Statistical regularity and linear response of Mather measures for Tonelli Lagrangian systems
Abstract
We study the statistical regularity of Mather measures associated with C1 perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish H\"older continuity of the perturbed Mather measure with respect to the perturbation parameter. The H\"older 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.
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