Among all computer-generated mathematical images, Julia sets of rational
maps occupy one of the most prominent positions. Their beauty and
complexity can be fascinating. They also hold a deep mathematical
content.
Computational hardness of Julia sets is the main subject of this book.
By definition, a computable set in the plane can be visualized on a
computer screen with an arbitrarily high magnification. There are
countless programs to draw Julia sets. Yet, as the authors have
discovered, it is possible to constructively produce examples of
quadratic polynomials, whose Julia sets are not computable. This result
is striking - it says that while a dynamical system can be described
numerically with an arbitrary precision, the picture of the dynamics
cannot be visualized.
The book summarizes the present knowledge (most of it from the authors'
own work) about the computational properties of Julia sets in a
self-contained way. It is accessible to experts and students with
interest in theoretical computer science or dynamical systems.