# The Mandelbrot Set and Julia Sets

## Combinatorics in the Mandelbrot Set - Lavaurs Algorithm

 Lavaurs' algorithm is a method for deciding which pairs of rationals to connect, and the abstract Mandelbrot set is obtained by drawing arcs between these pairs and collapsing each arc to a point. If a conjecture (the local connectivity conjecture, which implies the hyperbolicity conjecture) is true, the abstract Mandelbrot set is topologically equivalent to the Mandelbrot set. Lavaurs' algorithm: (1) Connect 1/3 and 2/3, (2) Assuming all the numbers of period < k have been connected, connect those of period k, starting with the smallest number not yet connected, and connecting it to the next smallest number not yet connected, always making the choices so no connecting arc crosses any other arc. The picture illustrates Lavaurs' algorithm for features through period 5. Here the point of period 1 (0), the points of period 2 (1/3 and 2/3) and of period 3 (1/7, 2/7, 3/7, 4/7, 5/7, 6/7), along with some of period 4 and 5, are shown on the circle. The points of period 4 are those fractions with denominator 15, excluding 0, 5/15 = 1/3, and 10/15 = 2/3, fractions already in the list because they have period 1 (x = 0) or 2 (x = 1/3 and x = 2/3). (Note, a fixed point constitutes a 4-cycle, and the points of a 2-cycle are also points of a 4-cycle.) The points of period 5 are those fractions with denominator 31. All of these, except 0, are new points because 5 is prime.