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?

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?

This paper "Local connectivity of Julia sets and bifuraction loci: three theorems of J.C.Yoccoz"  (link at Hubbard's page) 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 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  $0$, then 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?

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?

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 deoth by hand(see Remark 5.3). I wonder that for the 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, then the topology of depth-1 can be got by mapping each puzzle at depth-0 under $f^{-1}$, now I am a little confuse that how do we know the position of the perimage 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?

This paper "Local connectivity of Julia sets and bifuraction loci: three theorems of J.C.Yoccoz" (link at Hubbard's page) 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 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 $0$, then 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?