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Aubry-Mather theory is a family of mathematical results within the theory of dynamical systems. The theory characterizes the set of trajectories, which are globally minimal with respect to a class of discrete one-dimensional actions. These actions arise in the context of monotone twist maps of the annulus, the Frenkel–Kontorova model and geodesics on tori.

Variational problem

In Aubry-Mather theory, one considers bi-infinite sequences $x\colon \,\mathbb {Z} \to \mathbb {R}$ together with an "action" given by a function $H\colon \,\mathbb {R} ^{2}\to \mathbb {R}$. Finite segments $(x_{j},\dots ,x_{k})$ are evaluated by summing over the contributions of next neighbours:

$H(x_{j},\dots ,x_{k}):=\sum _{i=j}^{k-1}H(x_{i},x_{i+1})$

A segment $(x_{j},\dots ,x_{k})$ is said to be minimal with respect to $H$, if $H(x_{j},\dots ,x_{k})\leq H(y_{j},\dots ,y_{k})$ for all segments $(y_{j},\dots y_{k})$ with identical endpoints $x_{j}=y_{j}$ and $x_{k}=y_{k}$. If all finite segments of a sequence $x$ are minimal, then $x$ is said to be globally minimal. Let the corresponding set of all globally minimal sequences be denoted by ${\mathcal {M}}\equiv {\mathcal {M}}(H)$. Aubry-Mather theory makes statements about the structure of ${\mathcal {M}}$ for functions $H$, that satisfy the following conditions:

1. periodicity: $H(\xi +1,\eta +1)=H(\xi ,\eta )$ for all $(\xi ,\eta )\in \mathbb {R} ^{2}$
2. coercivity: $\lim _{|\eta |\to \infty }H(\xi ,\xi +\eta )=\infty$ uniformly in $\xi$
3. ordering: if $\xi _{1}<\xi _{2}$ and $\eta _{1}<\eta _{2}$, then $H(\xi _{1},\eta _{1})+H(\xi _{2},\eta _{2})<H(\xi _{1},\eta _{2})+H(\xi _{2},\eta _{1})$
4. transversality: if $(x_{-1},x_{0},x_{1})\neq (y_{-1},y_{0},y_{1})$ with $x_{0}=y_{0}$, then $(x_{-1}-y_{-1})(x_{1}-y_{1})<0$

These properties are not as restrictive as they may seem. In fact, if $H$ is twice differentiable, then conditions 2. to 4. are implied by

$D_{1}D_{2}H(\xi ,\eta )\leq -\delta <0$ for all $(\xi ,\eta )\in \mathbb {R} ^{2}$.

In that case, a trajectory is said to be stationary , if $D_{2}H(x_{i-1},x_{i})+D_{1}H(x_{i},x_{i+1})=0$ for all $i\in \mathbb {Z}$, which includes all elements of ${\mathcal {M}}$.

Example

If there exists a strictly convex function $h\in C^{2}$ with $h''>0$, such that $H$ can be written as $H(\xi ,\eta )=h(\xi -\eta )$, then

$D_{1}D_{2}H(\xi ,\eta )=-h''(\xi -\eta )<0$

and all stationary trajectories satisfy

$D_{2}H(x_{i-1},x_{i})+D_{1}H(x_{i},x_{i+1})=-h'(x_{i-1}-x_{i})+h'(x_{i}-x_{i+1})=0\Leftrightarrow h'(x_{i-1}-x_{i})=h'(x_{i}-x_{i+1}){\text{ for all }}i\in \mathbb {Z} .$

Since the derivate of a strictly convex function is injective, the difference between consecutive elements of a stationary $x\in {\mathcal {M}}$ must be the same for all $i\in \mathbb {Z}$. On the other hand, for every $\ x_{0}\in \mathbb {R}$ and $\rho \in \mathbb {R}$ the trajectory given by $\ x_{i}=x_{0}+i\rho$ is stationary. In conclusion, all globally minimal trajectories are of this form, i.e.

