$\alpha$-limits by reversing field

Question

I want to determine the $\alpha$-limit for all points of $\mathbb{R}^2$ for a given gradient field $$F(x,y)=-\nabla V$$
I think I know how to prove what the $\omega$-limits are, using arguments in which I trap the (positive time) orbits of the points in compact sets and argue that this forces the $\omega$-limit to be connected and non-empty. This, together with the fact that $\omega$-limit contains only singularities when dealing with gradient fields is usually enough to prove what the $\omega$-limits are.
For $\alpha$-limits, I don't have such powerful theorems to use: all the theorems I know are about the $\omega$-limit. I feel like I can just treat the case $F=\nabla V$ instead and argue that the $\omega$-limits of this case will be the $\alpha$-limits of our original problem.
Is this correct? Why exactly?

Answer 1

It is easy to check that if $\gamma(t)$ is a trajectory of $\dot{x} = v(x)$ then $\hat{\gamma}(t) = \gamma(-t)$ is a trajectory of $\dot{x} = -v(x)$. It is also easy to show that $\omega$-limit set of $\hat{\gamma}(t)$ concides with $\alpha$-limit set of $\gamma(t)$ -- just use the definition of $\omega$-limit set of $\hat{\gamma}(t)$ and use the same sequences and points for $\alpha$-limit set of $\gamma(t)$.