\documentstyle{amsart}
\title[Aubry--Mather theory]{On the Aubry--Mather theory in statistical mechanics }
\author{A. Candel}
\address{ Department of Mathematics
\\ California Institute of Technology \\ Pasadena, CA 91125 }
\author{ R. de la Llave}
\address{ Department of Mathematics
\\ University of Texas at Austin\\ Austin, TX 78712}
\thanks{R. Ll. partially supported by research grants from the NSF}
\newtheorem{theorem}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{defin}{Definition}
\newcommand{\Ll}{\Lambda}
\renewcommand{\qedsymbol}{ }
\begin{document}
\begin{abstract}
We generalize Aubry-Mather theory for configurations on the
line to general sets with a group action. Cocycles on the group play
the role of rotation numbers. The notion of Birkhoff configuration
can be generalized to this setting.
Under mild conditions on the group, we show how to find Birkhoff ground states
for many-body interactions which are ferromagnetic, invariant under the
group action and having periodic phase space.
\end{abstract}
\date{}
\maketitle
\section{Introduction}
In the motivation from Solid State Physics, Aubry-Mather theory describes the
structure of solutions to the following problem. Frenkel and Kontorova
proposed a very simple model for a one-dimensional crystal to describe
the structure of dislocations. They considered a one-dimensional chain
of identical atoms, connected by springs, placed in a periodic
substrate potential $V$ with period $p$. The potential energy of such
system or configuration $u$ (indexed by the integers ${\mathbb Z}$)
describes the interaction of a particle $u(j)$ with its neighbors, and
is given by the formal expression
$$S(u)= \sum_{j\in {\mathbb Z}} \frac{1}{2}(u(j)-u(j+1)-a)^2 +V(u(j)).$$
The parameter $a$ is the length of the connecting springs. The problem
is then to find configurations $u$ which minimize this potential, and
to describe their structure.
The first part of the sum above is the energy of the internal
interaction between particles, the second is the external energy. In
the absence of substrate potential, that is, when $V=0$, the
configurations of minimal energy
are given by $u(j) = a j +k$. The mean spacing is
the rest distance of the spring.
(This situation corresponds to the integrable case in the
dynamical systems interpretation.)
When $V$ is large, it is reasonable to expect -- and indeed it can be
proved -- that the particles settle around the minima of
the potential $V$, hence that the configurations
that minimize the energy
are spaced with period $p$. Thus
ground states appear as a compromise between two different periods,
and one is then interested in describing their structure.
The various possible configurations of the atoms in the chain are
characterized by trajectories in phase space, which turns out to be
orbits of a twist map of an annulus. This is obtained by looking at
the variational equations that define the map. In general one finds
periodic orbits, to which one can assign a rational rotation
number. Other orbits have irrational rotation number, corresponding to
incommensurate structures.
Several authors (see \cite{2} , \cite{6}, \cite{9})
have considered extensions of the Aubry-Mather theory to higher
dimensions, that is, configurations on lattices ${\mathbb Z}^n$.
In \cite{2}, a study is made of minimal configurations of the
variational problem with nearest neighbor potentials; while in
\cite{6} the authors study variational solutions which are
well-ordered (Birkhoff configurations). In fact, it is shown in
\cite{2} that in higher dimensions, there may be minimal
configurations that are not well ordered.
In \cite{9} the Aubry-Mather theory is discussed for nearest neighbor
potential for lattices in the plane. One can find there some
interesting physical applications of this theory.
For a more detailed survey of the physical significance of the theory
of incommensurate crystals in several dimensions, the reader may
consult \cite{8}.
The main goal of this paper is to generalize the setting of
Aubry-Mather theory to configurations on sets more
general than ${{\mathbb Z}}^d$ and for general many-body interactions.
We consider a set
$\Ll$, together with potentials $H_B$ associated to finite subsets $B$
of $\Ll$.
We will need to assume that $\Ll$ admits a
group action by a group $G$ satisfying some
mild assumptions. The potentials $H_B$ describe the interaction of a
particle lying in $\Ll$ with its neighbors. The potential energy of
such a system can be modeled by a formal sum
$$ S(u)= \sum_{B\subset \Ll} H_B(u),$$
where $u:\Ll \to {\mathbb R}$ is a configuration on $\Ll$.
