On the rigidity of the stable norm and Mather's β-function for geodesic flows
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| Format: | Preprint |
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2025
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| _version_ | 1866915620485857280 |
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| author | Florio, Anna Leguil, Martin Sorrentino, Alfonso |
| author_facet | Florio, Anna Leguil, Martin Sorrentino, Alfonso |
| contents | We investigate rigidity phenomena associated to the stable norm and Mather's $β$-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 $β$-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 $β$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the rigidity of the stable norm and Mather's β-function for geodesic flows Florio, Anna Leguil, Martin Sorrentino, Alfonso Dynamical Systems We investigate rigidity phenomena associated to the stable norm and Mather's $β$-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 $β$-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 $β$-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. |
| title | On the rigidity of the stable norm and Mather's β-function for geodesic flows |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2511.12226 |