${\mathcal {M}}=\lbrace x\in \mathbb {R} ^{\mathbb {Z} }\ |\ x_{i}=x_{0}+i\rho ,\ x_{0}\in \mathbb {R} ,\ \rho \in \mathbb {R} \rbrace .$

These sequences can be interpreted as orbits of iterated functions $f_{\rho }$, meaning each element $x_{i}$ is mapped to the next one by $x_{i+1}=f_{\rho }(x_{i}):=x_{i}+\rho$. Under the projection onto the circle $\mathbb {R} \to \mathbb {R} /\mathbb {Z} \cong \mathrm {S} ,x\mapsto \mathrm {frac} (x)$, the functions $f_{\rho }$ generating ${\mathcal {M}}$ can be interpreted as lifts of rotations by an angle $\rho$.

Minimal trajectories

As a main result, Aubry-Mather theory identifies the elements of ${\mathcal {M}}$ as trajectories of lifted orientation-preserving circle homeomorphisms. That is, for every $x\in {\mathcal {M}}$, there exists a continous strictly increasing map $f_{x}\colon \,\mathbb {R} \to \mathbb {R}$, with $f_{x}(\xi +1)=f_{x}(\xi )+1$ for all $\xi \in \mathbb {R}$, such that $x_{i+1}=f_{x}(x_{i})$ for all $i\in \mathbb {Z}$. This allows to classify the elements of ${\mathcal {M}}$ by their corresponding rotation number $\omega (x):=\omega (f_{x})$. Moreover, the map $\omega \colon \,{\mathcal {M}}\to \mathbb {R}$ is onto, meaning for every real number $\rho$ the set ${\mathcal {M}}_{\rho }:=\lbrace x\in {\mathcal {M}}\ |\ \omega (x)=\rho \rbrace$ is non-empty. According to the Poincaré classification theorem, there is a topological distinction between homeomorphisms with rational and irrational rotation number, which is also reflected in the structure of the ${\mathcal {M}}_{\rho }$.

For both rational and irrational rotation numbers, certain subsets of ${\mathcal {M}}_{\rho }$ have the property of being totally ordered by elementwise comparison ($x<y$ iff $x_{i}<y_{i}$ for all $i\in \mathbb {Z}$). Then, if $M\subseteq {\mathcal {M}}_{\rho }$ is such a set, the projection $p_{0}\colon \,M\to A,x\mapsto x_{o}$ maps $M$ onto a subset $A\subseteq \mathbb {R}$ homeomorphically.

Trajectories with rational rotation number

Let $\rho =p/q\in \mathbb {Q}$ with $p,q$ coprime and ${\mathcal {M}}_{\rho }^{per}\subseteq {\mathcal {M}}_{\rho }$ the subset of periodic trajectories, that is $x_{i}=x_{i-q}+p$ for all $i\in \mathbb {Z}$. Then ${\mathcal {M}}_{\rho }^{per}$ is totally ordered and its image $A_{\rho }^{per}$ under $p_{0}$ is either equal to $\mathbb {R}$, in which case ${\mathcal {M}}_{\rho }={\mathcal {M}}_{\rho }^{per}$, or there exist neighboring orbits ${\underline {x}},{\overline {x}}\in {\mathcal {M}}_{\rho }^{per}$, such that ${\underline {x}}<{\overline {x}}$ and no $x\in {\mathcal {M}}_{\rho }^{per}$ between them. Then, for each pair of neighboring orbits, there exist trajectories whose forward and backward orbits converge to ${\overline {x}}$ and ${\underline {x}}$ respectively:

${\mathcal {M}}_{\rho }^{\pm }:=\left\lbrace x\in {\mathcal {M}}_{\rho }\ {\bigg |}\ \lim _{i\to \pm \infty }|x_{i}-{\overline {x}}_{i}|=0{\text{ and }}\lim _{i\to \mp \infty }|x_{i}-{\underline {x}}_{i}|=0{\text{ for neighboring }}{\underline {x}},{\overline {x}}\in {\mathcal {M}}_{\rho }^{per}\right\rbrace$

The elements of ${\mathcal {M}}_{\rho }^{+}$ and ${\mathcal {M}}_{\rho }^{-}$ are the only heteroclinic trajectories in ${\mathcal {M}}_{\rho }$ and are the only occurrences of trajectories crossing each other. In particular, each of the unions ${\mathcal {M}}_{\rho }^{per}\cup {\mathcal {M}}_{\rho }^{+}$ and ${\mathcal {M}}_{\rho }^{per}\cup {\mathcal {M}}_{\rho }^{-}$ are totally ordered. This concludes the structure of ${\mathcal {M}}_{\rho }$, since it is the disjoint union of ${\mathcal {M}}_{\rho }^{+}$, ${\mathcal {M}}_{\rho }^{-}$ and ${\mathcal {M}}_{\rho }^{per}$.

