Singularity of dynamical maps

For a dynamical map Lambda(t,0), which sends a state rho(0) of a quantum open system to a state rho(t) = Lambda(t, 0)rho(0), the decomposition law Lambda(t,0) = Lambda(t, t(c))Lambda(t(c),0) may break down at a specific time tc. In this paper, we present a method to find the singular points tc and propose a measure for the singularity of the dynamical map. Two examples are portrayed to illustrate the method, and the measure of singularity for these singular points is calculated and discussed. An extension to a high-dimensional system is presented.