Abstract

We will look at the dynamics of the quadratic maps P(z)=z^2+c on the complex plane. The Complex plane is decomposed into two parts, The Julia and fatou sets. The Connectedness locus or the Mandelbrot set is the set of parameters c for which the corresponding Julia set is connected. Some Combinatorial data is associated to each parameter in this locus. The Combinatorial Rigidity is a fundamental phenomenon in holomorphic dynamics which means that the metric properties of these dynamical systems are determined by their combinatorics. We will talk about conjectures and known results in this direction.



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