I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of the metric (so there is no conjugate points on the universal covering as well).

Suppose that there exists two geodesics on the covering wich are strongly asymptotic on the future (the distance between them goes to zero as $t\rightarrow\infty$). Is there any hope of obtainning estimatives for the distance between them on the past (as $t\rightarrow -\infty$)?

I see that on the hyperbolic plane if the geodesics get closer on the future, they deviate on the past. I'm wondering if there is some similar "deviating behavior" (even if it was not monotonically increasing) in abscense of conjugate points too.

Thanks on advance.