Abstract: The study of iterated maps on the complex plane (such as the quadratic map z mapsto z^2 + c) has become very popular, partly because of the groundbreaking work of Douady and Hubbard in their `Orsay Notes'. Still there are many questions, even pertaining to the more basic notions introduced in the Orsay Notes, that are not completely answered.

The structure of the Julia set can be well-understood by means of a

finite tree (or dendrite), connecting the postcritical points: this is

the Hubbard tree. Hubbard trees encode all combinatorial information

of the Julia set, and it enables us to study this using kneading theory

(= symbolic dynamics).

In this talk I want to address the `reverse' question of constructing

the Hubbard tree starting from the kneading sequence. In other words, given a (pre)periodic 0-1-sequence $nu$, how can one construct a tree with a single critical point, and at most two-to-one dynamics, such that the itinerary of the critical point is exactly $nu$?

Interestingly, there exist such trees that don't correspond to any

quadratic polynomial, which leads us to the so-called complex

Admissibility Condition. Whereas `admissibility' for real kneading

sequences dates back to at least Milnor & Thurston monograph from the early 1980s, my joint work with Dierk Schleicher solves the question of `complex admissibility'.