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In this minicourse, we will discuss various topological and analytic
properties of the Mandelbrot set M, which is the connectedness locus of quadratic
complex polynomials (i.e., parameters c for which the polynomial z^2+c has a
connected Julia set). The main features of M that will be touched upon include:
(1) connectedness of M,
(2) combinatorial rigidity of geometrically finite parameters in M (analogue of
Mostow rigidity),
(3) topological model of M in terms of external parameter rays, and
(4) renormalization and existence of small copies of M within itself. |