# Exercises for Chaos Under Control

## Chapter 7: Some Geometry of the Mandelbrot Set

10. The Figure shows a part of the cardioid
of the Mandelbrot Set from Seahorse Valley. The sequence generated by iterating
the Mandelbrot map using the control parameters, (a, b), corresponding to the
point A, converges to an n-cycle. What is n? (Hint: read the antenna.) Repeat,
taking (a, b) to be given by B and by C. Answer

11. The Figure shows some buds attached to the
cardioid of the M Set. The sequences generated by iterating the Mandelbrot map
using the control parameters, (a, b), corresponding to A, B, and C, converge to
n-cycles. Using what you know about the geometry of the Mandelbrot Set, what is
the value of n for each of A, B, and C? Answer

12. (a) The North Principal Sequence for the head of the M Set begins 4, 6, 8.
Fill in the Farey sequence two levels below this 4, 6, 8.
Answer

(b) For the top 3-cycle bud, the Principal Sequences are oriented east and west,
not north and south. The East Principal Sequence, say, begins 6, 9, 12. Fill in
the Farey sequence two levels below this 6, 9, 12.
Answer

13. (a) Starting at the second level of the
Farey sequence,
find a rule giving the smallest number in each level.
Answer

(b) Finding a rule for the largest number is a bit trickier, since all levels
contain arbitrarily large entries. So restrict your attention to the part of the
diagram lying between the 2 and 3. Find a rule giving the largest number in each
level for this part of the diagram. Find a similar rule for the part of the diagram
between the 3 and 4.
Answer

(This exercise gives a cartoon idea of why the M Set boundary has dimension two:
everywhere around the boundary of the M Set are buds that are very small, are deep
in the Farey sequence, and consequently have many-branched antennas attached to them.
The closer we get to the boundary of the M Set, the denser is this thicket of
antennas. The real argument is much more subtle than this, relying on relations
between the M Set and Julia sets, and on the delicate selection of an appropriate
sequence of Julia sets. Nevertheless, this little cartoon helps to make the result
plausible.)

Return to Chapter 7 Exercises

Return to Chapter 7 Exercises: A Quick Tour of the
Mandelbrot Set

Go to Chapter7 exercises: Julia Sets

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