# Topology of the Yoccoz puzzles at depth-$n$

This paper "Local connectivity of Julia sets and bifuraction loci: three theorems of J.C.Yoccoz" (see http://pi.math.cornell.edu/~hubbard/Yoccoz.pdf) tells that it is quite easy to construct the topology of a puzzle to a given depth by hand (see Remark 5.3). I wonder that for the \frac{1}{3}-limb, we know that there are three external rays landing at the repelling fixed points $$\alpha$$, then it is easy to know the knowledge at depth-$$0$$, and the topology of depth-$$1$$ can be got by mapping each puzzle at depth-$$0$$ under $$f^{-1}$$. Now I am a little confused that how do we know the position of the preimage of $$\alpha$$, i.e. how do we know which piece does $$f^{-1}(\alpha)$$ belong to? Furthermore, how do we know that the topology of a puzzle at depth-$$2$$, or depth-$$n$$? Are there any more details can I get from here?

• What is a puzzle in this context? – Vincent Sep 11 at 8:18
• It is a theory which can be used in study the local connectivity of Julia sets, you can find the details in pi.math.cornell.edu/~hubbard/Yoccoz.pdf – Yee Neil Sep 11 at 10:29

The "rabbit" (1/3-limb) puzzle has three pieces, let's call them 0 (the piece containing the critical point $$0$$), 1 (the piece containing the critical value $$c$$), and 2 (the remaining piece). Combinatorially speaking, 1 is mapped 1-1 to 2, 2 is mapped 1-1 to 0. (More precisely, 1 is mapped to a piece bounded by the same rays as 2, but with the equipotential somewhat further out.) 0 is mapped over the whole puzzle, covering 1 twice, and the remaining two pieces once.
In particular, the non-periodic preimage of the $$\alpha$$ fixed point must necessarily be in the $$0$$ sector. The next level of the puzzle will therefore have one piece contained in 1, one piece contained in 2, but 3 pieces contained in the original piece 0, touching at the preimage of the $$\alpha$$ fixed point.
You can continue inductively to work out the structure of the level $$n+1$$ pieces from that of the level $$n$$ pieces, assuming that you know the position of the critical value with respect to the latter. Equivalently, you should know how the critical value maps through the original pieces under iteration. (I am being slightly vague here, but trust you can work out the details.)
In particular, the structure of the puzzle at higher levels does depend on where you are in the Mandelbrot set - it will not be the same for all points in the 1/3-limb. Indeed, for non-renormalisable maps, Yoccoz's theorem shows that the structure of how the puzzle develops determines the parameter $$c$$ uniquely.