Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:

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Homework Assignment 22

Assigned Tuesday March 9; due Friday March 19, 2PM, at SS 1071

This document in PDF: HW.pdf

Required reading. All of Spivak's Chapter 22.

To be handed in. From Spivak Chapter 22: 1 (odd parts), 2 (odd parts), 5, 13.

Recommended for extra practice. From Spivak Chapter 22: 1 (even parts), 2 (even parts), 9, 27, 28, 29.

Just for fun. For some constant number $c$, consider the function $f_c(x)=4cx(1-x)$. Let $A$ be the set of all pairs $(c,y)$ so that $0\leq c\leq 1$ and $y$ is a limit of a subsequence of the sequence $f_c(\frac12)$, $f_c(f_c(\frac12))$, $f_c(f_c(f_c(\frac12)))$, .... Write a computer program to draw the set $A$ in the plane whose axes are $c$ and $y$, and if your program and picture are nice, they'll find their place on this class' web site.

It's a hard one, but it's well worth it. The set $A$ is way more complex than you would expect, with parts that scream ``structure'' and parts that scream ``mess''. If you've ever heard the word ``chaos'' in a mathematical context before, this is it. And if you've ever seen pictures of the beautiful ``Mandelbrot Set'', our $A$ is a close relation.

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