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Fractal geometry is used to model complicated natural and technical phenomena in various disciplines like physics, biology, finance, and medicine. Since most convincing models contain an element of randomness, stochastics enters the area in a natural way. This book documents the establishment of fractal geometry as a substantial mathematical theory. As in the previous volumes, which appeared in 1998 and 2000, leading experts known for clear exposition were selected as authors. They survey their field of expertise, emphasizing recent developments and open problems. Main topics include multifractal measures, dynamical systems, stochastic processes and random fractals, harmonic analysis on fractals.
List of contents
1. Fractal Sets and Measures.- Markov Operators and Semifractals.- On Various Multifractal Spectra.- One-Dimensional Moran Sets and the Spectrum of Schrödinger Operators.- 2. Fractals and Dynamical Systems.- Small-scale Structure via Flows.- Hausdorff Dimension of Hyperbolic Attractors in$${mathbb{R}^3}$$.- The Exponent of Convergence of Kleinian Groups; on a Theorem of Bishop and Jones.- Lyapunov Exponents Are not Rigid with Respect to Arithmetic Subsequences.- 3. Stochastic Processes and Random fractals.- Some Topics in the Theory of Multiplicative Chaos.- Intersection Exponents and the Multifractal Spectrum for Measures on Brownian Paths.- Additive Lévy Processes: Capacity and Hausdorff Dimension.- 4. Fractal Analysis in Euclidean Space.- The Fractal Laplacian and Multifractal Quantities.- Geometric Representations of Currents and Distributions.- Variational Principles and Transmission Conditions for Fractal Layers.- 5. Harmonic Analysis on Fractals.- Function Spaces and Stochastic Processes on Fractals.- A Dirichlet Form on the Sierpinski Gasket, Related Function Spaces, and Traces.- Spectral Zeta Function of Symmetric Fractals.
Summary
Fractal geometry is used to model complicated natural and technical phenomena in various disciplines like physics, biology, finance, and medicine. Since most convincing models contain an element of randomness, stochastics enters the area in a natural way. This book documents the establishment of fractal geometry as a substantial mathematical theory. As in the previous volumes, which appeared in 1998 and 2000, leading experts known for clear exposition were selected as authors. They survey their field of expertise, emphasizing recent developments and open problems. Main topics include multifractal measures, dynamical systems, stochastic processes and random fractals, harmonic analysis on fractals.
