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On the rigidity of the stable norm and Mather's β-function for geodesic flows

Anna Florio, Martin Leguil, Alfonso Sorrentino

TL;DR

The paper studies rigidity phenomena for the stable norm and Mather’s β-function in geodesic flows and, more generally, Tonelli Lagrangians. It introduces a normalization-based distortion constant $C_{g_1}(g_2)$ and proves a sharp pointwise inequality $\beta_{g_2}(h) \le C_{g_1}(g_2)\,\beta_{g_1}(h)$ for all homology classes $h$, with equality signaling a strong alignment of the dynamical and geometric structures: Mather sets coincide (up to inclusion) and the metrics are homothetic on the corresponding Mather sets. In conformal cases, equality at a nonzero $h$ forces a pointwise homothety on the projected Mather set, yielding a pointwise analogue of Bangert’s global rigidity on $\mathbb{T}^2$ and a criterion for flatness in the presence of a normalized flat metric. The results extend to Mañé’s perturbations of Tonelli Lagrangians, showing analogous inequalities and rigidity statements, including a Liouville-integrable torus example where equality forces the perturbation to vanish, thereby connecting pointwise local data to global geometric/dynamical structure.

Abstract

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.

On the rigidity of the stable norm and Mather's β-function for geodesic flows

TL;DR

The paper studies rigidity phenomena for the stable norm and Mather’s β-function in geodesic flows and, more generally, Tonelli Lagrangians. It introduces a normalization-based distortion constant and proves a sharp pointwise inequality for all homology classes , with equality signaling a strong alignment of the dynamical and geometric structures: Mather sets coincide (up to inclusion) and the metrics are homothetic on the corresponding Mather sets. In conformal cases, equality at a nonzero forces a pointwise homothety on the projected Mather set, yielding a pointwise analogue of Bangert’s global rigidity on and a criterion for flatness in the presence of a normalized flat metric. The results extend to Mañé’s perturbations of Tonelli Lagrangians, showing analogous inequalities and rigidity statements, including a Liouville-integrable torus example where equality forces the perturbation to vanish, thereby connecting pointwise local data to global geometric/dynamical structure.

Abstract

We investigate rigidity phenomena associated to the stable norm and Mather's -function for Riemannian geodesic flows on closed manifolds. Given two metrics and , we compare these objects pointwise at individual homology classes. Our main result establishes that if Mather's -function (or the stable norm) of at a non-zero homology class h equals that of at multiplied by a suitable factor determined by the metrics, then the two metrics are homothetic on the Mather set of homology h associated to . 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 -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.

Paper Structure

This paper contains 3 sections, 5 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Main results for Riemannian geodesic flows
  3. Mañé's perturbations of Tonelli Lagrangians

Key Result

Theorem 1

When $M={ \mathbb T}^2$, $\beta_g$ is differentiable at a $1$-irrational homology class if and only if the Mather set $\widetilde{\mathcal{M}}_{g}^h$ consists of an invariant torus foliated by periodic orbits.

Theorems & Definitions (12)

  • Theorem 1: Bangert
  • Theorem 2: Bangert
  • Remark 3
  • Remark 4
  • Remark 5
  • Corollary 6
  • Remark 7
  • Remark 8
  • Corollary 9
  • proof : Proof of Main Theorem.
  • ...and 2 more