On the regularity of Mathers $beta$-function for standard-like twist maps

We consider the minimal average action (Mathers $beta$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $beta$-function associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.