Dynamical Systems and Fractals Lecture Notes

David J. Wright

• Contents
• Introduction to Dynamical Systems
  • Symbolic dynamical systems
  • Discrete-time systems
  • Continuous-time systems
  • Fractal long-term behavior
• Computational aids
  • Netscape
  • MAPLE
  • FRACTINT
  • Supplementary program files
• L-systems
  • Basic definitions
  • Fibonacci L-system
  • Types of L-systems
  • Thue-Morse L-system
  • Paperfolding and the Dragon curve
  • Turtle graphics and L-systems
  • FRACTINT conventions
  • Branching and bracketed L-systems
  • Famous L-systems of mathematical history
  • Self-similarity and scaling
• Tilings as Dynamical Systems
  • Basic concepts
  • Euclidean similarities
  • Examples of self-similar tilings
  • Periodic and recurrent tilings
  • Quasicrystals
  • Self-similar tilings arising from projections
  • Complex numbers and similarity constants
• Dynamical Systems and Fractal Basics
  • What is a fractal?
  • Metric spaces
  • Dynamical systems, orbits and attractors
  • Rotations of a circle
  • Topological dimension
  • Fractal dimension
  • Dimension computed by growth and by self-similarity
  • Some sample computations
• Iterated Function Systems
  • Self-similarity and sets of affine maps
  • Contractions and Fixed Points
  • Linear Contractions
  • Contraction mappings and hyperbolic IFS's
  • Calculating the attractor by random iteration
  • Plotting IFS's with FRACTINT
  • Designing IFS's: the Collage Theorem
• Interval Self-mappings
  • Iteration of a function f(x)
  • Web diagrams and fixed points
  • Population dynamics and iteration of functions
  • Quadratic mappings of the unit interval
  • Down the waterfall
  • Falling all the way: Cascade diagrams
  • Sarkovskii's theorem and the islands of stability in the cascade diagram
• Complex Iteration
• Outside work
  • Extra Readings
  • Problems
• References
• About this document ...