Course Outline

Course Lecture Notes

L-systems

Through the concept of Lindenmayer systems we will learn about the behavior of discrete symbolic systems. We will see how the symbolic rule $F \to F+F--F+F$ leads to the picture at the right. We will also see how tilings of the plane arise from symbolic dynamical systems, including the nonperiodic Penrose tiling at right.

Self-replicating tilings

We will discuss tilings of space arising from simple dynamical systems known as ``inflation-and-deflation.''

Fractal basics

Most dynamical systems lead to geometric and physical behavior that defies conventional smooth calculus and geometry. We will introduce the basic concepts of ``fractal geometry,'' including the definition and numerical computation of fractal dimension and the concept of self-similarity.

Iterated Function Systems

Here we discuss examples of ``stochastic'' dynamical systems where several rules of evolution are given and at any given stage each rule has a certain probability of being applied. Our main examples will be Iterated Functions Systems where the rules are linear functions on the plane. The classic example is the ``fern.'' Lately, these systems have been advanced as a possible method of compression of photographic and video images.

Interval self-mappings

We will study the classic example of the one-dimensional dynamical system $x\mapsto \lambda x(1-x)$ where $x$ belongs to the unit interval and $\lambda$ is a fixed constant between 0 and 4. This mapping maps the unit interval $[0,1]$ back into itself. We think of the number $x$ as the ``state'' and the function as the rule determining the state at the next stage. Many of the basic concepts of dynamical systems: ``fixed point,'' ``periodic cycle,'' ``attractor,'' ``bifurcation,'' ``chaos,'' etc., can be clearly defined in this simple example.

Complex Iteration

If time permits, we will discuss very simple aspects of the theory of conformal dynamical systems, particularly iteration of complex quadratic polynomials. I will try to map out the structure of the Mandelbrot set in terms of the symbolic dynamical behavior.