Starting from time evolution of wave function, quantum dynamics for a periodically kicked free top system is studied in this paper. For an initial spherical coherent state wave packet (localized) we find that 1) as the number of kicking is small, the speed and the direction of the diffusion for a time-evolving wave packet on a periodically kicked free top is related to the kicking strength: the stronger the kicking strength, the more chaotic for the diffusion (which means the more randomized in direction) is and the faster the speed of diffusion is, and then more quickly the full phase space is filled up; 2) as the kicking number is large, the time-evolving wave function will take on fine structure distribution in phase space, and the scope of the distribution for the fine structure will expand with the increase of the kicking strength, and the whole phase space will be filled up finally, and then the wave function will show multifractal property in phase space.#br#We study the multifractal behavior for a time-evolving wave function by partition function method: 1) for different kicking strengths and different q values, we study the scaling properties of partition function X(q), and find the power law relation between the partition function and the scaling L, i.e., X(q)-Lτ(q); 2) at different kicking strength, for a time-evolving wave function we calculate the singularity spectrum f(a)-a, and find that a maximum value of f(a) is 2.0 independent of the kicking strength, but the width of the singularity spectrum becomes narrow with the increase of the kicking strength, which means that the scope of the distribution for a is widest for regular state (localized), and is narrower for transition state from regular to chaotic, and is narrowest for chaotic state; 3) in the time-evolving process, the fluctuation for the width of the singular spectrum is smallest for chaotic state, intermediate for transition state of regular to chaotic, and the largest for regular state; 4) we calculate the generalized fractal dimension Dq-q for different kicking strengths, and find D0 = 2 independent of the kicking strength.#br#We study the mutifractal behaviors for the mean propbability amplitude distribution for a sequence of time-evolving wave functions and find that the result is similar to that of the single wave function type but has the difference: the width of the spectrum is reduced for each kicking strength.