On the rigidity of the stable norm and Mather's $ \beta $ -function for geodesic flows

We investigate rigidity phenomena associated to the stable norm and Mather's $ \beta $-function for Riemannian geodesic flows on closed manifolds. Given two metrics $ g_1 $ and $ g_2 $, we compare pointwise the corresponding $ \beta $-functions. Our main result establishes that they satisfy an inequality, which becomes an equality at an homology class $ h $ if and only if the two metrics are homothetic on the Mather set of homology $ h $. In the case of conformally equivalent metrics, this yields a pointwise criterion for homothety on the projected Mather set. Some consequences are discussed, including a pointwise rigidity result on the 2-torus implying that if a metric has the same Mather's $ \beta $-function at some non-zero homology class as a normalized flat metric in the same conformal class, then the metric must be flat. This result can be considered a pointwise version of a similar global result by Bangert. Finally, an extension of these results to Mañé's perturbations of general Tonelli Lagrangians is discussed.