  Chaotic scattering is an important topic in nonlinear dynamics and chaos with
applications in several fields in physics and engineering. The study of this
phenomenon in relativistic systems has receivedlittle attention as compared to
the Newtonian case. Here, we focus our work on the study of some relevant
characteristics of the exit basin topology in the relativistic H\'enon-Heiles
system: the uncertainty dimension, the Wada property and the basin entropy. Our
main findings for the uncertainty dimension show two different behaviors
insofar we change the relativistic parameter $\beta$, in which a crossover
behavior is uncovered. This crossover point is related with the disappearance
of KAM islands in phase space that happens for velocity values above the
ultra-relativistic limit, $v>0.1c$. This result is supported by numerical
simulations and also by qualitative analysis, which are in good agreement. On
the other hand, the computation of the exit basins in the phase space suggests
the existence of Wada basins for a range of $\beta<0.625$. We also studied the
evolution of the exit basins in a quantitative manner by computing the basin
entropy, which shows a maximum value for $\beta \approx 0.2$. This last
quantity is related to the uncertainty in the prediction of the final fate of
the system. Finally, our work is relevant in galactic dynamics and it also has
important implications in other topics in physics as in the St\"ormer problem,
among others.
