A criterion for loxodromicity in Gromov-hyperbolic spaces

Question

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other "repelling". This terminology is not completely standard but I am following Das–Simmons–Urbański in their book (available on arxiv), see Definition 6.1.2 there.

I am wondering whether there is a reference for the following criterion for an isometry $f$ of $X$ to be loxodromic : $f$ is loxodromic iff there is an open subset $U \subset \partial X$ such that $\overline {fU} \subsetneq U$. Moreover if this is the case then $f$ has an attracting fixed point in $fU$.

In the "classical" case ($X$ is proper and geodesic) this follows from the classification of isometries together with standard facts (parabolic and elliptic isometries preserve metrics on the boundary or boundary minus one point so this contracting behaviour is impossible). I think this should also work for the spaces which Das--Simmons--Urbański call "strongly hyperbolic" (Definition 3.3.6) where there are also nice metrics on the boundary (Observation 3.6.7 and Proposition 3.6.19). I am unsure about the general non-proper case.

The motivation to establish this criterion is to be able to deduce from the usual ping-pong argument the following more precise result : if $f, g$ are two loxodromic isometries of $X$ with disjoint fixed points sets in $\partial X$ and sufficiently large minimal translation then the subgroup $\langle f, g \rangle$ is freely generated by $f, g$, all its non-trivial elements are loxodromic and two elements either have disjoint fixed point sets or they have a common power. If there is a reference for this in full generality I would also be interested.

Answer 1

An answer was given in Ashot Minasyan's comment, i'm writing it here so the question does not remain unanswered.

In Lemma 2.5 of "Group actions on metric spaces: fixed points and free subgroups" by Mathias Hamann (on arxiv : https://arxiv.org/abs/1301.6513) he gives a criterion for an element of a group acting on a "contractive completion" to be hyperbolic (in a sense he defines for these actions). In the next section he proves that the Gromov bordification of a geodesic hyperbolic space is a contractive completion for any group of isometries, and his terminology in the previous section agrees with the usual one (so an element to which the ctiterion applies will be loxodromic in the terminology of my question).

The conditions in the criterion are a bit stronger that what i wrote in the question but they are sufficiently lax for the application to ping-pong and in fact Hamann does that in the paper (Theorem 2.7).