Aubry Set in Variational & Hamiltonian Dynamics

• Aubry set is an invariant set defined by action minimization, marking the threshold where calibrated or minimal trajectories emerge in various dynamical systems.
• It appears in diverse formulations including weak KAM theory, Hamilton–Jacobi equations, and lattice variational problems, establishing a uniqueness set for critical solutions.
• It serves as a bridge between larger minimizing ensembles and measure-theoretic objects, reinforcing rigidity phenomena and regularity in both conservative and dissipative models.

The Aubry set is an action-minimizing invariant set that appears in Aubry–Mather theory, weak KAM theory, lattice variational problems, Hamilton–Jacobi equations, and several dissipative or control-theoretic extensions. In the undriven Frenkel–Kontorova model, the Aubry set is the set of globally minimizing Birkhoff configurations; in Tonelli Hamiltonian dynamics, points of the Aubry set lie on globally minimizing orbits; in many weak KAM formulations it is characterized by a zero-diagonal condition such as $h(x,x)=0$ for the Peierls barrier or $S(x,x)=0$ for the Mañé potential (Slijepčević, 2013, Zhang, 2017, Cannarsa et al., 2022, Kajihara et al., 1 Apr 2025). Across these settings, the Aubry set typically sits between larger minimizing or semi-static sets and smaller measure-theoretic objects such as the Mather set, and it often serves as a uniqueness set for critical equations or as the locus where strict subsolutions fail to exist (Davini et al., 2012, Pozza, 2024).

1. Classical variational and dynamical formulations

In the classical Frenkel–Kontorova setting, configurations $u=(u_i)_{i\in\mathbb Z}$ minimize an energy

$$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$

with $V$ periodic and $W$ convex. The rotation number is

$\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$

when the limit exists, and ordered, or Birkhoff, configurations are those whose family of spatial and vertical translates is totally ordered. In the undriven case, the Aubry set is the set of globally minimizing Birkhoff configurations, while the Mather set is the union of supports of invariant probability measures supported on minimizing orbits of the associated monotone twist map of the cylinder (Slijepčević, 2013).

For Tonelli Hamiltonians on a compact manifold $M$, the Aubry set can be written as

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$

where $(u,w)$ ranges over conjugate weak KAM pairs, $S(x,x)=0$0, and

$S(x,x)=0$1

Points of $S(x,x)=0$2 lie on globally minimizing orbits of the Hamiltonian flow, each $S(x,x)=0$3 is invariant, and each $S(x,x)=0$4 is a Lipschitz Lagrangian graph. The Mather set $S(x,x)=0$5, defined as the union of supports of invariant probability measures minimizing $S(x,x)=0$6, satisfies $S(x,x)=0$7 (Zhang, 2017).

This classical distinction between Aubry and Mather sets remains important in later generalizations. A common misconception is that the two sets always coincide. The data show that equality holds in special situations, but not in general: the Aubry set may contain globally minimizing or static orbits beyond the supports of minimizing invariant measures (Zhang, 2017).

2. Weak KAM, Peierls barriers, and uniqueness mechanisms

A recurrent characterization of the Aubry set is through barrier-type functions. In the sub-Riemannian control setting, for

$S(x,x)=0$8

the projected Aubry set is

$S(x,x)=0$9

The same framework introduces dominated functions, calibrated curves, and critical solutions $u=(u_i)_{i\in\mathbb Z}$0 of

$u=(u_i)_{i\in\mathbb Z}$1

and proves that critical solutions are horizontally differentiable on $u=(u_i)_{i\in\mathbb Z}$2 and satisfy $u=(u_i)_{i\in\mathbb Z}$3 there in the classical sense (Cannarsa et al., 2022).

For weakly coupled Hamilton–Jacobi systems on $u=(u_i)_{i\in\mathbb Z}$4, the Aubry set is defined using a Mañé matrix

$u=(u_i)_{i\in\mathbb Z}$5

It is the set of points $u=(u_i)_{i\in\mathbb Z}$6 for which $u=(u_i)_{i\in\mathbb Z}$7 is a critical solution, independently of $u=(u_i)_{i\in\mathbb Z}$8. In this setting the Aubry set is characterized as the region where the obstruction to the existence of globally strict critical subsolutions concentrates. There exist critical subsolutions that are strict and $u=(u_i)_{i\in\mathbb Z}$9 on $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$0, and comparison on $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$1 yields a uniqueness principle for the critical system (Davini et al., 2012).

