This book is devoted to a phenomenon of
fractal sets, or simply fractals. Topics covered includes: Sierpinski gasket,
Harmonic functions on Sierpinski gasket, Applications of generalized numerical
systems, Apollonian Gasket, Arithmetic properties of Apollonian gaskets,
Geometric and group-theoretic approach.
This PDF book covers the following
topics related to Fractals in Probability and Analysis : Minkowski and Hausdorff
dimensions, Self-similarity and packing dimension, Frostman’s theory and
capacity, Self-affine sets, Graphs of continuous functions, Brownian motion,
Random walks, Markov chains and capacity, Besicovitch–Kakeya sets, The Traveling
Author(s): Christopher J. Bishop Stony Brook
University, Yuval Peres Microsoft Research
The term fractal
usually refers to sets which, in some sense, have a self-similar structure. This
PDF book covers the following topics related to Random Fractals : Representing
fractals by trees, Fine properties of stochastic processes, More on the planar
Brownian path, etc.
This book is an introduction to the theory of iteration of
expanding and nonuniformly expanding holomorphic maps and topics in geometric
measure theory of the underlying invariant fractal sets. Major topics covered:
Basic examples and definitions, Measure preserving endomorphisms, Ergodic theory
on compact metric spaces, Distance expanding maps, Thermodynamical formalism,
Expanding repellers in manifolds and Riemann sphere, preliminaries, Cantor
repellers in the line, Sullivan’s scaling function, application in Feigenbaum
universality, Fractal dimensions, Sullivan’s classification of conformal
expanding repellers, Conformal maps with invariant probability measures of
positive, Lyapunov exponent and Conformal measures.
This note covers the following topics: Rigidity and inflexibility in conformal dynamics, Hausdorff
dimension and conformal dynamics: Strong convergence of Kleinian groups,
Geometrically finite rational maps and Computation of dimension.