  We propose a quantum system in which the winding number of rotations of a
particle around a ring can be monitored and emerges as a physical observable.
We explicitly analyze the situation when, as a result of the monitoring of the
winding number, the period of the orbital motion of the particle is extended to
$n>1$ full rotations, which leads to changes in the energy spectrum and in all
observable properties. In particular, we show that in this case, the usual
magnetic flux period $\Phi_0=h/q$ of the Aharonov-Bohm effect is reduced to
$\Phi_0/n$.
