Final Exam --- 2001
Do all problems and show all work.
- a. F(x) = x2+2 is chaotic on the
b.F(z) = z2 - 1 has a connected
filled Julia set.
c. F(x) = kx(1-x) has a saddle node
bifurcation at k=1.
d. The subset of the sequence space
that consists of all sequences that contain infinitely many 0's is dense.
e. If a continuous function on the real
line has a periodic point of period 60, then it also ha a periodic
point of period 40.
2. Quickies. Answers only -- no partial credit.
a. Give an example of a sequence in Sigma whose distance
from the sequence (010101...) is exactly 1/8.
b. Sketch the 2/7-bulb in the Mandelbrot set.
c. List all periodic points of period 3 for the doubling
function on the unit circle (i.e., theta --> 2 theta).
d. Give an example of a point in Sigma whose orbit under
the shift is dense.
e. The complex function F(z) = z2 + c has an attracting
fixed point for which complex c-values?
f. Consider the Cantor set obtained by removing middle
fifths subintervals from the unit interval at each stage. What is the
fractal dimension of this set?
3. Definitions. Give the precise definitions of each of
a. The filled Julia set of z2 + c.
b. The function F is continuous at x0.
c. H is a conjugacy between F and G.
d. Q is a dense subset of the set T
e. The function F is chaotic.
4. Give exact formulas for the fixed points and the
period two points of the function F(z) = z2 + c.
5. Cantor set. In an essay discuss the Cantor Middle
Thirds set. Prove that this set is uncountable. Discuss the fractal
dimension of the set.
6. In an essay, describe why the filled Julia set of
F(z) = z2 + c consists of infinitely many pieces
when the orbit of
0 tends to infinity.
7. In an essay, discuss the role of the critical
orbit (the orbit of 0) in the dynamics of F(x) = x2 +
both the real and complex case.