Paths of zeros of analytic functions describing finite quantum systems

Quantum systems with positions and momenta in Z(d), are described by the d zeros of analytic functions on a torus. The d paths of these zeros on the torus, describe the time evolution of the system. A semi-analytic method for the calculation of these paths of the zeros, is discussed. Detailed analysis of the paths for periodic systems, is presented. A periodic system which has the displacement operator to a real power t, as time evolution operator, is studied. Several numerical examples, which elucidate these ideas, are presented.