Why is local connectivity important for polynomial Julia sets?

Question

I'm trying to understand why local connectivity is important. I seem to remember a result that if the Julia set is locally connected then every external ray lands. I think this should mean we can get a complete description of the dynamics by considering the identification of $\mathbb{R}/\mathbb{Z}$ with the $\Sigma_2^+$ for quadratic maps, that is, the dynamics is semi-conjugate to the shift on two symbols, or conjugate to some quotient of the shift space. Is this true? Is there more to local connectivity than this?

Answer 1

Bingo. A theorem of Caratheodory states that if $K \subset \mathbb C$ is a full compact and locally connected set, then external rays land and the landing map $\ell: \mathbb R / \mathbb Z \to \partial K$ is continuous.

In the case of Julia set, this gives a continuous (finite to one) semi-conjugation between the dynamics on the Julia set and the full shift. Moreover, there are results on the points where more than one rays land, so essentially there is a good topological understanding of the Julia set and its dynamics.

As you probably know, the big question is the local connectedness of the Mandelbrot set. This would imply that we have a good topological understanding that set, and in turn would imply several other big conjectures, such as the conjecture that hyperbolic parameters are dense.