Homework 3
Math 323, Spring 2011
Due Date: Friday, February 18
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
You can get Mathematica code for this function from the Homework3Functions.nb notebook
on the class web page. As with the tent map, we can make itineraries for this function, with
L = [0, 4] and R = [4, 8].
In this assignment, you will be collecting evidence that f (x) is conjugate to the tent map.
1. (a) This function 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 . What is the Lyapunov number of this
cycle?
(b) Which point in [0, 1] has the same itinerary under the tent map?
4. (a) This function 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. Evidence suggests that f and T are conjugate. That is, there exists 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. (Get two from
question 1, nine from question 2, four from question 3, and six from question 4.)
(b) Find the formula for C. Make sure to check that C ◦ T = f ◦ C.