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Math 323, Fall 2012

Due Date: Thursday, November 1

1. Let f : [0 , 1] → [0 , 1] be the following piecewise-linear function: f ( x ) =

3 x

if 0 ≤ x ≤ 1 / 3 ,

2 − 3 x if 1 / 3 < x ≤ 2 / 3 ,

3 x − 2 if 2 / 3 < x ≤ 1 .

Find a cubic polynomial g : [0 , 1] → [0 , 1] and a homeomorphism h : [0 , 1] → [0 , 1] such that g ◦ h = h ◦ f .

2. Consider the functions f , g , h , and k whose graphs are shown below:

3

2

1

0

0 1 f

H x

L

2 3

5

4

3

2

1

0

0 1 2 3 4 5 g

H x

L

4

3

2

1

0

0 1 2 h

H x

L

3 4

3

2

1

0

0 1 k

H x

L

2 3

Exactly two of these functions are conjugate. Determine which two are conjugate, and find a formula for the conjugating homeomorphism.

3. Let f : [0 , 3] → [0 , 3] be the following piecewise-linear function:

3

2

1

0

0 1 2 3

Let L = [0 , 1], M = [1 , 2], and R = [2 , 3].

(a) Use the GraphPlot command in Mathematica to draw the transition graph for f .

Make sure to label the vertices L , M , and R , and include arrows along the edges to show the direction of each transition.

(b) The map f has three four-cycles. Find the itinerary of one point from each of these cycles.

(c) Find the point in [0 , 3] whose itinerary under f is RLRMMM .