  Magnetic topological insulators (MTI) constitute a novel class of materials
where the topologically protected band structure coexists with long-range
ferromagnetic order, which can lead to the breaking of time-reversal symmetry
(TRS), introducing a bandgap in the Dirac cone-shaped topological surface state
(TSS). The gap opening in MITs has been predicted to be accompanied by a
distortion in the TSS, evolving its warped shape from hexagonal to trigonal. In
this work, we demonstrate such a transition by means of angle-resolved
photoemission spectroscopy after the deposition of low concentrations of
magnetic rare earths, namely Er and Dy, on the ternary three-dimensional
prototypical topological insulator Bi$_2$Se$_2$Te. Signatures of the gap
opening occurring as a consequence of the TRS breaking have also been observed,
whose existence is supported by the observation of the aforementioned
transition. Moreover, increasing the Er coverage results in a tunable p-type
doping of the TSS. As a consequence, the Fermi level (E$_{\textrm{F}}$) of our
Bi$_2$Se$_2$Te crystals can be gradually tuned towards the TSS Dirac point, and
therefore to the magnetically induced bandgap; thus fulfilling two of the
necessary prerequisites for the realization of the quantum anomalous Hall
effect (QAHE) in this system. The experimental results are rationalized by a
theoretical model where a magnetic Zeeman out-of-plane term is introduced in
the hamiltonian governing the TSS band dispersion. Our results offer new
strategies to control magnetic interactions with TSSs based on a simple
approach and open up viable routes for the realization of the QAHE.
