Homework Assignment #2
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
5.5.
Assume that F (x0 ) = x0 , F (x0 ) = 1, and F (x0 ) > 0.
Then the graph of the function near x0 is the following.
y = F (x)
y=x
x0
From here we can conclude that
Homework Assignment #5
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 9.5.
1
If s satises d[s, 0] = 2 , then
i=0
1
si
= ,
i
2
2
so s0 = 0. We then have two cases:
Case 1.
If s1 = 1, then si = 0, for all i = 2, 3, . . .
Case 2.
If
Homework Assignment #3
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Extra Question.
(a) Let p < x0 < p+ . We prove by induction that xn > p . Assume that xn > p . Then
xn+1 = x2 + c > p2 + c = p .
n
By induction, we deduce that xn > p
Homework Assignment #4
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 7.15.
In this solution, I use the notation x (0.a1 a2 a3 . . .) to write x has the ternary expansion
(0.a1 a2 a3 . . .).
Let x = K, which has ternary expansion
Homework Assignment #6
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 11.8.
Part 1 .
This question has two parts.
Show that this function has a 6-cycle.
First the function can be written as
2 + 2x
F (x) = 6 2x
4 x
Then take x0 =
MAT335 - Chaos, Fractals, and Dynamics - Fall 2014
Solution of Term Test 1 - October 21, 2014
Time allotted: 60 minutes.
1.
Consider the function F : R R dened by
x
2(x 1)
F (x) = 0
2(x + 1)
x
Aids permitted: None.
if x
2
if 1 < x < 2
if 1
x
1
if 2 < x <