I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions.
2.5. Phase space for the flow.
The state of a moving particle at any time is specified by its position $q \in \mathcal{D}$ and velocity $v \in S^1$. Thus the phase is $\Omega = \mathcal{D} \times S^1$.
It then goes on to say that $\Omega$ is a doughnut with cross section of $\mathcal{D}$. This is intuitive, but what do phase space trajectories look like? Presumably if for every $\mathcal{D}$ we associate a velocity in $S^1$, then each transversal 'slice' of $\Omega$ will show line segments that are all oriented in the same direction. Is this the correct picture?
At each regular boundary point $q$ it is convenient to identify the pairs $(q, v^-)$ and $(q, v^+)$ related by the collision rule $v^+ = v^- - 2\langle v^-, n \rangle n$ ($v^\pm$ and $n$ are the incoming, outgoing and inward normal vectors at the moment of collision, respectively), which amounts to gluing $\Omega$ along its boundary.
What does it mean to glue $\Omega$ along its boundary? I understand that the motivation for doing this is to allow the billiard flow $\Phi^t$ to be smooth. But if that is the case, then my 'picture' of what a phase space trajectory should look like must be wrong: if we collapse $S^1$ into a half circle, making $\Omega$ a half doughnut, nothing really becomes 'smoother' - there are still discrete jumps between one 'slice' of the doughnut and another.
Finally, I become really confused when the collision map is introduced.
Given a flow $\Phi^t : \Omega \to \Omega$, one finds a hypersurface $M \in \Omega$ transversal to the flow so that each trajectory crosses $M$ infinitely many times.
Won't $M$ be always equal to $\mathcal{D}$, bar the singular points? Moreover, saying that the flow will pierce $M$ suggests to me that phase space trajectories are actually little curves moving 'through' the doughnut $\Omega$.
It goes on to redefine $\Omega$ as $$\Omega = \{ (x,s) : x \in M, 0 \le s \le L(x) \}$$
where $L(x) = \min \{ s > 0 : \Phi^s(x) \in M\}$ is a return time function and thus the collision map $F : M \to M $ is expressed as $F(x) = \Phi^{L(x)}(x)$. This new $\Omega$ makes the flow a suspension flow.
