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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 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.

Algebra. --- Algorithms. --- Computer science. --- Computer software. --- Information theory. --- Julia sets. --- Julia sets --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Computer Science --- Data processing --- Fractals. --- Data processing. --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Sets, Julia --- Mathematics. --- Computer programming. --- Computers. --- Computer science --- Programming Techniques. --- Theory of Computation. --- Algorithm Analysis and Problem Complexity. --- Mathematics of Computing. --- Dimension theory (Topology) --- Fractals --- Software, Computer --- Computer systems --- Communication theory --- Communication --- Cybernetics --- Informatics --- Science --- Mathematical analysis --- Algorism --- Algebra --- Arithmetic --- Foundations --- Computer science—Mathematics. --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Machine theory --- Calculators --- Cyberspace --- Computers --- Electronic computer programming --- Electronic data processing --- Electronic digital computers --- Programming (Electronic computers) --- Coding theory --- Programming