Statistical regularity and linear response of Mather measures for Tonelli Lagrangian systems

We study the statistical regularity of Mather measures associated with $C^1$ perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish Hölder continuity of the perturbed Mather measure with respect to the perturbation parameter. The Hölder exponent is shown to depend explicitly on the Diophantine index of the frequency. We also discuss the possibility of achieving Lipschitz regularity using KAM theory.