A multifractal analysis for escaping trajectories on free groups

We consider a free group extension of a subshift of finite type $\sigma :\mathrm{\Sigma }\to \mathrm{\Sigma }$, and consider three sets of points in $\mathrm{\Sigma }$ to which the corresponding trajectories on the free group escape to a given point in the Gromov boundary of the free group in three different senses. Under very mild conditions, we provide a common positive lower bound for the $u$-dimensions of these three sets. We also show that the lower bound is equal to the $u$-dimensions of these three sets when the extension and $u$ have certain symmetries. Moreover, we apply our results to geodesic flows on free group covers of Schottky surfaces, and show that there exists a dimension gap between the three sets we considered and the entire escaping set.