1.1: Rotation Numbers and Internal angles of the Mandelbrot bulbs

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Page ID: 101023

Robert L. Devaney

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"The Mandelbrot cactus" ("quadratic" parametrization).

The Mandelbrot set consists of many small decorations or bulbs (or limbs or atoms). A decoration directly attached to the main cardioid in M is called a primary bulb. This bulb in turn has infinitely many smaller bulbs attached. It is known that if c lies in the interior of a bulb, then the orbit of z_o=0 is attracted to a cycle of a period n. For the "quadratic" parametrization c = ¹/₄ - a²

z_n+1 = z_n² + ¹/₄ - a²

the main cardioid of the M-set turns into a circle with radius r = 1/2. A primary bulb attaches to the main circle at an internal angle

φ = 2 π ^m/_n

where ^m/_n is rotation number (e.g. ¹/₂ → 180^o, ¹/₃ → 120^o and ¹/₄ → 90^o).

1.1.1.jpg — Figure \(\PageIndex{1}\): -0.7500 0.0000 1.5100 440ms

1.1.2.png — Figure \(\PageIndex{2}\): 0.0500 0.0000 2.6700 72ms

1.1.3.jpg — Figure \(\PageIndex{2}\): 0.0500 0.0000 2.6700 72ms

1. One can count rotation number of a bulb by its periodic orbit star on dynamical plane. An attracting period n cycle z₁ → z₂ →...→ z_n → z₁ hops among z_i as f_c is iterated. If we observe this motion, the cycle jumps exactly m points in the counter-clockwise direction at each iteration. Another way to say this is the cycle rotates by a ^m/ _n revolution in the counter-clockwise direction under iteration.
2. The J_c-set contains infinitely many "junction points" at which n distinct black regions in J-set are attached, because c- value lies in a primary period n (3 or 5 for these images) bulb in the M-set. And the smallest black region is located m revolutions in the counter-clockwise direction from the largest central region.
3. The number of spokes in the largest antenna attached to a primary decoration of M-set is equivalent to the period of that decoration. And the shortest spoke is located m revolutions in the counter-clockwise direction from the main spoke ("C" parametrization here).

1.1.4.png — Figure \(\PageIndex{4}\): -0.1175 0.9000 0.5000 246ms

1.1.5.png — Figure \(\PageIndex{5}\): -0.5400 0.6100 0.2000 370ms

The rule follows from similarity of an antenna near a Mizurevich point and corresponding J-set (see The M and J-sets similarity).

1.1.6.jpg — Figure \(\PageIndex{6}\): -0.0500 0.0000 2.8000 315ms

1.1.7.png — Figure \(\PageIndex{7}\): -0.0500 0.0000 2.8000 341ms