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In Aubry-Mather theory, one considers bi-infinite sequences together with an "action" given by a function . Finite segments are evaluated by summing over the contributions of next neighbours:
A segment is said to be minimal with respect to , if for all segments with identical endpoints and . If all finite segments of a sequence are minimal, then is said to be globally minimal. Let the corresponding set of all globally minimal sequences be denoted by . Aubry-Mather theory makes statements about the structure of for functions , that satisfy the following conditions:
If there exists a strictly convex function with , such that can be written as , then
and all stationary trajectories satisfy
Since the derivate of a strictly convex function is injective, the difference between consecutive elements of a stationary must be the same for all . On the other hand, for every and the trajectory given by is stationary. In conclusion, all globally minimal trajectories are of this form, i.e.
These sequences can be interpreted as orbits of iterated functions, meaning each element is mapped to the next one by . Under the projection onto the circle , the functions generating can be interpreted as lifts of rotations by an angle .
As a main result, Aubry-Mather theory identifies the elements of as trajectories of lifted orientation-preserving circlehomeomorphisms. That is, for every , there exists a continousstrictly increasing map , with for all , such that for all . This allows to classify the elements of by their corresponding rotation number. Moreover, the map is onto, meaning for every real number the set 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 .
For both rational and irrational rotation numbers, certain subsets of have the property of being totally ordered by elementwise comparison ( iff for all ). Then, if is such a set, the projection maps onto a subset homeomorphically.
Let with coprime and the subset of periodic trajectories, that is for all . Then is totally ordered and its image under is either equal to , in which case , or there exist neighboring orbits , such that and no between them. Then, for each pair of neighboring orbits, there exist trajectories whose forward and backward orbits converge to and respectively:
The elements of and are the only heteroclinic trajectories in and are the only occurrences of trajectories crossing each other. In particular, each of the unions and are totally ordered. This concludes the structure of , since it is the disjoint union of , and .
Let , then is totally ordered and generated by a single function . 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 with integers :
The image of under is equal to the set of recurrent points and by the Poincaré classification theorem, it either holds , in which case , or is a Cantor set. This has implications regarding sequences of trajectories: in the former case, convergence in is always uniform, while in the latter case of the Cantor set, convergence never is uniform. Moreover, there is the countable subset of trajectories , where is the endpoint of a component of . 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 , even if is a Cantor set, but it is in general possible for to be non-empty. In that case, for every , there are asymptotic trajectories , with and .
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.
satisfies all conditions imposed above, given that . 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 is a minimal trajectory generated by , then the trajectory generated by the inverse function is also minimal, because is symmetric in its variables. It furthermore holds , 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:
Applying the results of Aubry-Mather theory, there exists a minimal energy configuration for every atomic mean distance. Recurrent trajectories[b] from are called ground states of the model, while a trajectory in the complement is an elementary defect. Configurations with rational or irrational rotation number, are called commensurate and incommensurate respectively. The notion of heteroclinic trajecoties in and is translated to the physical context as advanced and delayed elementary discommensurations, respectively.
From a given monotone twist map, one can construct a 1-form, that is exact on a subset of and hence is the exterior derivative of a function . This generating function can be extended to all of and satisfies all conditions required above. This in turn allows to extend and to resp. the cylinder. Let , then is unique up to an additive constant and its relation to can be expressed through[e]
and.
To a trajectory one can thus associate another trajectory given by and is stationary if and only if is a trajectory of . To return to the original domain of , one can utilize the preservation of the boundary components. The orbits of generated by and completely lie in and , respectively, and their rotation numbers and bound the rotation numbers of trajectories realized on the annullus. That is, it holds and for every real number inside the twist interval a trajectory in has a corresponding orbit of contained in . In particular, if , its orbits lie in .
^By projecting a sphere onto by sending the equator to and doubly covering the interior, one can identify geodesics on the sphere w.r.t a Riemannian metric with orbits of the billiard.