Volume preservation Liouville's Theorem , explanation of proof

Question

I am trying to understand the following proof of Liouville's Theorem , that states that trajectories generated by Hamiltonian equations are volume preserving.The proof can be found in the following link http://omega.albany.edu:8008/cdocs/summer99/lecture7/l7.pdf in page 4.

The goal is to prove that the volume

$\upsilon(0)$ at time $0$ is equal to the volume $\upsilon(t)$ at time $t$, after the use of the Hamiltonian dynamics.

So it starts straightforward with the calculation of

$\upsilon(t)=\int_{D_{t}}dq^{'}dp^{'}=\int_{D_{0}}det(I+tM)dqdp+O(t)$

where the following change of variable(Hamiltonian Dynamics) is used

$q^{'}=q+t\dot{q}+o(t)$

$p^{'}=p+t\dot{p}+o(t)$

With

$\dot{q}=\frac{dH}{dp}$ ,

$\dot{p}=-\frac{dH}{dq}$

and $H(q,p)=K(p,q)+V(q)$ is the Hamilotnian energy function with $K$ the kinetic energy and $V$ the potential energy. The matrix M is the part of the Jacobian with derivatives $\dot{q}$ and $\dot{p}$.

My questions are:

1. What's the interpretation of $O(t)$ and $o(t)$?
2. For region $D_{t}$ we made a transformation into a region $D_{0}$ , does this mean that we calculate the Hamiltonian trajectory backwards (because $t>0$)??