In the paper KNOTTED PERIODIC ORBITS IN DYNAMICAL SYSTEMS-I: LORENZ’S EQUATIONS Birman and Williams researched the periodic orbits of the Lorenz equations and more specifically what kind of structure they have. The first part of the paper is devoted to the proof that there exist an infinite amount of in-equivalent orbits. I think that is the content of the following theorem: The collapsing map $q$ is bijective on the union of all the periodic orbits.
I get the first part of the proof, that the strongly stable manifold $W^{ss}(x)$ ($x$ is a periodic point) does not intersect any other periodic orbits or itself. But then the proof talks about this process of collapsing and deformation. The collapsing has been touched upon in the section leading up to this theorem (bottom of page 52 of the paper), but I don't understand how we get a mapping from this process.
I have no clue how I should interpret this deformation ($H_s$) they talk talk about. It hasn't been mentioned before and I couldn't find it in the references for the section. Did I completely miss it?
Finally, I am having difficulties with how exactly the proof proves that it is a bijection. I feel that the proof is really hand-wavy and since I am not really familiar with the material this makes it quite difficult to correctly parse the proof, as there are so many details missing.
I would be really happy with a reference that has a little more detail. Most papers that I have found just refer to the original paper and offer no additional insights.
The "collapsing" in the statement of 2.3.1 is a quotient map $q : N \to B$ whose domain $N$ is a neighborhood of the attractor, so you'll need to brush up on quotient maps in a topology textbook such as Munkres "Topology".
In brief, as with any quotient map: $q$ is a surjective; the sets $q^{-1}(b)$ (for $b \in B$) form a decomposition of $N$ into disjoint subsets; and the topology on $B$ is the finest topology which makes the map $q$ continuous. And so as with any quotient map, the topological space $B$ is completely determined once you describe the decomposition of $N$. As described just before 2.3.1, the decomposition elements of $N$ are just the connected components of $W^{ss}(x) \cap N$ for points $x \in N$.
Take a look at Figure 1.2(b) of this paper by Tao Li to get a feel for what these quotient maps look like locally. I also recommend reading the earlier papers of Williams.
One more suggestion, take a look at the book of Ghrist, Holmes and Sullivan.