In the symbolic XY model, for a Lipschitz potential $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$2, the Mañé potential $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$3 and Peierls barrier $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$4 satisfy

$$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$5

For any $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$6, the map $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$7 is a Lipschitz calibrated subaction, and

$$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$8

defines an equivalence relation on $$E(u)=\sum_{i\in\mathbb Z}\big[W(u_{i+1}-u_i)+V(u_i)\big],$$9 (Kajihara et al., 1 Apr 2025).

On networks, the critical semidistance

$V$0

plays the same role. If $V$1 is prescribed on the Aubry set $V$2, then

$V$3

is the unique viscosity solution of the critical eikonal equation with trace $V$4 on $V$5. This makes $V$6 a uniqueness set in a precise weak KAM sense (Pozza, 2024).

3. Twist maps, lattices, and ordered configurations

In driven generalized elastic chains,

$V$7

with $V$8 and $V$9, the paper defines

$W$0

the union of supports of all $W$1-invariant probability measures. There is a continuous injective projection

$W$2

from $W$3 to $W$4, and for every rotation number $W$5 there exists an ergodic invariant measure supported on rotationally ordered configurations with rotation number $W$6. In the DC case, $W$7 consists entirely of equilibria and uniformly sliding solutions, while in both AC and DC cases $W$8 is a spatio-temporal attractor (Slijepčević, 2013).

The paper on ghost circles studies variational, monotone lattice recurrence relations and defines an Aubry–Mather set as a nonempty, closed, strictly ordered, shift-invariant family consisting of global minimizers. Its main addition is the ghost circle: a nonempty, closed, connected, strictly ordered, shift-invariant, gradient-flow invariant family that interpolates the Aubry–Mather set. Ghost circles classify gaps: if an Aubry–Mather set has a gap, then that gap is either foliated by minimizers or contains a stationary non-minimizing solution (Mramor et al., 2011).

For 2-locally constant twist potentials in the XY model, $W$9, the minimizing value is

$\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$0

and the minimizing periodic measures are exactly

$\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$1

In this case,

$\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$2

so the Aubry set is explicitly a union of fixed points determined by the diagonal minimizers of $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$3. The associated Mañé set is much larger: $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$4 and contains cubes of any finite dimension (Kajihara et al., 3 Oct 2025).

4. PDE, control, and graph-based generalizations

The weakly coupled Hamilton–Jacobi literature adds a second characterization of the Aubry set through cycles. In the random frame determined by the coupling matrix, the paper proves that a point $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$5 belongs to the Aubry set if and only if the infimum of the expected critical action over $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$6-cycles based at $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$7 is zero. This extends to systems a cycle condition already known in the scalar case (Ibrahim et al., 2016).

In sub-Riemannian control systems, the Aubry set is compact, nonempty, and invariant under the dynamics of calibrated minimizers. It is built from the Peierls barrier, the critical constant

$\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$8

and a class of $\rho(u)=\lim_{n\to\infty}\frac{u_n-u_0}{n},$9-closed measures satisfying

$M$0

This replaces Euler–Lagrange invariant measures in a control-affine setting where the classical Tonelli formalism is not available (Cannarsa et al., 2022).

On networks, the Aubry set

$M$1

consists of points incident to closed simple curves $M$2 with

$M$3

together with points $M$4 where

$M$5

It is nonempty, closed, partitioned into static classes, and any critical subsolution is a viscosity solution on $M$6. The comparison principle uses any closed $M$7 as a hidden boundary (Pozza, 2024).

These PDE and control variants show that the Aubry set is not tied to a single geometric model. What persists is the combination of a critical value, a minimizing or calibrated action, and a rigidity phenomenon concentrated on a distinguished invariant set.