We will require that
$H_B$ satisfy a ferromagnetic or twist
condition, that they are invariant under the action of $G$
and also satisfy some other periodicity assumption.
Physically, we could think of the points of $\Ll$ as
atoms, whose state is characterized by
one number. The configurations describe what is the
situation of all the atoms.
We look for configurations $u$
on $\Ll$ which are stationary points or minimizers
for $S$ and which have a
prescribed rotation number. In our situation, the rotation number is a
cocycle for the group $G$.
Solutions to the variational equations with given rotation number
always exists. But in order to find ground state configurations we
have to require that the group $G$ satisfies a (very natural)
property. Essentially, this property says that we can exhaust $\Ll$ by
fundamental domains for subgroups of $G$ of finite index.
We will discuss the well-order properties of the
solutions that we find. These properties are the exact analog of the
non-crossing properties of codimension one minimal surfaces of
Riemannian manifolds, which obviously is the natural generalization of
the relation between solutions to the one-dimensional
Frenkel-Kontorova model and geodesics on a torus.
We think that the generality emphasizes the relevant features of
the theory and that it would be of interest even if
we were interested only in the one dimensional case.
Nevertheless, we had several concrete examples that
we wanted to consider.
One of them was the following.
Consider a completely homogeneous discrete subset of the
hyperbolic plane, something we may call a hyperbolic crystal. This
set, say $\Ll$, is assumed to be left invariant by a discrete
cocompact group $G_0$ of hyperbolic isometries, that is $G_0 \subset
\text{PSl}(2,{\mathbb R})$ is its isotropy group. There is a natural
potential in this situation, namely the one whose solutions are
discrete harmonic functions. Our theory applies to this case, and to
perturbations of it.
Although hyperbolic crystals may not seem very realistic, let us
mention the work of Kleiman and Saduc \cite{5}. They suggest that
amorphous structures could be regarded as projections of ordered
patterns in hyperbolic space. They could be seen as models for
disclinations in ordered Euclidean structures. Imagine we add wedges
to a crystal. This operation decreases the curvature, thus the new
crystal resembles a periodic pattern in hyperbolic space.
Finally let us mention that the theory presented here
applies to the Bethe lattice. Besides its applications
in statistical mechanics, this seems a reasonable model
for hierarchical networks, for example computer networks.
It is not hard to imagine situations in which there
is a penalty for computers to be out of phase and also
for not being in phase with the environment that they
are occupying.
In the next section, we will describe the properties of
the geometry of the lattice that we will need,
in the following one, we will describe the
properties of interactions. Then, we will
state and prove the main theorem that states that there
are solutions for the variational equations satisfying
extra properties. Finally, we will briefly discuss
some other problems.
\section{Generalities on groups and cocycles}
The situation we will consider is as follows. There is a countable
space $\Ll$
on which a finitely generated group $G$ which acts. We impose
some conditions on the action of $G$ on $\Ll$. First, it is effective, that is only
the identity element of $G$ acts trivially (this is no restriction
because we can always pass to a quotient of $G$ satisfying this
property). Second, there is a finite fundamental domain for the
action, that is, a finite subset $F$ of $\Ll$ which intersects each
orbit of $G$ in exactly one point.
For instance, $\Ll$ could be the set of vertices of some
graph (we would like to keep in mind the edges as well, as we may use
them for the path metric). Perhaps to avoid unnecessary complications
we should assume that the graph is locally finite or uniformly locally
finite (it does not became more and more populated as one goes to
infinity).
Now this more general situation includes the Cayley graph of finitely
generated group, graphs which are lifts of the 1-skeleton of a compact
manifold to its universal covering, graphs which are quasi-isometric
to manifolds of bounded geometry, etc. For example, it includes the
Bethe lattice as the universal cover of the figure 8. Consult \cite{3}
for the thermodynamic properties of the Bethe lattice.
In this non-commutative situation that we will consider, the role of
rotation numbers in Aubry-Mather theory would be played by cocycles
on the group $G$, that is maps
$$\sigma:G \rightarrow {\mathbb R}$$ such that $\sigma(\gamma
\gamma')=\sigma(\gamma) + \sigma(\gamma')$. The space of cocycles on
$G$ forms a real vector space, which will be denoted by $H^1(G;{\mathbb R})$.
