On the regularity of Mather's β-function for standard-like twist maps

Abstract

We consider the minimal average action (Mather's β 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 β-function associated to a standard-like twist map admits a unique C1-holomorphic (canonical) complex extension, which coincides with this function on the set of real diophantine frequencies. In particular, we deduce a uniqueness result for Mather's β function.