Local similarity between the Mandelbrot set and Julia sets
Get acquainted with the phenomenon of local similarity in an interactive session with the Java applet Juliette below. Click the gray column to give it keyboard focus (do not click the images). Use the following commands or the buttons below:
- The current parameter c, which is indicated by the yellow cross in the left window, belongs to a primitive small Mandelbrot set of period 16 within the Mandelbrot set M. The filled Julia set K_{c} in the right window contains a corresponding small Julia set. When it is rescaled and rotated appropriately, its decorations will look similar to those of the small Mandelbrot set. Hit the key z repeatedly to zoom into the parameter plane, and hit p to rescale the dynamic plane.
- You may either hit z many times before hitting p, or hit z and p alternately to zoom into both planes simulataneously. Hit v to zoom out again. (If you hit h to restart, you must use x afterwards before hitting p again.)
- When you have zoomed in far enough for the small Mandelbrot set to become prominent, you may change the parameter c by clicking into the left window, or move it with the arrow keys. Hit p to rescale the new dynamic plane. Although the small Julia set is changing, its rescaled decorations remain unchanged except for bending. (Try to hit p immeadiately after hitting the arrow key.)
- This kind of local similarity happens for many primitive small Mandelbrot sets, which are found everywhere in the antennae of M. First set the current parameter by clicking inside its cardioid, hit x and Enter to accept the period shown, which moves the parameter to the center. Afterwards you may hit p to rescale the dynamic plane according to local similarity at the specified center, zoom out with v and in again with z, or change the parameter before hitting p. Note that the period will be cleared when you hit h or use any command referring to asymptotic similarity: then p will use the latest Misiurewicz point for rescaling.
Download Juliette. See the
context menu for an overview of commands and keyboard shortcuts.
Or read this introduction.