Mather Set in Aubry–Mather Theory

• Mather set is defined as the closure of supports of action-minimizing measures, forming the measure-theoretic core of minimizing dynamics.
• It typically lies within the Aubry set and exhibits properties such as invariance, compactness, and, under convexity, a graph structure.
• Extensions of this concept apply to contact, non-convex, and discrete systems, preserving the principle of support-of-extremal-measures.

Searching arXiv for relevant papers on Mather sets and related Aubry–Mather theory. The Mather set is the invariant geometric locus determined by action-minimizing invariant probability measures in Aubry–Mather theory and its extensions. In the classical Tonelli setting, for a Lagrangian $L$ on a compact manifold, a Mather measure minimizes the averaged action among appropriate closed or invariant probability measures, and the Mather set is the closure or union of the supports of all such minimizing measures. Across the literature, this object functions as the measure-theoretic core of minimizing dynamics, typically lying inside the Aubry set and often inheriting compactness, invariance, and graph-type properties. The notion has been generalized to contact Hamiltonian systems, conformally symplectic flows, graph-based dynamics, driven lattice systems, non-convex Hamiltonians, and even symbolic matrix cocycles, where the common structural theme is that the Mather set organizes the extremal invariant dynamics selected by a variational or subadditive principle (Camilli et al., 27 Apr 2026, Ni et al., 2023, Marò et al., 2016, Siconolfi et al., 2021, Siconolfi et al., 2021, Morris, 2011).

1. Classical variational definition and basic meaning

In the Tonelli framework on the flat torus, a probability measure $\mu$ on $T^d\times \mathbb{R}^d$ is called a Mather measure if it is closed, meaning

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$

and if it minimizes the action among all closed probability measures: $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$ The minimal value is

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$

The Mather set is then defined as

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$

In this formulation, points of the Mather set lie on globally minimizing trajectories (Camilli et al., 27 Apr 2026).

A related formulation for Tonelli Lagrangians on compact manifolds is the minimization over invariant measures. Let $I(L)$ denote the compactly supported Borel probability measures on $TM$ invariant under the Euler–Lagrange flow. A Mather measure is a measure $\mu\in I(L)$ minimizing

$\mu$0

For a perturbation by a closed $\mu$1-form $\mu$2, one considers $\mu$3, meaning $\mu$4, and its minimizing measures. In this perspective, Mather measures are the minimizers of a “universal” infinite-dimensional linear programming problem; more precisely, Mather measures of a Tonelli Lagrangian are precisely the measures minimizing $\mu$5 among all compactly supported closed measures (Bernard, 2010).

This measure-theoretic definition explains the standard role of the Mather set. It is not merely an invariant set of orbits, but the union of supports of those invariant or closed measures that realize the least possible average action. A plausible implication is that the set retains only the recurrent or statistically persistent part of globally minimizing dynamics, discarding minimizing curves that do not carry invariant measures.

2. Position inside Aubry–Mather and weak KAM structures

A persistent structural statement is that the Mather set sits inside the Aubry set. In the Tonelli semi-discrete approximation paper, the continuous Mather set satisfies

$\mu$6

and the discrete analogue satisfies

$\mu$7

The discrete proof uses the calibration defect

$\mu$8

showing that minimizing discrete holonomic measures are supported where $\mu$9, hence on calibrated edges (Camilli et al., 27 Apr 2026).

In conformally symplectic systems, the Mather set is defined by

$T^d\times \mathbb{R}^d$0

where an invariant probability measure is minimizing if

$T^d\times \mathbb{R}^d$1

with

$T^d\times \mathbb{R}^d$2

The paper proves

$T^d\times \mathbb{R}^d$3

because equality in

$T^d\times \mathbb{R}^d$4

holds iff $T^d\times \mathbb{R}^d$5 (Marò et al., 2016).

