Semi-Discrete Approximation of Aubry and Mather sets

Abstract: We study the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians on the flat torus. Starting from the discrete Lax--Oleinik equation, we introduce natural discrete analogues of these sets and analyze their convergence, as the time step tends to zero, in the sense of Kuratowski. Our results show that the semi-discrete variational framework captures not only the ergodic constant, but also the minimizing invariant geometry of the continuous dynamics. In full generality, we prove upper Kuratowski limit inclusions for both the Aubry and Mather sets. For the Aubry set, we establish full convergence under a hyperbolicity assumption on the continuous Aubry set. For the Mather set, we prove full convergence under a genericity assumption ensuring that the Lagrangian admits finitely many ergodic Mather measures. This provides a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets in the Tonelli setting, and clarifies how discrete variational models recover the central geometric objects of weak KAM and Aubry--Mather theory.

• The paper introduces a semi-discrete variational model that accurately approximates Aubry-Mather sets by preserving calibrated trajectories and holonomic probability measures.
• The paper applies Kuratowski limits to prove convergence of the discrete approximations for both Aubry and Mather sets under hyperbolicity and genericity assumptions.
• The paper's framework overcomes traditional discretization challenges by retaining key geometric invariants fundamental to Tonelli Lagrangian dynamics.

Semi-Discrete Approximation of Aubry and Mather Sets: Analysis and Convergence

Background and Motivation

The study addresses the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians defined on the flat torus. These sets, central to weak KAM theory and Aubry-Mather theory, encode the geometry of globally minimizing dynamics in stationary Hamilton-Jacobi equations. The classical Aubry set is structured in terms of calibrated curves for critical viscosity solutions, while the Mather set comprises supports of action-minimizing invariant probability measures. Both sets exhibit strong invariance properties and play a vital role in ergodic and variational dynamical analysis.

Numerical approximation of these sets has been historically difficult due to the lack of regularity—in particular, viscosity solutions typically develop singularities and the minimization structure is highly sensitive to discretization errors. Standard PDE-based schemes often fail to capture correct invariant geometry, especially for critical (ergodic) regimes. There is a need for discretizations preserving the variational skeleton and minimizing invariants intrinsic to Aubry-Mather theory.

Semi-Discrete Variational Framework

The paper leverages a semi-discrete approach that originates from the time discretization of the Lax-Oleinik semigroup. This framework replaces continuous trajectories with discrete configurations (sequences of points), while substituting the action integral with a discrete sum. Crucially, the resulting discrete variational model retains meaningful notions of calibrated trajectories and holonomic probability measures, mirroring continuous weak KAM structures.

For fixed time step $\tau > 0$, the authors construct discrete analogues of the Aubry and Mather sets, denoted $\widetilde{\mathcal{A}}^\tau_L$ and $\widetilde{\mathcal{M}}^\tau_L$. The discrete Lax-Oleinik equation admits unique solutions for each $\tau$, and the discrete Aubry set admits an analogue of global calibration via bi-infinite sequences. Discrete holonomic measures are defined in a manner parallel to the continuous closed measure concept, and discrete Mather minimizers are established as action-minimizing holonomic probability measures.

The authors rigorously prove the inclusion $$\widetilde{\mathcal{M}}^\tau_L \subset \widetilde{\mathcal{A}}^\tau_L$$ for all $\tau>0$, conserving the continuous hierarchy. Compactness and non-emptiness are established for both sets.

Convergence Analysis via Kuratowski Limits

The centerpiece of the work is the analysis of convergence of the discrete sets to their continuous counterparts as $\tau\to0$, employing Kuratowski limits.

For Aubry sets:

Upper Kuratowski limit inclusion $$\limsup_{\tau\to0}\widetilde{\mathcal{A}}^\tau_L \subset \widetilde{\mathcal{A}}_L$$ is derived via asymptotic analysis of calibrated configurations. Lower Kuratowski limit convergence is proven under a hyperbolicity assumption on the continuous Aubry set, with shadowing arguments for pseudo-orbits in the discrete dynamics. Ferromagnetic nature assumptions on the Lagrangian ensure the discrete Euler-Lagrange map is a smooth perturbation of the continuous flow, crucial for applying hyperbolic shadowing.

For Mather sets:

Upper Kuratowski limit inclusion $$\limsup_{\tau\to0}\widetilde{\mathcal{M}}^\tau_L \subset \widetilde{\mathcal{M}}_L$$ is shown using tightness and weak convergence arguments on lifted discrete measures. Lower Kuratowski limit convergence is achieved under a genericity criterion (finite ergodic Mather measures), using viscosity selection techniques and penalized variational minimization.

The main results can be succinctly stated as: $$\limsup_{\tau\to0}\widetilde{\mathcal{A}}^\tau_L \subset \widetilde{\mathcal{A}}_L, \qquad \limsup_{\tau\to0}\widetilde{\mathcal{M}}^\tau_L \subset \widetilde{\mathcal{M}}_L$$ with full Kuratowski convergence (equality of liminf and limsup) under the stated structural assumptions.

Numerical and Structural Implications

The paper provides a rigorous structure-preserving foundation for discretizations of Aubry-Mather sets. Unlike conventional schemes, semi-discrete variational models are shown to capture not only the ergodic constant, but also the underlying minimizing invariant sets in the Tonelli class. The results clarify that capturing the value function (ergodic constant) is insufficient; the critical challenge is to recover the minimizing sets that represent the skeleton of global dynamics.

From a computational perspective, these findings open the path for robust and faithful numerical algorithms for invariant sets in Hamiltonian dynamics. The semi-discrete framework, with its proven convergence properties, stands as a blueprint for further discretizations—possibly in fully discrete space-time models—and for computational approximation of singular geometric objects in variational dynamical systems.

Theoretical Implications and Future Directions

On the theoretical side, the work provides new links between discrete weak KAM theory and the geometry of minimizing sets for Tonelli Lagrangians. It extends the domain of variational approximation theory to non-smooth structures and solidifies the connection between discrete calibrated configurations, holonomic measures, and the weak KAM framework.

The convergence proofs rely on sophisticated dynamical systems arguments, including hyperbolicity, shadowing lemmas, and viscosity selection principles. The bridge between discrete and continuous geometric invariants is established via detailed measure-theoretic and variational constructions.

Future directions include relaxing structural assumptions (e.g., exploring non-hyperbolic Aubry sets or Lagrangians with infinitely many ergodic Mather measures), extension to more general manifolds, and the development of efficient computational algorithms based on these semi-discrete models. The structural robustness of discrete variational schemes suggests applicability to multidimensional and stochastic extensions.

Conclusion

The paper "Semi-Discrete Approximation of Aubry and Mather sets" (2604.24148) rigorously establishes a semi-discrete variational framework for the approximation of Aubry and Mather sets in Tonelli Lagrangian dynamics. Convergence results are proven in the sense of Kuratowski limits, both for the Aubry set (under hyperbolicity) and the Mather set (under genericity). The discrete model preserves the essential invariants and geometric structures of the continuous theory, thereby providing a foundation for future numerical and theoretical developments in weak KAM and Aubry-Mather analysis.