Homework 7
Math 323, Fall 2012
Due Date: Thursday, November 8
1. Let f : [0, 1] → [0, 1] be the function f (x) = cx(1 − x), where c is the smallest positive
value for which 1/2 is periodic with period 7.
(a) Determine the value of c. Your answer must be correct to six decimal places.
(b) The seven points in the orbit of 1/2 partition [0, 1] into eight intervals. Use the
Plot command to draw a graph of f , adding GridLines so that it is possible to
see how these intervals map to one another.
(c) Use GraphPlot to draw the transition graph for f with respect to these eight
intervals. (Label the intervals A, B, C, D, E, F, G, H in order from left to right.)
(d) Which periods exist for f ? Justify your answer completely by describing a sample
itinerary for each possible period.
2. The following picture shows a tree T ⊆ R2 , which is the union of four line segments A,
B, C, and D:
H0, 1L
B
H0, 0L
A
C
H1,0L
D
H3,0L
H0,-1L
Let F : T → T be the continuous function with the following properties:
• F (0, 0) = (0, 0), F (0, −1) = (3, 0), F (1, 0) = (0, 1), F (0, 1) = F (3, 0) = (0, −1).
• F acts as a rotation on each of the segments B, C, and D.
• On A, the map F acts as a dilation by a factor of 3, followed by a 90◦ rotation.
(a) Use GraphPlot to draw a transition graph for F with respect to the line segments
A, B, C, and D.
(b) Find a formula for F (x, y) on each of the segments A, B, C, and D.
(c) Find the itineraries and the coordinates of all the period-three points for F .
(d) Find the itineraries and the coordinates of all the period-five points for F .
(e) Which periods exist for F ? Justify your answer completely by describing a sample
itinerary for each possible period.