  We investigate the effect of conditional null measurements on a quantum
system and find a rich variety of behaviors. Specifically, quantum dynamics
with a time independent $H$ in a finite dimensional Hilbert space are
considered with repeated strong null measurements of a specified state. We
discuss four generic behaviors that emerge in these monitored systems. The
first arises in systems without symmetry, along with their associated
degeneracies in the energy spectrum, and hence in the absence of dark states as
well. In this case, a unique final state can be found which is determined by
the largest eigenvalue of the survival operator, the non-unitary operator
encoding both the unitary evolution between measurements and the measurement
itself. For a three-level system, this is similar to the well known shelving
effect. Secondly, for systems with built-in symmetry and correspondingly a
degenerate energy spectrum, the null measurements dynamically select the
degenerate energy levels, while the non-degenerate levels are effectively wiped
out. Thirdly, in the absence of dark states, and for specific choices of
parameters, two or more eigenvalues of the survival operator match in
magnitude, and this leads to an oscillatory behavior controlled by the
measurement rate and not solely by the energy levels. Finally, when the control
parameters are tuned, such that the eigenvalues of the survival operator all
coalesce to zero, one has exceptional points that corresponds to situations
that violate the null measurement condition, making the conditional measurement
process impossible.
