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
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
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
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