We will assume that $G$ acts on $\Ll$ with small stabilizers, meaning
that if an element $\gamma$ of $G$ fixes a point of $\Ll$, then
$\sigma(\gamma)=0$ for any cocycle $\sigma$ on $G$. For this to hold
it is sufficient that the stabilizer of each point of $\Ll$ is a
torsion group, or, more generally, that it is a group with trivial
first cohomology).
One technical result that we will need is the possibility of
approximating arbitrary cocycles by simpler ones, namely
\begin{prop} Let $\sigma:G\rightarrow {\mathbb R}$ be a
cocycle. Then there is a sequence of cocycles $\sigma_n $ which
converges point-wise to $ \sigma$ and such that $\sigma_n$ has
integral values when restricted to some subgroup $G_n$ of finite index
in $G$.
\end{prop}
\begin{pf} Since $G$ is finitely generated, $H^1(G,{\mathbb R})$ is a
finite dimensional real vector space. Take a basis for it formed by
cocycles with integral values, say $\{\tau_1,\ldots,\tau_e\}$. Then
we can write $\sigma=a_1\tau_1+\ldots+a_n\tau_n$, some real numbers
$a_j$. For each $ k =1,2,\ldots$, choose rational numbers $b_{i,k}$,
$i=1,\ldots,e$, such that $|a_i-b_{i,k}|< 1/k^2$. Then:
$$|\sigma (\gamma)-
b_{1,k}\tau_1(\gamma)-\ldots-b_{e,k}\tau_e(\gamma)|< |\gamma|_1/k^2,$$
where $|\gamma|_1$ is the $l_1$-norm of $\gamma$ in $H^1(G;{\mathbb R})$
with respect to the basis $\{ \tau_i\}$. Furthermore, this cocycle
$\sigma_k=\sum_ib_{i,k}\tau_i$ takes on rational values only. Since
$G$ is finitely generated, its image by $\eta$, say $P_k$, is a
finitely generated subgroup of $\Bbb Q$. Thus there is an integer $m$
such that $mP_k = {\mathbb Z}$, and $P_k/mP_k$ is a finite group. In
conclusion, there is a finite index subgroup of $G$ such that
$\sigma_k$ has integer values when restricted to it.
Finally, if $\gamma \in G$ has $|\gamma|_1k$ we get
$$|\sigma(\gamma)-\sigma_n(\gamma)| < |\gamma|_1/k^2 <1/k,$$ so that
$\sigma_n$ converges point-wise to $\sigma$ as required.
Also,
$$\|\sigma-\eta\|=\sup_{\gamma\in G}
|\sigma(\gamma)-\sigma_n(\gamma)|/|\gamma| \leq \sup_{\gamma \in
G}|\sigma(\gamma)-\sigma_n(\gamma)|/|\gamma|_1 < 1/k^2,$$ because
$|\gamma|_1\leq|\gamma|$ as one easily verifies.
\end{pf}
There is one property of the group that we will need to consider when
proving existence of ground states, and it is that of being residually
finite. This means that for each element of $G$, other than the
identity, there is a finite index subgroup of $G$ which does not
contain it. Most familiar groups are residually finite, for
instance, a theorem of Selberg asserts that, if $R$ is a field and $n\geq 1$, then
every finitely generated subgroup of $SL(n,R)$ is residually finite.
The reason for requiring this property is that it would allow us to do
a kind of renormalization needed later in our discussion. More
specifically we need the following:
\begin{prop} Let $G$, $\Ll$ be as above with $G$ residually
finite. Then there is a sequence $G_1 \supset G_2\supset\dots $ of
finite index subgroups of $G$ whose fundamental domains exhaust $\Ll$.
\end{prop}
\begin{pf} If $F$ is a fundamental domain for $G$, we can write
$$\Ll =F\cup g_1F\cup\dots$$ where $g_i$ are the nontrivial elements
of $G$. By induction, choose a subgroup $G_n$ of finite index in
$G_{n-1}$ which does not contain $g_n$. (Recall that residually finite
is a property inherited by finite index subgroups.) Since a
fundamental domain for $G_n$ may be constructed by adding the
translates of a fundamental domain of $G_{n-1}$ by representatives of
elements of $G_{n-1}/G_n$, the proof is complete.
