Holographic complexity growth for a charged AdS-dilaton black holes with fixed and dynamical boundary respectively

The holographic complexity conjectures are considered in a Einstein-Maxwell-Dilaton gravity, by using the "Complexity-Volume" proposal. Specifically, we calculate the growth rate of complexity for an eternal charged AdS-dilaton black holes with fixed and dynamical boundaries respectively. The dynamical boundary is achieved by introducing a moving self-graviting brane on which the induced metric has an exact FLRW form. In case of fixed AdS boundary, there exists a bound for evolution of growth rate on late time, while this bound will become larger as the dilaton coupling constant $\alpha$ increases. In large $\alpha$ limit, we analytically prove that this bound is a finite value which is proportional to the black hole mass. In case of dynamical boundary, namely the brane-bulk system, the growth rate decreases monotonously on late time, after reaching a maximum value at a certain time. We find that the evolution of growth rate for brane-bulk system on late time is dominated by the velocity of the moving brane. We guess this result is model-independent.