Download the .ipynb file for this notebook, and place your solutions where indicated (you can make more cells for each problem), keeping the original problem descriptions. Upload only one file, which contains all your work; it should be named "HW09_firstname_lastname". Please include comments in your code; this can also help you get partial credit if your code doesn't work.
Then upload it to Blackboard under the Assignments tab. See Collaboration Policy in Homework section of course webpage (it's the same as it was for previous homeworks).
Choose a point $u$ in the Mandlebrot set and a point $v$ not in the Mandlebrot set. Draw the Julia sets for $f_u(z) = z^2+u$ and $f_v(z) = z^2 + v$. Also draw the Julia sets $f_w$ for several $w$ on the line segment connecting $u,v$. Observe the change from a connected Julia set to a disconnected Julia set.
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Julia sets have a self-similarity property that is a bit different from the fractals we studied earlier. Features that appear in a small region near some point $u$ in the Julia set $J(f)$ will also appear near any other point $v$ in $J(f)$. However the features may appear somewhat rotated, scaled, and distorted relative to each other. Demonstrate this phenomenon with the quadratic polynomial $f(z) = z^2 + (0.3893 + 0.2179*i)$. Choose some point $u$ in $J(f)$, zoom in near $u$, and identify some feature. Then choose another point $v$ in $J(f)$ and find a version of that feature near $v$.
Remember that as you zoom in, you may have to increase the number of iterations in order to get a good picture.
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The set $M\cap \mathbb{R}$ is the intersection of the Mandlebrot set $M$ with the real axis $\mathbb{R}$. Figure out exactly what this set is by studying the picture of the Mandlebrot set. Then give a mathematical justification for this in terms of the definition of the Mandlebrot set.
(It is not possible to give such an simple description for the whole $M$.)
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Recall that the Julia set $J(f)$ was actually defined as the boundary of the set $$\{ z \in \mathbb{C}: \lim_{n\to\infty} f^{\circ n} (z) =\infty\}.$$ Our drawings have been approximations to the above set (called the Filled Julia Set), rather than its boundary. Modify the function draw_julia (from lecture 04/04) so that it draws just the boundary of the set of escaping points. That is, color a pixel black if and only if it escapes and one of its 4 neighboring pixels does not escape.
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