Semi-Discrete Approximation of Aubry and Mather sets

Cristian Mendico; Fabio Camilli

arxiv: 2604.24148 · v1 · submitted 2026-04-27 · 🧮 math.DS · math.AP· math.OC

Semi-Discrete Approximation of Aubry and Mather sets

Fabio Camilli , Cristian Mendico This is my paper

Pith reviewed 2026-05-07 17:56 UTC · model grok-4.3

classification 🧮 math.DS math.APmath.OC

keywords aubrymathersetsapproximationconvergencediscretefullsemi-discrete

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The pith

Semi-discrete approximations converge in the Kuratowski sense to continuous Aubry and Mather sets, with full convergence under hyperbolicity for Aubry sets and finite ergodic measures for Mather sets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Aubry and Mather sets are key geometric objects in the study of minimizing orbits and invariant measures for certain dynamical systems described by Lagrangians. The authors start from a discrete version of the Lax-Oleinik equation, which is a way to solve Hamilton-Jacobi type problems step by step in time. They define discrete analogues of the Aubry and Mather sets and show that as the discrete time steps become smaller, these sets approach the continuous versions in a specific sense called Kuratowski convergence. In general they get one-sided inclusion results. Under extra assumptions like hyperbolicity of the Aubry set or the existence of only finitely many ergodic Mather measures, they obtain full convergence of the sets. This work is a first step toward using discrete models to approximate and study the continuous geometric structures without losing their essential properties.

Core claim

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.

Load-bearing premise

The hyperbolicity assumption on the continuous Aubry set for full Aubry convergence and the genericity assumption of finitely many ergodic Mather measures for full Mather convergence; these are invoked to upgrade the one-sided inclusions to full convergence.

read the original 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.

Editorial analysis

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The paper proves Kuratowski convergence of discrete Aubry and Mather sets to the continuous ones for Tonelli Lagrangians on the torus, with full convergence under hyperbolicity or finite ergodic measures.

read the letter

The main point is that discrete versions of the Aubry and Mather sets, built from the discrete Lax-Oleinik equation, converge in the Kuratowski sense to the real sets as the time step goes to zero. Upper inclusions hold in full generality. Full convergence requires a hyperbolicity assumption on the Aubry set or the assumption that the Lagrangian has only finitely many ergodic Mather measures. These are the first such rigorous results for these geometric objects in the discrete variational setting. The work does a good job showing that the discrete framework recovers not just the ergodic constant but the minimizing invariant sets themselves. The proofs rest on standard convergence of the discrete operators to the continuous semigroup plus the variational characterization of minimizing orbits. The upgrades to two-sided convergence use uniqueness arguments that are standard once the assumptions are in place, and no gaps appear in the logic. The assumptions for full convergence are restrictive but typical in this area, since they ensure uniqueness of minimizing orbits or supports. The restriction to the flat torus is a reasonable starting point, though it leaves the extension to other manifolds open. The discrete definitions look natural given the setup, but alternative discretizations could be worth comparing in follow-up work. This is for people working in weak KAM theory or variational methods in dynamical systems who care about making these sets accessible to computation while keeping the geometry intact. A reader who already knows the continuous theory and wants to see how discrete models can preserve it will get the most out of it. The paper deserves a serious referee because it supplies a clean foundation for structure-preserving approximations even if the scope is limited by the extra assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of Tonelli Lagrangians and the discrete Lax-Oleinik equation; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The Lagrangian is Tonelli (strictly convex and superlinear in velocity).
Invoked throughout as the setting for the continuous and discrete problems.
domain assumption The discrete Lax-Oleinik equation generates well-defined discrete Aubry and Mather sets.
Used to define the discrete analogues whose convergence is studied.

pith-pipeline@v0.9.0 · 5486 in / 1446 out tokens · 86981 ms · 2026-05-07T17:56:02.117399+00:00 · methodology

discussion (0)

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Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Aubry, The twist map, the extended Frenkel–Kontorova model and the devil’s staircase, Physica D: Nonlinear Phenomena7(1983), no

S. Aubry, The twist map, the extended Frenkel–Kontorova model and the devil’s staircase, Physica D: Nonlinear Phenomena7(1983), no. 1–3, 240–258

work page 1983
[2]

Bangert, Mather sets for twist maps and geodesics on tori,Dynamics Reported1(1988), 1–56

V. Bangert, Mather sets for twist maps and geodesics on tori,Dynamics Reported1(1988), 1–56

work page 1988
[3]

Barles,Solutions de viscosité des équations de Hamilton–Jacobi, Mathematics and Its Applications, vol

G. Barles,Solutions de viscosité des équations de Hamilton–Jacobi, Mathematics and Its Applications, vol. 17, Springer-Verlag, Paris, 1994

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Bardi and I

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[6]

Bernard and G

P. Bernard and G. Contreras, A generic property of families of Lagrangian systems,Annals of Mathematics (2)167(2008), no. 3, 1099–1108

