  Coherent steering of a quantum state, induced by a sequence of weak
measurements, has become an active area of theoretical and experimental study.
For a closed steered trajectory, the underlying phase factors involve both
geometrical and dynamical terms. Furthermore, considering the reversal of the
order of the measurement sequence, such a phase comprises a symmetric and an
antisymmetric term. Superseding common wisdom, we show that the symmetric and
the antisymmetric components do not correspond to the dynamical and geometrical
parts respectively. Addressing a broad class of measurement protocols, we
further investigate the dependence of the induced phases on the measurement
parameters (e.g., the measurement strength). We find transitions between
different topologically distinct sectors, defined by integer-valued winding
numbers, and show that the transitions are accompanied by diverging dephasing.
We propose experimental protocols to observe these effects.