\end{pf}
\section{Configurations, the Birkhoff property and rotation
cocycles}
A configuration on $\Ll$ is a map $u:\Ll \rightarrow {\mathbb R}$. The set
of all configurations on $\Ll$ has the structure of a real vector
space with the obviously defined operations, and we denote it by ${\cal C}
(\Lambda)$. The space of configurations is partially ordered in the
following sense. We say that two configurations $u$, $v$ are such that
$u\leq v$ if $u(p)\leq v(p)$ for all $p$ in $\Lambda$. This makes
${{\cal C}}(\Ll)$ into a partially ordered vector space.
One metric we want to consider on the space of configurations is the
one induced by the family of semi-norms $|u(p)|$, $p\in \Ll$, which gives rise
to the pointwise convergence topology. An explicit form for this metric is the
following: if we enumerate the points of
$\Lambda$ as $p_1,p_2,\cdots,p_n,\cdots $, an explicit form is
$$d(u,v)=\sum_{p_n \in \Ll}
\frac{1}{2^n}\frac{|u(p_n)-v(p_n)|}{1+|u(p_n)-v(p_n)|}.$$
We denote
$\|u\|=d(0,u)$ and note that $\|tu\|\leq (1+|t|)\|u\|$ for any real
number $t$.
The action of $G$ on $\Lambda$ extends to one on ${\cal C}(\Lambda)$,
defined in the following manner: if $\gamma\in G$ and $u\in {\cal C}(\Lambda)$,
then
$${\cal T}_\gamma u(p)=u(\gamma(p))$$
for any element $p$
of $\Lambda$. It is evident that this action of $G$ on ${\cal C}{\Ll}$ preserves the
partial order on configurations.
We will denote by $\chi_A$ the characteristic configuration associated with a
subset $A$ of $\Lambda$, and which is defined as $\chi_A(p)=1 \,
\,\text{if}\, \, p \in A$, $\chi_A(p)=0$ otherwise.
We define operators ${\cal R}_A: {\cal C}(\Ll) \longrightarrow {\cal
C}(\Ll)$ by
$${\cal R}_A (u) = u - \chi_A.$$ It is clear that they are invertible
and furthermore that if $A$, $B$ are subsets of $\Ll$, then the
operators ${\cal R}_A$ and ${\cal R}_B$ commute.
We denote by $\Ll_\alpha$, $\alpha=1,\ldots,l$ the equivalence classes
of $\Ll$ modulo $G$, and denote by ${\cal R}_\alpha$ the operators
corresponding to the sets $\Ll_\alpha$. Given any $s$ in ${\mathbb Z}^l$,
$s=(s_1,\ldots,s_l)$, we denote by ${\cal R}_s$ the operator
${\cal R}_1^{s_1}\ldots {\cal R}_l^{s_l}.$
A cocycle $\sigma$ defines a configuration $u_\sigma$ on $\Ll$ as
follows. Let $F$ be a fundamental domain for the action of $G$ on
$\Ll$. Any $p$ of $\Ll$ can be written as $p=\gamma q$ for a unique
$q$ in $F$. Then define $u_\sigma(p)=\sigma(\gamma)$. It is elementary
that $u_\sigma$ is well-defined, for if also $p=\gamma_1q$, then
$\gamma^{-1}\gamma_1$ stabilizes $q$, so
$\sigma(\gamma)=\sigma(\gamma_1)$ by our hypothesis. Note that
$u_\sigma(q)=0$ for all the elements of $F$.
Then, with these conventions, we say that a configuration $u$ is of
type $\sigma$ if
$$ \sup_{p\in\Ll}|u(p)-u_\sigma(p)| < \infty.$$ The set of
configurations of type $\sigma$ will be denoted by ${\cal O}_\sigma$.
Put in another way, once a fundamental domain $F\subset\Ll$ is fixed,
any cocycle $\sigma$ on $G$ defines a configuration and ${\cal O}_\sigma$
is the subspace of ${\cal C}(\Ll)$ formed by those
configurations $u$ at bounded distance from $u_\sigma$. Thus $\cal
O_\sigma$ is an affine space modeled on $\ell^\infty(\Ll,{\mathbb R})$.
