  Strong symmetries in open quantum systems lead to broken ergodicity and the
emergence of multiple degenerate steady states. From a quantum jump
(trajectory) perspective, the appearance of multiple steady states is related
to underlying dynamical phase transitions (DPTs) at the fluctuating level,
leading to a dynamical coexistence of different transport channels classified
by symmetry. In this paper, we investigate how strong symmetries affect both
the transport properties and the activity patterns of a particular class of
Markovian open quantum system, a three-qubit model under the action of a
magnetic field and in contact with a thermal bath. We find a pair of twin DPTs
in exciton current statistics, induced by the strong symmetry and related by
time reversibility, where a zero-current exchange-antisymmetric phase coexists
with a symmetric phase of negative exciton current. On the other hand, the
activity statistics exhibits a single DPT where the symmetric and antisymmetric
phases of different but nonzero activities dynamically coexists. The presence
of a strong symmetry under non-equilibrium conditions implies non-analyticities
in the dynamical free energy in the dual activity-current plane, including an
activity-driven current lockdown phase for activities below some critical
threshold. Finally, we also study the effect of a symmetry-breaking,
ergodicity-restoring dephasing channel on the coupled activity-current
statistics for this model. Interestingly, we observe that while this dephasing
noise destroys the symmetry-induced DPTs, the underlying topological symmetry
leaves a dynamical fingerprint in the form of intermittent, bursty on/off
dynamics between the different symmetry sectors.
