  Equilibrium topological phases are robust against weak static disorder but
may break down in the strong disorder regime. Here we explore the stability of
the quench-induced emergent dynamical topology in the presence of dynamical
noise. We develop an analytic theory and show that for weak noise, the quantum
dynamics induced by quenching an initial trivial phase to Chern insulating
regime exhibits robust emergent topology on certain momentum subspaces called
band inversion surfaces (BISs). The dynamical topology is protected by the
minimal oscillation frequency over the BISs, mimicking a bulk gap of the
dynamical phase. Singularities emerge in the quench dynamics, with the minimal
oscillation frequency vanishing on the BISs if increasing noise to critical
strength, manifesting a dynamical topological transition, beyond which the
emergent topology breaks down. Two types of dynamical transitions are
predicted. Interestingly, we predict a sweet spot in the critical transition
when noise couples to all three spin components in the same strength, in which
case the dynamical topology survives at arbitrarily strong noise regime. This
work unveils novel features of the dynamical topology under dynamical noise,
which can be probed with control in experiment.
