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 these objects pointwise at individual homology classes. Our main result establishes that if Mather's $\beta$-function (or the stable norm) of $g_2$ at a non-zero homology class h equals that of $g_1$ at $h$ multiplied by a suitable factor determined by the metrics, then the two metrics are homothetic on the Mather set of homology h associated to $g_1$. 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 Ma\~n\'e's perturbations of general Tonelli Lagrangians is discussed.