Theorem 1.2 to Additive Combinatorics......Page 243 3.3 Getting a Contradiction......Page 246 3.4 Sums with Self-similar Sets......Page 249 4.1 Entropy......Page 250 4.2 Inverse Theorems for Entropy......Page 251 4.3 Reduction of Theorem 1.2 to a Convolution Inequality......Page 252 4.4 Getting a Contradiction......Page 254 References......Page 255 1 Introduction......Page 257 2 Basic Notions......Page 260 3 Quasisymmetric Metrics and Scales......Page 262 4 Sierpinski Carpet and Its Invisible Sets......Page 267 5 Metric Associated with Invisible Set......Page 273 6 Construction of Invisible Sets......Page 276 7 Generalized Sierpinski Carpet......Page 283 References......Page 286 1 Introduction......Page 287 2 Marstrand's Projection Theorem......Page 288 3 Projection Theorems in Heisenberg Groups......Page 291 4 Generalized Projections......Page 296 5 Restricted Families of Projections......Page 298 6 Minkowski and Packing Dimensions......Page 300 7 Constancy Results for Projections......Page 301 8 Slicing Theorems......Page 302 References......Page 304 11 The Geometry of Fractal Percolation......Page 306 2.1 An Informal Description of Fractal Percolation......Page 307 2.2 Fractal Percolation Set in More Details......Page 308 2.3 The Corresponding Probability Space and Statistical Self-similarity......Page 310 3.1 The Homogeneous Case......Page 311 3.2 The Inhomogeneous Case......Page 312 4.1 The Arithmetic Sum and Its Visualization......Page 313 4.2 The Product of Two One Dimensional Fractal Percolation Versus a Two Dimensional Fractal Percolation......Page 314 4.4 The Lebesgue Measure of the Arithmetic Difference Set......Page 315 5 General Projections: The Opaque Case......Page 316 5.2 Condition A......Page 318 5.4 Checking Condition A(α)......Page 321 6 General Projections: The Transparent Case......Page 322 7 The Arithmetic Sum of at Least Three Fractal Percolations......Page 323 References......Page 325 1 Introduction......Page 327 2.1 Definition and Basic Properties of the SVF......Page 329 2.2 SVF Topological Pressure......Page 331 2.3 Affinity Dimension and Self-affine Sets......Page 332 3.1 Subadditive Thermodynamic Formalism......Page 334 3.2 Oseledets' Multiplicative Ergodic Theorem......Page 336 3.3 The Cone Condition......Page 337 4.1 General Strategy and the Case of Equal Lyapunov Exponents......Page 339 4.2 The Case of Distinct Lyapunov Exponents......Page 341 4.3 Some Remarks on the Higher-Dimensional Case......Page 343 References......Page 344 1 Introduction......Page 345 2.1 Normal Cycles and Curvatures......Page 346 2.2 Distance Estimates for Curvature Measures......Page 349 3 Stability of Fractal Curvatures Under Approximate Perturbations......Page 353 References......Page 355 Participants......Page 357