Currently investigating #ExternalRays in #EmbeddedJuliaSets in the #Mandelbrot set. Have rediscovered (I had forgotten) that the complex dynamics of a pre-periodic #Misiurewicz point under z→z²+c don't necessarily exactly match the dynamics of its rays under angle doubling (but there is a relationship).

For example: take an embedded Julia set with 5-fold hubs, with an influencing island of period 3. Then the main tips of the EJS where two rays land have period 3 (both point and rays), but the hubs with five spokes have period 3 in complex dynamics, but period 15 = 3 × 5 in ray dynamics.

What I'm doing now is tracing some of the rays to calculate their binary expansions, to see how they are made up from the blocks of digits from the influencing island and central island.

Later I want to see how the external angles are related when moving about inside the structure of an embedded Julia set in different ways.

Trying to figure out the formula for the (pre)periods of the principal #Misiurewicz points of embedded #Julia sets in the #Mandelbrot set.

Here's some context for the following toot: numbers are periods of hyperbolic components, fractions are rotation numbers of bond points, and in the next toot there will be some (preperiod)p(period) labels too.