The induced metric in ${\cal O}_\sigma$ is
$$|u(\gamma)-v(\gamma)|_\infty =\sup_{\gamma \in
G}|u(\gamma)-v(\gamma)|,$$
which makes it into a Banach space.
\begin{prop} The space ${\cal O}_\sigma$ is independent of
the fundamental domain.
\end{prop}
This is an elementary consequence of the cocycle property. Also, the
same property implies:
\begin{prop} The operators ${\cal T}_\gamma$, $\gamma \in
G$, and ${\cal R}$ leave ${\cal O}_\sigma$ invariant.
\end{prop}
Next we introduce the notion of Birkhoff configuration. A
configuration $u$ of type $\sigma$ is called a Birkhoff configuration
for the group $G$ if $\sigma(\gamma) \geq (\leq) s_\alpha$ for $\gamma
\in G$, $s\in {\mathbb N}^l$ only when ${\cal T}_\gamma(u) \geq (\leq)
{\cal R}_s(u)$. The set of Birkhoff configurations in ${\cal O}_\sigma$ is
denoted by ${\cal B}_\sigma$.
Since ${\cal R}_s(u)(p) = u(p)-s_\alpha$ if $p$ belongs to the
equivalence class $\Ll_\alpha$ of $\Ll/G$, we see that $u$ is a
Birkhoff configuration if either ${\cal T}_\gamma u + \sigma(\gamma)
\geq u$ or ${\cal T}_\gamma u +\sigma(\gamma) \leq u$ for the partial
order of configurations. It also follows immediately that:
\begin{prop} For every $\gamma$ in $G$ there exists
$s^+$, $s^-$ such that
$$ {\cal R}_{s^-} (u) \leq {\cal T}_\gamma (u) \leq {\cal R}_{s^+}(u)$$ for
any $u$ in ${\cal B}_\sigma$.
\end{prop}
Although we have defined the Birkhoff property of a configuration by
referring to a given cocycle, we could have avoided it. We
show next that the non-intersecting property of a configuration
implies that there is a cocycle for which it is Birkhoff. That is,
suppose that $u:\Ll \rightarrow {\mathbb R}$ is a configuration which
satisfies the following property: for each $\gamma \in G$ and any
integer $k\in {\mathbb Z}$, we have
$$u(\gamma p)+k\leq u(p)$$ for all $p\in \Ll$, or
$$ u(\gamma p) +k \geq u(p)$$ for all $p\in \Ll$. We are going to
define a cocycle $\sigma:G \rightarrow {\mathbb R}$ such that
$u\in {\cal O}_\sigma$.
First, let
$$\tau^+(\gamma)=\sup_p (u(\gamma p)-u(p))$$ and
$$\tau^-(\gamma)=\inf_p (u(\gamma p)-u(p)).$$ Note that they are
finite numbers. Furthermore, $\tau^+$ is a sub-cocycle, that is,
$$\tau^+(\gamma_1\gamma_2)\leq \tau^+(\gamma_1)+\tau^+(\gamma_2),$$
and $\tau^- $ is a super-cocycle. This implies that we can define
$$\sigma(\gamma)=\lim_{n\to\infty} \frac{\tau^+(\gamma^n)}{n} = \lim_{n\to\infty}
\frac{\tau^-(\gamma^n)}{n}.$$ because both limits
exist and they are equal. We also see that
$$\sigma(\gamma_1\gamma_2)=\lim_{n\to\infty}\frac{\tau^+((\gamma_1\gamma_2)^n)}{n}
= \lim_{n\to\infty}\frac{\tau^+(\gamma_1(\gamma_2\gamma_1)^{n-1}\gamma_2)}{n}
\leq \sigma(\gamma_2\gamma_1)$$
by elementary use of the sub-cocycle and
super-cocycle properties and of the definition of $\sigma$. Therefore
$\sigma(\gamma_1\gamma_2)=\sigma(\gamma_2\gamma_1)$.
It is also elementary to check that
$\sigma(\gamma^{-1})=-\sigma(\gamma)$, and that, if $\gamma_1$ and
$\gamma_2$ commute, then
$$\sigma(\gamma_1\gamma_2)=\sigma(\gamma_1)+\sigma(\gamma_2).$$ From
all these properties it follows that $\sigma $ is a cocycle.