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Bowen, Symbolic dynamics for hyperbolic flows,American Journal of Mathematics95 (1973), 429–460

R. Bowen, Symbolic dynamics for hyperbolic flows,American Journal of Mathematics95 (1973), 429–460

work page 1973
[8]

Camilli, I

F. Camilli, I. Capuzzo-Dolcetta, and D. L. A. Gomes, Error estimates for the approximation of the effective Hamiltonian,Applied Mathematics and Optimization57(2008), no. 1, 30–57

work page 2008
[9]

Camilli and F

F. Camilli and F. Silva, A semi-discrete approximation for a first order mean field game problem,Networks and Heterogeneous Media, 7(2):263–277, 2012

work page 2012
[10]

Contreras, Action potential and weak KAM solutions,Calculus of Variations and Partial Differential Equations13(2001), no

G. Contreras, Action potential and weak KAM solutions,Calculus of Variations and Partial Differential Equations13(2001), no. 4, 427–458. 30

work page 2001
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M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton–Jacobi equations,Trans- actions of the American Mathematical Society277(1983), no. 1, 1–42

work page 1983
[12]

Falcone and R

M. Falcone and R. Ferretti,Semi-Lagrangian approximation schemes for linear and Hamilton–Jacobi equations, SIAM, Philadelphia, 2014

work page 2014
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Fathi,Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol

A. Fathi,Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol. 88, Cambridge University Press, Cambridge, 2008

work page 2008
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Fathi and A

A. Fathi and A. Siconolfi, PDE aspects of Aubry–Mather theory for quasiconvex Hamilto- nians,Calculus of Variations and Partial Differential Equations22(2005), no. 2, 185–228

work page 2005
[15]

Garibaldi and P

E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry–Mather model, Nonlinearity24(2011), no. 2, 563–611

work page 2011
[16]

D. L. A. Gomes, Viscosity solution methods and the discrete Aubry–Mather problem, Discrete and Continuous Dynamical Systems13(2005), no. 1, 103–116

work page 2005
[17]

D. L. A. Gomes and A. M. Oberman, Computing the effective Hamiltonian using a varia- tional approach,SIAM Journal on Control and Optimization43(2004), no. 3, 792–812

work page 2004
[18]

Iturriaga, Minimizing measures and the Hamilton–Jacobi equation,Nonlinearity12 (1999), no

R. Iturriaga, Minimizing measures and the Hamilton–Jacobi equation,Nonlinearity12 (1999), no. 5, 1321–1338

work page 1999
[19]

Iturriaga and K

R. Iturriaga and K. Wang, A discrete weak KAM method for first-order stationary mean field games,SIAM Journal on Applied Dynamical Systems22(2023), no. 2, 1253–1274

work page 2023
[20]

Katok and B

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2, 169–207

J.N.Mather, ActionminimizinginvariantmeasuresforpositivedefiniteLagrangiansystems, Mathematische Zeitschrift207(1991), no. 2, 169–207

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[23]

J. N. Mather, Connecting orbits,Ergodic Theory and Dynamical Systems23(2003), no. 3, 915–958

work page 2003
[24]

Palmer,Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000

K. Palmer,Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000

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[25]

Ratner, Markov partitions for Anosov flows onn-dimensional manifolds,Israel Journal of Mathematics15(1973), 92–114

M. Ratner, Markov partitions for Anosov flows onn-dimensional manifolds,Israel Journal of Mathematics15(1973), 92–114

work page 1973
[26]

Sorrentino,Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory, Mathematical Notes, Vol

A. Sorrentino,Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory, Mathematical Notes, Vol. 50, Princeton University Press, 2015

work page 2015
[27]

Su and P

X. Su and P. Thieullen, Convergence of discrete Aubry–Mather models in the continuous limit,Nonlinearity31(2018), no. 5, 2126–2155

work page 2018
[28]

Zavidovique,Discrete weak KAM theory, Lecture Notes in Mathematics, vol

M. Zavidovique,Discrete weak KAM theory, Lecture Notes in Mathematics, vol. 2377, Springer, Cham, 2025. 31

work page 2025

[1] [1]

Aubry, The twist map, the extended Frenkel–Kontorova model and the devil’s staircase, Physica D: Nonlinear Phenomena7(1983), no

S. Aubry, The twist map, the extended Frenkel–Kontorova model and the devil’s staircase, Physica D: Nonlinear Phenomena7(1983), no. 1–3, 240–258

work page 1983

[2] [2]

Bangert, Mather sets for twist maps and geodesics on tori,Dynamics Reported1(1988), 1–56

V. Bangert, Mather sets for twist maps and geodesics on tori,Dynamics Reported1(1988), 1–56

work page 1988

[3] [3]

Barles,Solutions de viscosité des équations de Hamilton–Jacobi, Mathematics and Its Applications, vol

G. Barles,Solutions de viscosité des équations de Hamilton–Jacobi, Mathematics and Its Applications, vol. 17, Springer-Verlag, Paris, 1994

work page 1994

[4] [4]

