AC conductivity sigma(omega) of single crystals of Ag beta-alumina and Na beta-alumina has been studied in the temperature range from approximately 100 K to room temperature. The DC regime of sigma(omega), which had a close relation to the ion dynamics for the long-range diffusion, was found to have the small power-law dependence sigma(omega) proportional to omega(n) with frequency exponent n = 0.11-0.15, not sigma(omega) proportional to omega(0) = const. In higher frequencies, the sigma(omega) was monotonically increasing with Log-frequency and was put into Jonscher's universal law (sigma(omega) proportional to omega(0.6)). The beta-alumina is an ideal two-dimensional super ionic conductor. Its low-dimensionality and the scaling theory for a random walk enable us to understand the small power-law dependence of ionic conductivity. We suggested that a "scale-invariance" would hold true behind the super ionic conduction in the self-similarity point of view. (C) 2015 Elsevier B.V. All rights reserved.