How close can closed geodesics be? A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such that for any such manifold $N$ and any two distinct closed geodesics loops $c_1,c_2$ on $M$ both of length less than $R$ we have 
$$
d_H(c_1,c_2) \ge c \quad (\ast)
$$
where $d_H$ is Hausdorff distance. 
This is false for general locally symmetric spaces, since the higher-rank ones can contain flat tori. My question is the following: if we fix a real-rank one symmetric space $X$ (i.e. $X$ is a real, complex or quaternionic hyperbolic space, or octonionic hyperbolic plane) is there a c>0 (depending on the space) such that $(\ast)$ holds for all quotients of $X$ and all pairs of distinct closed geodesics of bounded length on such? (More generally one could ask the same question for Riemannian $n$-manifolds with sectional curvature between $-1$ and $\kappa<0$, with a $c$ depending on $\kappa$ and $n$ and the bound on the length). 
Edit: As noted in Anton's comments the opening remark is false if one does not bound the length of the geodesics; I edited to add that.  
 A: I think such a bound exists (depending only on pinched curvature constant $\kappa$, dimension $n$, and $R$). Suppose one has an infinite sequence of pinched negatively curved manifolds where geodesics of length $\leq R$ have Hausdorff distance approaching $0$. By the generalized Margulis Lemma (see Ballmann-Schroeder), there is an $\epsilon$ so that in a $\kappa$-pinched $n$-dim. manifold, $\epsilon$-thin regions are tubular neighborhoods about short geodesics of radius $>R$. Thus, we may assume that the curves pass through the thick part of the manifold, otherwise they must be homotopic to short geodesics in the thin part, in which case it follows. One may take a pointed Cheeeger-Gromov limit (with basepoint on either curve in the thick part), to get a limit in which the Hausdorff distance between the two curves is zero. This gives a contradiction, since these curves are homotopic in the limit, but this implies they will be homotopic in the approximates.  
A: This is really a question about the fundamental group, since two such geodesics necessarily belong to distinct free homotopy classes of loops. One immediate answer is given by the injectivity radius of the space.  The distance between the loops must be at least the injectivity radius, since otherwise one can be homotoped into the other.