Trajectories with irrational rotation number

Let $\rho \in \mathbb {R} \setminus \mathbb {Q}$, then ${\mathcal {M}}_{\rho }$ is totally ordered and generated by a single function $f$. While there are no perdiodic trajectories with irrational rotation number, the more general set of recurrent trajectories plays a central role. These trajectories are limits of periodic ones and can be approximated using sequences $(p_{k},q_{k})$ with integers $p_{k},q_{k}\neq 0$:

${\mathcal {M}}_{\rho }^{rec}:=\left\lbrace x\in {\mathcal {M}}_{\rho }\ {\bigg |}\ {\text{there exists a sequence }}(p_{k},q_{k}){\text{, such that }}x_{i}=\lim _{k\to \infty }x_{i-q_{k}}+p_{k}{\text{ for all }}i\in \mathbb {Z} \right\rbrace$

The image of ${\mathcal {M}}_{\rho }^{rec}$ under $p_{0}$ is equal to the set of recurrent points $\mathrm {Rec} (f)$ and by the Poincaré classification theorem, it either holds $\mathrm {Rec} (f)=\mathbb {R}$, in which case ${\mathcal {M}}_{\rho }={\mathcal {M}}_{\rho }^{rec}$, or $\mathrm {Rec} (f)$ is a Cantor set $C$. This has implications regarding sequences of trajectories: in the former case, convergence in ${\mathcal {M}}_{\rho }^{rec}$ is always uniform, while in the latter case of the Cantor set, convergence never is uniform. Moreover, there is the countable subset of trajectories $x$, where $x_{0}$ is the endpoint of a component of $\mathbb {R} \setminus C$. Such trajectories can only be approximanted from above or below, in contrast to the uncountable set of trajectories not corresponding to endpoints, which can be approximated from both sides.

In most cases, one will have ${\mathcal {M}}_{\rho }={\mathcal {M}}_{\rho }^{rec}$, even if $\mathrm {Rec} (f)$ is a Cantor set, but it is in general possible for ${\mathcal {M}}_{\rho }\setminus {\mathcal {M}}_{\rho }^{rec}$ to be non-empty. In that case, for every $x\in {\mathcal {M}}_{\rho }\setminus {\mathcal {M}}_{\rho }^{rec}$, there are asymptotic trajectories ${\underline {x}},{\overline {x}}\in {\mathcal {M}}_{\rho }^{rec}$, with ${\underline {x}}<x<{\overline {x}}$ and $\lim _{i\to \pm \infty }|{\underline {x}}-{\overline {x}}|=0$.

Relevance

In the grander scheme of Hamiltonian mechanics, the significance of Aubry-Mather theory becomes apparent in its application to non-integrable systems. In general, the question of stable orbits is linked to the existence of invariant subsets. While for an integrable system, phase space is foliated by such invariant tori, the Kolmogorov–Arnold–Moser theorem makes statements about which of these tori survive under a weak nonlinear pertubation. Aubry-Mather theory then completes this picture as it guarantees the existance of so called Cantori, invariant remnands of those tori which are destroyed.

Applications

Frenkel-Kontorova model

For a classical Frenkel-Kontorova model with an arbitrary differentiable 1-periodic substrate potential $U_{sub}(\xi )=U_{sub}(\xi +1)$, the function

$H(\xi ,\eta )={\frac {1}{2}}\left(g(\xi -\eta -a_{o})^{2}+U_{sub}(\xi )+U_{sub}(\eta )\right)$

satisfies all conditions imposed above, given that $g\geq 0$. Summing over nearstest neighbours then gives the total potential energy, where the set of stationary trajectories corresponds to the equilibrium states of the system, each sequence representing the atoms positions. If $x\in {\mathcal {M}}$ is a minimal trajectory generated by $f_{x}$, then the trajectory $y$ generated by the inverse function $f_{x}^{-1}$ is also minimal, because $H$ is symmetric in its variables. It furthermore holds $\omega (x)=-\omega (y)$, hence it firstly suffices to consider trajectories with a positve rotation number and secondly allows to interpret the rotation number as the atomic mean distance:

$\omega (x)=\lim _{i\to \infty }{\frac {x_{i}-x_{-i}}{2i}}$

Applying the results of Aubry-Mather theory, there exists a minimal energy configuration for every atomic mean distance. Recurrent trajectories from ${\mathcal {M}}^{rec}$ are called ground states of the model, while a trajectory in the complement ${\mathcal {M}}\setminus {\mathcal {M}}^{rec}$ is an elementary defect. Configurations with rational or irrational rotation number, are called commensurate and incommensurate respectively. The notion of heteroclinic trajecoties in ${\mathcal {M}}_{\rho }^{+}$ and ${\mathcal {M}}_{\rho }^{-}$ is translated to the physical context as advanced and delayed elementary discommensurations, respectively.

