# 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.)