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.
Goal of this course
note is primarily to develop the foundations of geometric measure theory, and
covers in detail a variety of classical subjects. A secondary goal is to
demonstrate some applications and interactions with dynamics and metric number
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, Sullivanís scaling function, application in Feigenbaum
universality, Fractal dimensions, Sullivanís 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.