In contact Hamiltonian systems, the same inclusion pattern appears: $T^d\times \mathbb{R}^d$6 where $T^d\times \mathbb{R}^d$7 is the Mañé set and $T^d\times \mathbb{R}^d$8 is the Aubry set (Siconolfi et al., 2021). A later paper in the same program refines this to

$T^d\times \mathbb{R}^d$9

placing the Mather set inside the Mané set and the closure of recurrent points, and below a new intermediate invariant set, the strongly static set (Ni et al., 2023).

These inclusion chains support a common interpretation. The Aubry set describes globally calibrated or static dynamics, the Mañé set captures a larger semi-static structure, and the Mather set is the most measure-theoretically rigid part among them. This suggests that the Mather set is best viewed as the recurrent minimizing skeleton inside weak KAM theory rather than as a generic minimizing relation.

3. Structural properties: invariance, graph phenomena, recurrence, and support

Several papers record standard structural properties of the Mather set: non-emptiness, compactness, invariance, and graph-type behavior.

In conformally symplectic systems, the Mather set is non-empty, compact, invariant under $$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$0, and satisfies a graph theorem: the projection

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$1

is injective, hence a bi-Lipschitz homeomorphism onto its image. Equivalently,

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$2

The set also lies in the maximal attractor

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$3

via

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$4

The Lyapunov-type quantity

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$5

satisfies

$$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$6

which explains why the Mather set belongs to the asymptotic core of the dissipative dynamics (Marò et al., 2016).

On graphs, the Mather set has a combinatorial graph property: the restriction of the projection $$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$7 to $$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$8 and $$\int_{T^d\times\mathbb{R}^d} v\cdot D\varphi(x)\,d\mu(x,v)=0 \qquad \forall \varphi\in C^1(T^d),$$9 is injective, and more explicitly

$$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$0

where

$$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$1

for any critical subsolution $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$2, independently of the chosen subsolution on the Aubry set (Siconolfi et al., 2021).

In contact systems, the Mather set is explicitly linked to recurrence. The inclusion

$$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$3

is deduced from the Poincaré recurrence theorem because Mather measures are invariant probability measures, so almost every point in their support is recurrent (Ni et al., 2023).

A related support theorem appears in the Tonelli genericity literature: Mather measures are supported on a Lipschitz graph, and the union of their supports is the Mather set (Bernard, 2010). In the non-convex literature, however, that property need not survive; one paper explicitly remarks that the classical theorem that minimizing measures are supported on a Lipschitz graph is not expected to remain true in the non-convex setting (Vichery, 2014).

4. Relations with minimizing measures, homology, cohomology, and generic multiplicity

The Mather set is inseparable from the convex duality encoded by the $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$4- and $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$5-functions and by minimizing measures indexed by homology or cohomology.

On graphs, for a homology class $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$6,

$$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$7

and a Mather measure with homology $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$8 is a closed probability measure $$\int L\,d\mu =\min\Big\{\int L\,d\nu:\nu \text{ closed probability measure}\Big\}.$$9 satisfying $\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$0. The corresponding Mather set is

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$1

For a cohomology class $\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$2,

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$3

and a Mather measure with cohomology $\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$4 satisfies $\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$5. The corresponding set is

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$6

Fenchel duality is expressed by

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$7

Moreover,

$\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$8

This preserves the classical link between convex duality and the supports of minimizing measures (Siconolfi et al., 2021).

The graph model also yields an unusually explicit description of Mather measures. The restriction of any Mather measure to an edge of its support is concentrated at a point, every such measure is a convex combination of occupation measures based on circuits, and irreducible Mather measures are exactly the occupation measures corresponding to parametrized circuits (Siconolfi et al., 2021). This discrete picture makes the Mather set a finite or finite-combinatorial object in a way that is generally absent on manifolds.

