Quadratic Hubbard trees, automata, and ray connections

Abstract

We describe an algorithm that, given a postcritically finite quadratic polynomial, outputs a finite state machine whose regular language is the set of external arguments for all points on the Hubbard tree, generalizing an example of Milnor. We do this by extracting orbit portraits from angled internal addresses, and constructing a suitable Markov partition of angle intervals. A main application is computing direct ray connections between Hubbard trees in matings of quadratic polynomials. We use this to understand ray connections in quadratic matings along veins of the Mandelbrot set, and prove that first ray connections between Hubbard trees are picked up at parabolic parameters or tips of veins.