Computability of Julia Sets

  • Mark Braverman
  • Michael Yampolsky

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 23)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Pages 37-63
  3. Pages 65-79
  4. Pages 81-118
  5. Back Matter
    Pages 149-151

About this book


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.


Computer Julia set complexity computability computational complexity computer science dynamische Systeme theoretical computer science

Authors and affiliations

  • Mark Braverman
    • 1
  • Michael Yampolsky
    • 2
  1. 1.Department of Computer ScienceUniversity of Toronto Sandford Fleming BuildingTorontoCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Bibliographic information