The parametric dependence of the eigenvalues and the eigenvectors of a Floquet operator that describes a unit step time evolution of quantum map under the influence of a rank-1 perturbation, is shown. As a function of the strength of the perturbation, the perturbed Floquet operator and its spectrum are shown to have a period. Contrary to this, I will explain examples that each eigenvalue doesn't obey the same periodicity but exhibits Cheon's eigenvalue anholonomy [T. Cheon, PLA 248, p.285 (1998)]. Accordingly each eigenvector exhibits an anholonomy in its direction. Note that this is completely different from the phase anholonomy known as geometric phases. A geometrical argument shows that Cheon's anholonomy are abundant in systems whose time evolutions are described by Floquet operators, including kicked rotors as well as periodically driven systems. If time permits, an application to quantum state manipulations will be discussed. This talk is based on collaboration with Dr. M. Miyamoto (Waseda Univ.).