Mather measures are probability measures supported on orbits of a Lagrangian system that minimize the action. They generalize KAM tori: when a KAM torus exists, the Mather measure sits on it, but Mather measures also exist when KAM tori have broken up. They are the dynamical system's way of remembering ordered motion even after the structures supporting it have been destroyed.
Perturb the Lagrangian and the Mather measure moves. The question is how smoothly: does a small perturbation produce a small change in the measure, or can the measure jump discontinuously?
For Mather measures on quasi-periodic tori with Diophantine frequency, the answer depends on the arithmetic of the frequency (arXiv:2603.11576). The perturbed measure is Hölder continuous in the perturbation parameter, and the Hölder exponent is determined by the Diophantine properties of the underlying frequency. Better Diophantine properties — frequencies that are harder to approximate by rationals — give better regularity. Worse Diophantine properties give weaker regularity.
The structural observation: the stability of a dynamical orbit under perturbation is controlled by a number-theoretic property of its frequency. The same orbit can be robustly or fragily stable depending on how rational its frequency is, and this arithmetic property is entirely extrinsic to the mechanics of the system. The dynamics doesn't know about number theory — but the number theory determines how the dynamics responds to perturbation. The arithmetic of the orbit's label controls the analysis of its persistence.