 1. Let F(x)=x^2. Compute the first five points on the oribit of 1/2.
F^1(1/2)=F(1/2)=1/4
F^2(1/2)=F(1/4)=1/16
F^3(1/2)=F(1/16)=1/256
F^4(1/2)=F(1/256)=1/65536
F^5(1/2)=F(1/65536)=1/4294967296
 3. Let F(x)=x^22. Computer F^2(x) and F^3(x).
The second iterate of F is
F^2(x) = F(F(x)) = F(x^22) = [x^22]^2  2 = x^4  4x^2 + 2
The third iterate of F is
F^3(x) = F(F^2(x)) = F(x^44x^2+2) =[x^44x^2+2]^2  2=x^8  8x^6 + 20x^4  16x^2 + 2
 5. Let F(x)=x^2. Compute F^2(x), F^3(x) and F^4(x). What is the formula for F^n(x)?
F^2(x) = F(F(x)) = F(x^2) = [x^2] ^2 = x^4
F^3(x) = F(F^2(x)) = F(x^4) = [x^4] ^2 = x^8
F^4(x) = F(F^3(x)) = F(x^8) = [x^8] ^2 = x^16
In general,
F^n(x) = [x]^(2^n)
 6. Let A(x) = x . Compute A^2(x) and A^3(x).
 Recall, by definition:
x={ x if x >= 0
or, {x if x <= 0
Therefore, since A(x) >= 0 for all x
A^2(x) = A(A(x)) = A(x) = A(x)
Similarly,
A^3(x) = A(A^2(x)) = A(A(x)) = A(x) = A(x)
In fact,
A^n(x) = A(x) for n>=1.

7. Find all real fixed points.
a) F(x)=3x+2
3x+2=x > x=1
b) F(x)=X^22
x^22=x > x^2x2=0 > x=2,1
c) F(x)=x^2+1
x^2+1=x > x^2x+1=0 > no real fixed points
d) F(x)=x^33x
x^33x=x > x^34x=0 > x(x^24)=0 > x=0,+2,2
e) F(x)=x
x=x > xx=0 > all x>=0
f)F(x)=x^5
x^5=x > x^5x=0 > x(x^41)=0 > x=0,+1,1
 11. For each of the following seeds, discuss the behavior of the resulting orbit under D.
To recall the definition and graph of the doubling function click
here.
a) x=0.3
D(0.3)=0.6 , D(0.6)=0.2 , D(0.2)=0.4 , D(0.4)=0.8 and D(0.8)=0.6
{ie. 0.3 > 0.6 > 0.2 > 0.4 > 0.8 > 0.6}
Therefore, the orbit of 0.3 is eventually periodic with preperiod 1.
b) x=0.7
0.7 > 0.4
Therefore, it follows by part a) that 0.7 is eventually periodic with preperiod 1.
c) x=1/8
1/8 > 2/8 > 4/8 > 0 which is fixed by D
Therefore, 1/8 is eventually fixed with preperiod 3.
d) x=1/16
1/16 > 2/16 > 4/16 > 8/16 > 0
Therefore, 1/16 is eventually fixed with preperiod 4.
e) x=1/7
1/7 > 2/7 > 4/7> 1/7
Therefore, 1/7 is periodic with period 3 (ie. lies on a three cycle)
f) x=1/14
1/14 > 2/14 =1/7
Therefore, by part e), 1/14 is eventually periodic(period 3) with preperiod 1.
g) x=1/11
1/11 > 2/11 > 4/11> 8/11 > 5/11 > 10/11> 9/11 > 7/11 > 3/11> 6/11 > 1/11
Therefore, 1/11 is periodic with period 10.
h) x=3/22
3/22 > 6/22 = 3/11
Therefore, by part g), 3/22 is eventually periodic(period 10) with preperiod 1.
 12. Give an explicit formula for D^2(x),D^3(x) and D^n(x).
Recall:
D(x)=
{ 2x for 0 <= x < 1/2
{ 2x1 for 1/2 <= x < 1
D^2(x)=
{ 4x for 0 <= x < 1/4
{ 4x1 for 1/4 <= x < 1/2
{ 4x2 for 1/2 <= x < 3/4
{ 4x3 for 3/4 <= x < 1
D^3(x)=
{ 8x for 0 <= x < 1/8
{ 8x1 for 1/8 <= x < 1/4
{ 8x2 for 1/4 <= x < 3/8
{ 8x3 for 3/8 <= x < 1/2
{ 8x4 for 1/2 <= x < 5/8
{ 8x5 for 5/8 <= x < 3/4
{ 8x6 for 3/4 <= x < 7/8
{ 8x7 for 7/8 <= x < 1
D^n(x)=
{ 2^n*x for 0 <= x < 1/(2^n)
{ 2^n*x1 for 1/(2^n <= x < 2/(2^n)
{ 2^n*x2 for 2/(2^n) <= x < 3/(2^n)
{ .
{ .
{ .
{ 2^n*x(2^n 1) for (2^n 1)/2^n <= x < 1
or
2^n*x K for K/2^n <= x < (K+1)/2^n
 13. Graphs of D^2(x),D^3(x) and D^n(x).
The graph of D^2 consists of 4 straight lines, each with slope
4. The graph of D^3 consists of 8 straight lines, each with
slope 8. The graph of D^n consists of 2^n straight lines,
each with slope 2^n.