The multiplicity of Mather measures has its own genericity theory. For a Tonelli Lagrangian $\min_{\nu\text{ closed}}\int L\,d\nu = -\alpha(H).$9, the exceptional set of perturbations producing many Mather measures is very small. If

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$0

then for $$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$1 this set is countably rectifiable of codimension $$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$2 in $$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$3, hence Baire meager, Aronszajn-null, and with prevalent complement (Bernard, 2010). Since the Mather set is the union of supports of these minimizing measures, this result implies that highly complicated minimizing measure decompositions are strongly nongeneric. A plausible implication is that, generically, the Mather set should often be supported by only a small number of ergodic components.

5. Extensions beyond classical Tonelli convexity

The modern literature extends the Mather-set paradigm beyond autonomous convex Hamiltonian dynamics, but the extensions preserve only part of the classical structure.

Non-convex Hamiltonians

In the adjoint-method construction of non-convex Aubry–Mather measures, the authors define analogues of Mather measures by solving the regularized cell problem

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$4

and using stochastic dynamics

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$5

The limiting measure $$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$6 is required to satisfy:

• energy concentration:

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$7

• momentum-flux identity:

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$8

• holonomy-type condition:

$$\widetilde{\mathcal M}_L := \overline{\bigcup \{\operatorname{spt}(\mu):\, \mu \text{ Mather measure}\} } \subset T^d\times\mathbb{R}^d.$$9

The fundamental new phenomenon is dissipation: $I(L)$0 where $I(L)$1 is a symmetric nonnegative matrix of Borel measures. If $I(L)$2, then $I(L)$3 is invariant; otherwise exact invariance fails. Uniform convexity or uniform quasiconvexity force $I(L)$4, restoring invariance (Cagnetti et al., 2010). This sharply distinguishes what survives from classical Mather theory in non-convex regimes: existence of extremal measures and holonomic identities persist, while invariance and graph structure may fail.

A different non-convex extension uses spectral invariants and symplectic homogenization. There the paper does not define a classical Mather set as a minimizing graph-type subset. Instead, it defines a generalized $I(L)$5-function by

$I(L)$6

and for Tonelli $I(L)$7,

$I(L)$8

Its main theorem states that for $I(L)$9, there exists an invariant measure $TM$0 with

$TM$1

This preserves the foundational Mather principle relating subdifferentials of $TM$2 to invariant measures, but not necessarily the classical graph picture (Vichery, 2014).

Contact and conformally symplectic systems

In conformally symplectic systems, the Mather set remains a discounted-action-minimizing invariant set, still contained in the Aubry set and still satisfying a graph theorem (Marò et al., 2016). In contact Hamiltonian systems, by contrast, the surrounding Aubry structure becomes more intricate because of the $TM$3-dependence. The Mather set remains the measure-theoretic minimizing core,

$TM$4

but the projected Aubry sets may depend on the chosen weak KAM solution, and a new strongly static set may sit strictly between Mather and Aubry (Siconolfi et al., 2021, Ni et al., 2023).

6. Discrete, lattice, driven, and symbolic analogues

The term “Mather set” has been successfully transplanted to several non-manifold settings, each preserving the support-of-extremal-measures principle.

Semi-discrete Tonelli approximation

For time step $TM$5, the discrete action is

$TM$6

and a discrete holonomic measure $TM$7 on $TM$8 satisfies

$TM$9

The discrete Mather set is

$\mu\in I(L)$0

The paper proves the upper Kuratowski limit

$\mu\in I(L)$1

and under a genericity assumption ensuring finitely many ergodic Mather measures,

$\mu\in I(L)$2

This is presented as a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets (Camilli et al., 27 Apr 2026).

