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Dynamical Systems and Fractals Course
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
-systems
Basic definitions
Fibonacci
-system
Types of
-systems
Thue-Morse
-system
Paperfolding and the Dragon curve
Turtle graphics and
-systems
FRACTINT conventions
Branching and bracketed
-systems
Famous
-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
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
Bibliography
David Wright
2015-01-29
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