  The parametric dependence of a quantum map under the influence of a rank-1
perturbation is investigated. While the Floquet operator of the map and its
spectrum have a common period with respect to the perturbation strength
$\lambda$, we show an example in which none of the quasienergies nor the
eigenvectors obey the same period: After a periodic increment of $\lambda$, the
quasienergy arrives at the nearest higher one, instead of the initial one,
exhibiting an anholonomy, which governs another anholonomy of the eigenvectors.
An application to quantum state manipulations is outlined.