Monotone twist maps of the annulus

An orientation preserving $C^{1}$-diffeomorphism $\phi$ from the annulus $S^{1}\times [0,1]$ to itself is called a monotone twist map, if its lift ${\tilde {\phi }}\colon \,\mathbb {R} \times [0,1]\to \mathbb {R} \times [0,1]$

1. is Lebesgue area-preserving: $\mu ({\tilde {\phi }}^{-1}(A))=\mu (A)$, which is equivalent to the Jacobian determinant being $\det(D{\tilde {\phi }})=1$
2. satisfies the twist condition: $D_{2}{\tilde {\phi }}_{1}(\xi ,\eta )>0$
3. preserves the boundary components: ${\tilde {\phi }}_{2}(\xi ,0)=0$ and ${\tilde {\phi }}_{2}(\xi ,1)=1$

Such maps form measure preserving dynamical systems and arise for example as Poincare maps of Hamiltonian systems with two degrees of freedom as they inherit their parents symplectic structure. A prominent case of such a map is the standard map. The billiard in a bounded, strictly convex domain $C\subset \mathbb {R} ^{2}$ is another system, that can be represented by a monotone twist map. There, the first coordinate $\xi$ is given by a natural parametrization $S^{1}\to \partial C$ of the boundary of $C$, while the second coordinate encodes the angle of reflection using $-\cos(\tau )\in (-1,1)$.

From a given monotone twist map, one can construct a 1-form, that is exact on a subset of $\mathbb {R} ^{2}$ and hence is the exterior derivative of a function $H(\xi ,\eta )$. This generating function can be extended to all of $\mathbb {R} ^{2}$ and satisfies all conditions required above. This in turn allows to extend ${\tilde {\phi }}$ and $\phi$ to $\mathbb {R} ^{2}$ resp. the cylinder $S^{1}\times \mathbb {R}$. Let ${\tilde {\phi }}(x_{0},y_{0})=(x_{1},y_{1})$, then $H$ is unique up to an additive constant and its relation to ${\tilde {\phi }}$ can be expressed through

$-D_{1}H(x_{0},x_{1})=y_{0}\quad$and$\quad D_{2}H(x_{0},x_{1})=y_{1}$.

To a trajectory $x$ one can thus associate another trajectory $y$ given by $y_{i}:=-D_{1}H(x_{i},x_{i+1})$ and $x$ is stationary if and only if $(x_{i},y_{i})_{i\in \mathbb {Z} }$ is a trajectory of ${\tilde {\phi }}$. To return to the original domain of ${\tilde {\phi }}$, one can utilize the preservation of the boundary components. The orbits of ${\tilde {\phi }}$ generated by $f_{0}(\xi ):={\tilde {\phi }}_{1}(\xi ,0)$ and $f_{1}(\xi ):={\tilde {\phi }}_{1}(\xi ,1)$ completely lie in $\mathbb {R} \times \lbrace 0\rbrace$ and $\mathbb {R} \times \lbrace 1\rbrace$, respectively, and their rotation numbers $\rho _{0}:=\omega (f_{0})$ and $\rho _{1}:=\omega (f_{1})$ bound the rotation numbers of trajectories realized on the annullus. That is, it holds $\rho _{0}<\rho _{1}$ and for every real number inside the twist interval $\rho \in [\rho _{0},\rho _{1}]$ a trajectory in ${\mathcal {M}}_{\rho }$ has a corresponding orbit of ${\tilde {\phi }}$ contained in $\mathbb {R} \times [0,1]$. In particular, if $\rho \in (\rho _{0},\rho _{1})$, its orbits lie in $\mathbb {R} \times (0,1)$.

See also

• Denjoy's theorem
• Ergodic theory