Alan F. Beardon.
Iteration of Rational Functions: complex analytic dynamical systems.
Springer. 1991
This book makes available a comprehensive, detailed,
and carefully organized treatment of the foundations of
the theory of iteration of rational functions of a complex variable.
The material covered extends from the original memoirs of Fatou and Julia
to the recent and important results and methods of Sullivan and Shishikura.
Many of the details of the proofs have not occurred in print before.
The theory of dynamical systems and chaos has recently undergone
a rapid growth in popularity,
in part due to the spectacular computer graphics of Julia sets,
fractals, and the Mandelbrot set.
This text focuses on the specialized area of complex analytic dynamics,
a subject that dates back to 1916 and is currently a very active area in mathematics.