Lattice Aubry–Mather theory

For monotone lattice recurrence relations, the paper defines an Aubry–Mather set $\mu\in I(L)$3 as a nonempty closed strictly ordered shift-invariant collection of global minimizers, minimal with these properties. For irrational rotation vectors, this is the familiar Aubry–Mather set and may have gaps. The paper proves that every such set can be interpolated by a connected ordered shift- and gradient-flow-invariant set called a ghost circle. The key consequence is that if an Aubry–Mather set has a gap, then either the gap is parametrized by minimizers or contains a non-minimizing stationary point (Mramor et al., 2011). Although the paper uses “Aubry–Mather set” rather than a separate “Mather set,” its minimizing ordered support plays the same organizational role.

Driven generalized elastic chains

For driven infinite elastic chains, the set

$\mu\in I(L)$4

is the union of supports of all $\mu\in I(L)$5-invariant probability measures on the bounded-spacing configuration space $\mu\in I(L)$6. The paper identifies this as the relevant measure-theoretic Aubry–Mather set and proves that

$\mu\in I(L)$7

gives a continuous injective projection

$\mu\in I(L)$8

It also proves existence of ergodic invariant measures supported on rotationally ordered configurations with arbitrary rotation number $\mu\in I(L)$9, rational or irrational. In the DC case, every ergodic invariant measure is supported either on equilibria or on a single totally ordered periodic orbit, identified with a uniformly sliding solution

$\mu$00

The key tool is the weak Lyapunov function

$\mu$01

which counts intersections and is non-increasing in time (Slijepčević, 2013).

Matrix cocycles and joint spectral radius

For a compact set of matrices $\mu$02, the paper defines the set of maximizing measures

$\mu$03

and the Mather set

$\mu$04

This is the union of supports of growth-maximizing invariant measures. A structure theorem states that $\mu$05 is the support of one invariant measure, every maximizing measure is supported on $\mu$06, every point in $\mu$07 is strongly extremal, and weakly extremal orbits spend asymptotically all their time near $\mu$08 (Morris, 2011). Here action minimization is replaced by maximizing asymptotic matrix growth, but the support-of-extremal-measures paradigm remains unchanged.

7. Special cases, examples, and conceptual limits

Several papers emphasize that the Mather set can collapse to very simple invariant objects under additional assumptions.

For twisted Brownian motions associated to

$\mu$09

on a compact surface of nonpositive curvature, one assumes that the Aubry–Mather measure of

$\mu$10

is unique and supported on a closed geodesic $\mu$11, and that the Aubry set equals the Mather set. In that situation, the Mather set is the minimizing closed geodesic $\mu$12, and large deviations of the stationary measures are controlled by distance to $\mu$13 (Lopes et al., 2010). This example shows that, in low-complexity situations, the Mather set may be a single periodic orbit rather than a large lamination or Cantor-like invariant set.

In contact systems, the later inclusion chain

$\mu$14

makes clear that the Mather set need not coincide with the Aubry set. The paper exhibits an example where

$\mu$15

so the Aubry set is much larger than the strongly static set, hence also larger than the Mather set (Ni et al., 2023). This directly contradicts the common misconception that the Mather and Aubry sets are generally the same.

Another recurrent misconception is that the Mather set must always be a graph. The data show that graph properties are classical and persist in several dissipative or discrete settings, but they are explicitly not expected in general non-convex symplectic extensions (Vichery, 2014), and may require convexity or quasiconvexity assumptions to recover (Cagnetti et al., 2010).

A final conceptual limit concerns terminology. In singularity theory and birational geometry, “Mather” appears in unrelated notions such as Mather stability or Mather minimal log discrepancy, but those are distinct from the dynamical Mather set (Sadykov, 11 Oct 2025, Gu, 2017). The dynamical notion is tied specifically to supports of extremal invariant or closed measures.

The cumulative picture across these papers is stable: the Mather set is the invariant support geometry selected by an extremal principle—usually action minimization, but in some analogues maximizing asymptotic growth. What changes from one setting to another is not that principle, but which properties survive: compactness and support characterizations are robust; graph structure, exact invariance, equality with Aubry sets, and uniqueness of supporting measures depend strongly on convexity, regularity, and the ambient dynamical framework.