Complex dynamics is the study of iteration of holomorphic functions on the Riemann
sphere. In this thesis, we will focus on dynamical systems obtained by iterating
rational functions, that is iterating a holomorphic endomorphism of the Riemann
sphere f : CP^1 -> CP^1. The family of iterates {f^n; n natural number} divides CP^1 into two
complementary sets; the Fatou set F , that is the set of points in the neighbourhood of
which the family {f^n; n natural number} is normal, and the Julia set J, which is its complement.
Each such map has fixed points, which are classified into five types according
to their multiplier. The main goal of this thesis is to locally linearize f in some
neighbourhood of fixed points and describe the behaviour of orbits in their vicinity.
We prove that locally near a fixed point, f is conformally conjugate to a linear
map z |-> kz in the case of attracting, repelling and some irrationally indifferent
fixed points. In the case of superattracting fixed points f is conjugate to z |-> z^p,
and in the case of rationally indifferent ones it is conjugate to z |-> z+1 in some
disk with the fixed point on its boundary.
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