When are phase space flows induced by the Hamilton equations homeomorphisms? Suppose I have a certain Hamiltonian of a system, with the corresponding Hamilton equations. The equations induce a certain flow in phase space. Since each point in space has a certain trajectory determined by the flow, for each point in time $t$, I can think of the function $f(t)$, which sends a point to its location (determined by the flow) at time $t$.
Given some set $A$ in phase space, I want to learn about the possible topological properties of $A(t)=f(t)(A)$. Since the solutions of the Hamilton equations are continuous, I know that at least some properties of $A$ must be preserved over time (compactness, connectedness...).
Moreover, assume that the Hamiltonian is 'nice' enough, so that trajectories in phase space don't cross each other (so $f_{|A}(t)$ is a continuous and injective map onto $A(t)$). Under what conditions is $A(t)$ a homeomorphism?
It sounds like it would always happen for a system for which time is "reversible", but could anyone give a more precise explanation of when this would happen? Also, if you have any more examples of interesting topological properties that are preserved over time by these flows, I'd be happy to hear (for instance - if $A$ is path connected, is the fundamental group of $A$ preserved?). If there are any interesting results that are not limited to just the Hamilton equations - also feel free to share them.
I have almost no knowledge in the subject, so I'm sorry if the question is obvious, or not well described. Feel free to suggest me of good books/articles on the subject (in a more introductory level).
Thank you.
 A: So it turns out that the answer for the case I presented is very simple - the phase flow of a set is always a homeomorphism (for any point in time). We already know that it is continuous, injective and surjective, so all we need to show that its inverse is also continuous.
It turns out that it is not only continuous - it is also a solution for the Hamilton equations! Assume some flow generated by a solution to the Hamilton equations (I'll assume 2-dimensional phase space, although the proof for higher dimensions is the same). We can express the inverse function of the flow by reversing the direction of time; We make the substitution - $$t\mapsto -t,   p\mapsto -p$$
So define - $$Q(t)=q(-t)\Rightarrow \dot Q=- \dot q$$ $$P(t)=p(-t)\Rightarrow \dot P=- \dot p$$ 
We thus get:
$$\dot Q=- \dot q = - \frac{\partial H}{\partial p}=\frac{\partial H}{\partial P}$$
$$\dot P=- \dot p =  \frac{\partial H}{\partial q}=- \frac{\partial H}{\partial Q}$$
And so the inverse of the flow is indeed a solution for the Hamilton equations, which naturally means it is continuous. This means that the flow is indeed a homeomorphism, and so pretty much any topological property of a set $A$ in phase space is preserved over time.
If anyone is interested, the motivation for this question came from of Liouville's theorem, which states (in some sense) that the volume of a set in phase space is conserved over time. This made me wonder if there are any interesting topological (and not geometric) properties that have to be conserved over time - turns out that pretty much all of them.