5. Contact and dissipative Hamiltonian systems

For contact Hamiltonians $M$8 satisfying Tonelli conditions in $M$9 and

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$0

the stationary equation

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$1

has a unique backward weak KAM solution $\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$2 and a maximal forward weak KAM solution $\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$3. The Aubry set satisfies

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$4

and the projection

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$5

is a bi-Lipschitz homeomorphism. Barrier functions

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$6

decrease along calibrated curves, and

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$7

is the zero set of the barrier attached to $\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$8 (Wang et al., 2018).

When the monotonicity is only non-decreasing,

$\mathcal A(H)=\bigcap_{(u,w)}A_{u,w},$9

the theory acquires additional invariant sets. The projected Aubry set can depend on the choice of backward weak KAM solution, projected Aubry sets are partially ordered, and the strongly static set $(u,w)$0 can be strictly smaller than the Aubry set: $(u,w)$1 in explicit contact examples on $(u,w)$2 (Wang et al., 2021).

A later paper organizes the main invariant sets by

$(u,w)$3

where $(u,w)$4 is the Mather set, $(u,w)$5 the Mañé set, $(u,w)$6 the recurrent set, and $(u,w)$7 the non-wandering set. In the strictly increasing case, the strongly static set is used in the representation of minimal forward weak KAM solutions and in the construction of transitive orbits around the Aubry set (Ni et al., 2023).

For conformally symplectic systems, the discounted equation

$(u,w)$8

has a unique viscosity solution $(u,w)$9, and the Aubry set

$S(x,x)=0$00

is nonempty, compact, invariant, and supported on the graph of $S(x,x)=0$01. The Mather set is contained in the Aubry set. The Aubry set is the maximal invariant subset of the zero level of the Lyapunov-like function

$S(x,x)=0$02

but the paper also notes that the Aubry set need not be itself an attractor (Marò et al., 2016).

6. Geometry, measures, and approximation

In Tonelli Hamiltonian dynamics, the local geometry of the Aubry set is constrained by Green bundles. The paratingent cone $S(x,x)=0$03 is contained in the symplectic cone bounded by the two Green bundles: $S(x,x)=0$04 This sharpens earlier results of M.-C. Arnaud and links second-order weak KAM regularity to the tangent geometry of minimizing invariant sets (Zhang, 2017).

For geodesic flows of conformal metrics, generic finiteness of ergodic $S(x,x)=0$05-minimizing measures implies finiteness of the quotient Aubry set $S(x,x)=0$06 for every nontrivial cohomology class $S(x,x)=0$07. In that setting the quotient Aubry set is the set of static classes defined by

$S(x,x)=0$08

(Contreras et al., 2018).

The Aubry set also enters probabilistic asymptotics. For magnetic Lagrangians on a compact surface with nonpositive curvature, under the hypotheses that the Aubry set equals the Mather set and both are the unique closed geodesic $S(x,x)=0$09, the Peierls barrier satisfies

$S(x,x)=0$10

and this barrier controls the large deviations of stationary measures of twisted Brownian motions (Lopes et al., 2010).

In an infinite-dimensional mean-field setting for a version of the Vlasov equation on $S(x,x)=0$11, the Aubry set is again characterized by a Peierls-type barrier: $S(x,x)=0$12 The lifted Aubry set is invariant, contains the projected Mather set, and has a graph property over the quotient configuration space (Bessi, 2016).

Finally, semi-discrete approximation theory introduces discrete Aubry sets $S(x,x)=0$13 and proves upper Kuratowski limits

$S(x,x)=0$14

Under a hyperbolicity assumption on the continuous Aubry set and a ferromagnetic hypothesis on the Lagrangian, one has full Kuratowski convergence

$S(x,x)=0$15

This suggests that semi-discrete variational models can recover not only the ergodic constant but also the minimizing invariant geometry of the continuous dynamics (Camilli et al., 27 Apr 2026).

Across these formulations, the Aubry set remains the place where minimizing dynamics become rigid: it is the zero set of a barrier, the support of calibrated or static trajectories, a graph over the base in many Tonelli-type theories, and a uniqueness set for critical equations in weak KAM and Hamilton–Jacobi settings. The details vary sharply with the model—static versus driven, conservative versus dissipative, finite-dimensional versus lattice or mean-field—but the common structure is the same: a critical action level selects an invariant set on which minimization becomes exact.