Homework 5
Math 323, Fall 2012
Due Date: Thursday, October 25
Let f : [0, 8] → [0, 8] be the following piecewise-linear function:
8
7
6
5
4
f (x) =
if 0 ≤ x < 1
if 1 ≤ x < 2
if 2 ≤ x < 4


12 − x if 4 ≤ x < 6






30 − 4x if 6 ≤ x < 7





16 − 2x if 7 ≤ x ≤ 8.
3
2
1
0


2x







4x − 2






x + 4
0
1
2
3
4
5
6
7
8
As with the tent map, we can make itineraries for f , with L = [0, 4] and R = [4, 8]. The goal of this
assignment is to show that f is conjugate to the tent map.
1. (a) The function f has one fixed point on the interval (0, 8). Find it.
(b) Find the point whose itinerary under f is LRRRRRR · · · .
(c) What are the corresponding points for the tent map?
2. Find the endpoints of the LL, LR, RR, and RL intervals for f . Do the same for the LLL, LLR,
LRR, LRL, RRL, RRR, RLR, and RLL intervals.
3. (a) Find the itinerary of the point 1.6 under f .
(b) Find the Lyapunov number of the periodic cycle you found in part (a)
(c) Which point in [0, 1] has the same itinerary under the tent map?
4. (a) The function f has two 3-cycles. Find them. (Express your answers as fractions.)
(b) Determine the Lyapunov numbers of the cycles that you found in part (a).
(c) Find the two 3-cycles for the tent map.
5. To prove that f and T are conjugate, we must find a continuous bijection c : [0, 1] → [0, 8]
such that c ◦ T = f ◦ c.
(a) Use questions 1–4 to list twenty-one data points for the function c (two from question 1,
nine from question 2, four from question 3, and six from question 4).
(b) Find the formula for a function c whose graph includes all 21 data points. Verify this by
plotting a graph of c together with the data points.
(c) Use Mathematica to verify that your formula for c satisfies c ◦ T = f ◦ c.