Chaos, Fractals, and Dynamical Systems
Assignment #6, due Thursday, October 28th.

A.  Proving the shift map is chaotic.  In this assignment, I'll take you through the steps needed to prove the shift map

S(x) = 10x (mod 1),

also known as S(x) = fPart(10x), is chaotic on the interval [0, 1).  First, take out your homework assignment #1 and remind yourself about the shift map.

1.  Periodic points are dense.  You were asked to describe the periodic points of S in a previous assignment.

  1. List the fixed points of S.  Find the distance between two adjacent fixed points.
  2. Characterize the fixed points of S2.  Find the distance between two adjacent fixed points of S2.
  3. Following the pattern, find the distance between two adjacent fixed points of Sk.
Your answers should show that the distance between adjacent fixed points of Sk goes to zero as k goes to infinity.  This implies that the periodic points of S are dense in [0, 1).

2.  The shift map is transitive.  We must show that, given any x and y in the domain, and a small distance e,there's another element z whose orbit comes within e of both x and y.

  1. I claim that, if x = 0.3 and y = 0.5, and if e = 0.001, then the orbit of the point z = 0.3005 comes within e of both x and y.  Show that this is true by listing the first 4 points in the orbit of z (include z0). Identify the points that are within e of x or y.
  2. Now suppose x = 0.312, y = 0.54, and e = 0.001.  Find a z whose orbit comes within e of both x and y.
  3. Champernowne's constant is the number C = 0.123456789101112131415....   Show that for any e > 0 and any x and y, the orbit of Champernowne's constant comes within e of both x and y.  (Hint:  given e > 0, you can always find a natural number k so that 10-k < e.  Then show that the orbit of C comes within 10-k of x and y.)
3.  Sensitive dependence on initial conditions.  To prove sensitive dependence, we have to find a distance b>0 such that for any given e>0 and point x in [0, 1) there's a point y within e of x such that Sk(x) and Sk(y) are at least distance b apart for some k.

Ok, let's break this down a little!  I'm going to choose b = 0.1 .

  1. Suppose x = 0.3 and e = 0.001.  Find a value y that satisfies the following:
    1. The distance from x to y is less than e.
    2. You can find a k so that Sk(x) and Sk(y) are at least distance b apart.  Write down your value of k.
    This isn't as hard as it looks--in fact, pretty much any y you pick that satisfies the first property will satisfy the second.
  2. Now suppose x = 1/3  and e = 0.001.  Find a value y that satisfies the same two conditions.  (Hint:  look at the decimal expansion of x + e/2 or x - e/2.)
  3. Describe how you could find y no matter what x and e I gave you.
B.  The Chaos Game revisited.  Play the Chaos Game (found at http://math.bu.edu/DYSYS/applets/) until you are able to win a game at the Master Level.  We'll be talking and working with this in class.  You also might want to familiarize yourself with Fractalinia, found on the same page.