Falconer K. Techniques in Fractal Geometry

John Wiley & Sons, 1997. — 274 p. — ISBN: 0471957240.Following on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals. Much of the material presented in this book has come to the fore in recent years. This includes methods for studying dimensions and other parameters of fractal sets and measures, as well as more sophisticated techniques such as thermodynamic formalism and tangent measures. In addition to general theory, many examples and applications are described, in areas such as differential equations and harmonic analysis. This book is mathematically precise, but aims to give an intuitive feel for the subject, with underlying concepts described in a clear and accessible manner. The reader is assumed to be familiar with material from Fractal Geometry, but the main ideas and notation are reviewed in the first two chapters. Each chapter ends with brief notes on the development and current state of the subject. Exercises are included to reinforce the concepts. The author′s clear style and up–to–date coverage of the subject make this book essential reading for all those who with to develop their understanding of fractal geometry.Preface.
Introduction.
Mathematical background.
Sets and functions.
Some useful inequalities.
Measures.
Weak convergence of measures.
Review of fractal geometry.
Review of dimensions.
Review of iterated function systems.
Some techniques for studying dimension.
Implicit methods.
Box-counting dimensions of cut-out sets.
Cookie-cutters and bounded distortion.
Cookie-cutter sets.
Bounded distortion for cookie-cutters.
The thermodynamic formalism.
Pressure and Gibbs measures.
The dimension formula.
Invariant measures and the transfer operator.
Entropy and the variational principle.
Further applications.
Why "thermodynamic" formalism?
The ergodic theorem and fractals.
The ergodic theorem.
Densities and average densities.
The renewal theorem and fractals.
The renewal theorem.
Applications to fractals.
Martingales and fractals.
Martingales and the convergence theorem.
A random cut-out set.
Bi-Lipschitz equivalence of fractals.
Tangent measures.
Definition and basic properties.
Tangent measures and densities.
Singular integrals.
Dimensions of measures.
Local dimensions and dimensions of measures.
Dimension decomposition of measures.
Some multifractal analysis.
Fine and coarse multifractal theories.
Multifractal analysis of self-similar measures.
Multifractal analysis of Gibbs measures on cookie-cutter sets.
Fractal and differential equations.
The dimension of attractors.
Eigenvalues of the Laplacian on regions with fractal boundary.
The heat equation on regions with fractal boundary.
Differential equation on fractal domains.
References.
Index.