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MA 471-671

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Final Exam --- 2000

Do FIVE of the following problems. In each case,
show all work.
Each essay should
be approximately 3 pages in length. It should be written in proper
English and accompanied by relevant graphs or pictures.

1. Bifurcations. In an essay, discuss the concept of
bifurcation. Describe in detail at least two different bifurcations
with specific examples. Include graphs and other relevant diagrams.

2. Conjugacy. In an essay, discuss the notion of
conjugacy. Why is conjugacy important in dynamical systems? In the
essay, you should show explicitly that the functions $F(z) = z^2$ on
the unit circle and **F(x) = x**^{2} -2 on the interval
**-2 <= x <= 2** are (semi)-conjugate.

3. Define dense subset. Give several examples of dense
and non-dense subsets. Then prove or disprove: The set of
all sequences in the sequence space that end with all zeroes is dense
in the sequence space.

4. The Mandelbrot Set. In an essay, discuss the
structure and meaning of the Mandelbrot set. Be sure to include a
discussion of how you can "read off" the periods of the primary
bulbs from these bulbs' geometry. Finally, find a formula for the
boundary of the main cardioid.

5. Cantor set. In an essay discuss the Cantor Middle
Thirds set. Describe its construction and other features of this
set. Prove that this set is uncountable. Discuss the fractal
dimension of the set.

6. In an essay, discuss the concept of a filled Julia
set of the quadratic function $F_c(z) = z^2 + c$. Discuss the escape
criterion for these functions. Include in your essay a clear
discussion of the Fundamental Dichotomy. Give a brief description of
the proof that, if the orbit of $0$ escapes, then the filled Julia set
consists of infinitely many pieces.

7. Chaos. Define chaos. Prove that the shift map on
the sequence space is chaotic.