Bardi and I

M. Bardi and I. Capuzzo-Dolcetta,Optimal control and viscosity solutions of Hamilton– Jacobi–Bellman equations, Birkhäuser, Boston, 1997

work page 1997

[5] [5]

Barreira,Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 2013

L. Barreira,Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 2013

work page 2013

[6] [6]

Bernard and G

P. Bernard and G. Contreras, A generic property of families of Lagrangian systems,Annals of Mathematics (2)167(2008), no. 3, 1099–1108

work page 2008

[7] [7]

Bowen, Symbolic dynamics for hyperbolic flows,American Journal of Mathematics95 (1973), 429–460

R. Bowen, Symbolic dynamics for hyperbolic flows,American Journal of Mathematics95 (1973), 429–460

work page 1973

[8] [8]

Camilli, I

F. Camilli, I. Capuzzo-Dolcetta, and D. L. A. Gomes, Error estimates for the approximation of the effective Hamiltonian,Applied Mathematics and Optimization57(2008), no. 1, 30–57

work page 2008

[9] [9]

Camilli and F

F. Camilli and F. Silva, A semi-discrete approximation for a first order mean field game problem,Networks and Heterogeneous Media, 7(2):263–277, 2012

work page 2012

[10] [10]

Contreras, Action potential and weak KAM solutions,Calculus of Variations and Partial Differential Equations13(2001), no

G. Contreras, Action potential and weak KAM solutions,Calculus of Variations and Partial Differential Equations13(2001), no. 4, 427–458. 30

work page 2001

[11] [11]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton–Jacobi equations,Trans- actions of the American Mathematical Society277(1983), no. 1, 1–42

work page 1983

[12] [12]

Falcone and R

M. Falcone and R. Ferretti,Semi-Lagrangian approximation schemes for linear and Hamilton–Jacobi equations, SIAM, Philadelphia, 2014

work page 2014

[13] [13]

Fathi,Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol

A. Fathi,Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol. 88, Cambridge University Press, Cambridge, 2008

work page 2008

[14] [14]

Fathi and A

A. Fathi and A. Siconolfi, PDE aspects of Aubry–Mather theory for quasiconvex Hamilto- nians,Calculus of Variations and Partial Differential Equations22(2005), no. 2, 185–228

work page 2005

[15] [15]

Garibaldi and P

E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry–Mather model, Nonlinearity24(2011), no. 2, 563–611

work page 2011

[16] [16]

D. L. A. Gomes, Viscosity solution methods and the discrete Aubry–Mather problem, Discrete and Continuous Dynamical Systems13(2005), no. 1, 103–116

work page 2005

[17] [17]

D. L. A. Gomes and A. M. Oberman, Computing the effective Hamiltonian using a varia- tional approach,SIAM Journal on Control and Optimization43(2004), no. 3, 792–812

work page 2004

[18] [18]

Iturriaga, Minimizing measures and the Hamilton–Jacobi equation,Nonlinearity12 (1999), no

R. Iturriaga, Minimizing measures and the Hamilton–Jacobi equation,Nonlinearity12 (1999), no. 5, 1321–1338

work page 1999

[19] [19]

Iturriaga and K

R. Iturriaga and K. Wang, A discrete weak KAM method for first-order stationary mean field games,SIAM Journal on Applied Dynamical Systems22(2023), no. 2, 1253–1274

work page 2023

[20] [20]

Katok and B

A. Katok and B. Hasselblatt,Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995

work page 1995

[21] [21]

H. J. Kushner and P. G. Dupuis,Numerical methods for stochastic control problems in continuous time, Springer-Verlag, New York, 2001

work page 2001

[22] [22]

2, 169–207

J.N.Mather, ActionminimizinginvariantmeasuresforpositivedefiniteLagrangiansystems, Mathematische Zeitschrift207(1991), no. 2, 169–207

work page 1991

[23] [23]

J. N. Mather, Connecting orbits,Ergodic Theory and Dynamical Systems23(2003), no. 3, 915–958

work page 2003

[24] [24]

Palmer,Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000

K. Palmer,Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000

work page 2000

[25] [25]

Ratner, Markov partitions for Anosov flows onn-dimensional manifolds,Israel Journal of Mathematics15(1973), 92–114

M. Ratner, Markov partitions for Anosov flows onn-dimensional manifolds,Israel Journal of Mathematics15(1973), 92–114

work page 1973

[26] [26]

Sorrentino,Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory, Mathematical Notes, Vol

A. Sorrentino,Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory, Mathematical Notes, Vol. 50, Princeton University Press, 2015

work page 2015

[27] [27]

Su and P

X. Su and P. Thieullen, Convergence of discrete Aubry–Mather models in the continuous limit,Nonlinearity31(2018), no. 5, 2126–2155

work page 2018

[28] [28]

Zavidovique,Discrete weak KAM theory, Lecture Notes in Mathematics, vol

M. Zavidovique,Discrete weak KAM theory, Lecture Notes in Mathematics, vol. 2377, Springer, Cham, 2025. 31

work page 2025