### Renormalization and embedded Julia sets within the Mandelbrot set

Get acquainted with the phenomenon of embedded Julia sets in an interactive session with the web application Juliette.
• Note that the current parameter c is indicated by the yellow cross in the left window. It is within a “tiny” Mandelbrot set of period 35, which is close to a small Mandelbrot set of period 4. The latter is barely visible in the first image.
• Hit the key i repeatedly to zoom in smoothly. When the small Mandelbrot set of period 4 becomes prominent, hit r to mark it in the color magenta. Enter “4” or “0,4” in the dialog box.
• Zoom in further into the decorations of the small Mandelbrot set by hitting i repeatedly. You will see a thicker part on the decoration, which resembles the full Julia set in the right window. To mark this embedded Julia set in the parameter plane, hit r and enter “35,4” in the dialog box. Remember to increase the number of iterations appropriately with n.
• Zoom in further with the key i, until you see the tiny Mandelbrot set of period 35. Mark it with r by entering “35” or “0,35”.
• You may see analogous structures in the dynamic plane by hitting p at any time, or by zooming in there. To zoom or to mark them with r, you must switch between the planes by left-clicking once into the inactive window, or by hitting z or c, respectively. Restart by hitting h or zoom out with o.
• Now change the parameter c by left-clicking into the left window. Choose any small Mandelbrot set of primitive type. Determine its period p with the key x. Zoom in and look for embedded Julia sets in the decorations, just outside of the small Mandelbrot set, which is marked magenta with r. Zoom in further to the tiny Mandelbrot set and determine its period m with x. Zoom out again with o and mark the embedded Julia set with r by entering the values of “m,p” in the dialog box.

To do: implement x e p t 0 1 2 3 m.

Mathematical explanation of renormalization is under construction ...

Embedded Julia sets were observed and named in the 1990s by Robert Munafo and Jonathan Leavitt \cite{mrob99, leav}. These are subsets of $\M$ resembling a quadratic Julia set. See the examples in Figures~\ref{F10b} and~\ref{F3wd}. It turns out that each embedded Julia set is associated to two small Mandelbrot sets: first, it is contained in a decoration of a small Mandelbrot set $\M_p$ and it looks similar to the small Julia sets $\K_c^p$ for parameters $c$ nearby. Second, it is somewhat symmetric about another small Mandelbrot set $\M_m$\,, which shall be called the tiny Mandelbrot set in this context. The embedded Julia set is denoted by $\K_\sM^{m,\,p}\subset\M$\,; of course the periods $m$ and $p$ do not specify $\M_m$\,, $\M_p$\,, and $\K_\sM^{m,\,p}$ uniquely, but they must be supplemented with parameter values or external angles. Now what is the mechanism producing $\K_\sM^{m,\,p}$\,? In 2008 the author obtained the combinatorial construction and the asymptotic geometry at non-parabolic parameters, as well as the similarity results in Theorem~\ref{T3}, but this was published only in preliminary form in Demo Section~5 of Mandel \cite{mandel} and remained unknown; the discoveries of \cite{kk} are completely independent. Probably the combinatorial approach is simpler for quadratic polynomials and gives more classes of examples, while the analytic approach of \cite{DBDS, kk} will be more easily adapted to general one-parameter families of rational maps. See demo 5 of Mandel.