Math 145, Winter 1999: Chaos Theory

Lecture 7W, 17 Feb 1999

Agenda

• Play second half of Warren Burt, tape #1
• Complex dynamics

Definitions

1. the quadratic map family, Q_c: C -> C; z |-> z² + c
2. the trajectory of z₀, (z₀, z₁, z₂, ...)
3. the basin of infinity, B_c(inf) = { z₀ | z_i -> inf}
4. the filled Julia set, K_c = C \ B_c(inf)
5. the Julia set, J_c = boundary of K_c
6. the Mandelbrot set, M = { c | K_c is connected }
7. the magic disk, D_c = { z | mod(z) < max {2, mod(c)} }
   where mod(x + iy) = sqr root (x² + y²)

Theorems

1. IF there is a positive integer k
   such that z_k is outside the magic disk,
   THEN z₀ is in the basin of infinity.
2. IF mod(c) > 2 THEN 0 is in the basin of infinity
3. IF mod(c) > 2 THEN K_c is a Cantor (dust) set
4. J_c contains all periodic repellors of Q_c

• Devaney, A Short Course, Chs. 16, 17
• Devaney, An Introduction
• H.-O. Peitgen and coworkers, Chaos and Fractals