  If frequent measurements ascertain whether a quantum system is still in its
initial state, transitions to other states are hindered and the quantum Zeno
effect takes place. However, in its broader formulation, the quantum Zeno
effect does not necessarily freeze everything. On the contrary, for frequent
projections onto a multidimensional subspace, the system can evolve away from
its initial state, although it remains in the subspace defined by the
measurement. The continuing time evolution within the projected "quantum Zeno
subspace" is called "quantum Zeno dynamics:" for instance, if the measurements
ascertain whether a quantum particle is in a given spatial region, the
evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian
with hard-wall (Dirichlet) boundary conditions. We discuss the physical and
mathematical aspects of this evolution, highlighting the open mathematical
problems. We then analyze some alternative strategies to obtain a Zeno dynamics
and show that they are physically equivalent.