Finally we show that the configuration $u$ has $\sigma$ as rotation
cocycle, that is, that
$$\sup_{p\in\Ll} |u(p)-\sigma(\tau_p)|<\infty,$$ which is the same as
to show that
$$ \sup_{\gamma\in G, q\in F}|u(\gamma q)-\sigma(\gamma)|<\infty.$$
Here $F\subset\Ll$ denotes a finite fundamental domain for the action
of $G$.
Fix $q\in F$ and $\gamma\in G$, and consider the configuration $n\in
{\mathbb Z} \mapsto u(\gamma^nq)$. Since
$$\tau^-(\gamma^n)\leq u(\gamma^nq)-u(q) \leq \tau^+(\gamma^n),$$ it
follows from the definitions that it has rotation number
$\sigma(\gamma)$. As is well-known, this implies that
$$u(q)+n\sigma(\gamma)\leq u(\gamma^nq) < u(q)+n\sigma(\gamma) +1$$
from what it follows that
$$|u(\gamma q)-\sigma(\gamma)|\leq 1+|u(q)|.$$ Since the fundamental
domain $F$ is finite, this implies that the rotation cocycle of $u$ is
$\sigma$, as claimed above.
\begin{prop} If $u$ is a configuration satisfying the
well-ordering property above, then there is a cocycle $\sigma$ on $G$
for which $u\in{\cal B}_\sigma$.
\end{prop}
In particular, \begin{cor} If the group has no nontrivial
cocycles, then every Birkhoff configuration is bounded.
\end{cor}
A simple example of a group without cocycles is
$$G=< a,b; a^p=b^q=(a^{-1}b)^r=1>$$ with $1/p+1/q+1/r <1$. This group
can be realized faithfully as a discrete group of isometries of the
hyperbolic plane. However, this groups have finite index subgroups
which do have nontrivial cocycles.
We also mention the following elementary property of Birkhoff
configurations.
\begin{prop} The space of Birkhoff
configurations is closed for the product topology on
${\mathbb R}^\Ll$. Furthermore, for any cocycle $\sigma$, the set ${\cal B}_\sigma$
is non-empty and convex.
\end{prop}
\begin{pf} For the first part, simply note that by the previous
proposition the space of Birkhoff configurations is characterized by
the inequalities $T_\gamma u +n \geq u$ or $T_\gamma u+n\geq u$, so it
can be written as an intersection of closed sets.
The configuration $u_\sigma$ associated to the cocycle is Birkhoff, so
${\cal B}_\sigma$ is non-empty. The convexity property is obvious.
\end{pf}
Similarly, we also have
\begin{prop} Let $\sigma$ be a cocycle in $G$, and consider
${\cal O}_\sigma$ with the Banach space topology given by uniform
convergence. Then the space ${\cal B}_\sigma$ of Birkhoff configurations
in ${\cal O}_\sigma$ is closed.
\end{prop}
In analogy with rational rotation numbers, we say that a configuration
$u$ has rational rotation cocycle if there is a normal finite index
subgroup $G'$ of $G$ and an integer valued cocycle
$\sigma:G'\rightarrow {\mathbb Z}$ such that
$$u(\gamma(p))=u(p )+\sigma(\gamma)$$ for all $p\in \Ll$.
This definition could have been formulated in a slightly different way
by using the following two propositions.
\begin{prop} Let
$\sigma:G' \rightarrow {\mathbb Z}$, where $G'$ is a finite index normal
subgroup of $G$ with $G/G'$ commutative. Then $\sigma$ extends to a
rational cocycle on $G$.
\end{prop}
\begin{pf} For $\gamma$ in $G$, let $n$ be any nonzero integer such
that $\gamma^n\in G'$. Then define
$\sigma(\gamma)=\sigma(\gamma^n)/n$. It is immediate to check that
$\sigma:G\rightarrow \Bbb Q$ is well defined and, because $G/G'$ is
commutative, it is a homomorphism.
\end{pf}
\begin{prop} Let $u\in {\cal C}(\Ll)$ be a configuration such
that $u(\gamma(p))=u(p)+\sigma(\gamma)$ for all $p$ in $\Ll$ and all
elements $\gamma$ of a finite index normal subgroup $G'$ of $G$, and
for $\sigma:G'\rightarrow {\mathbb Z}$. Suppose that $\sigma$ extends to a
cocycle on $G$. Then $u\in {\cal O}_\sigma$, $\sigma $ the extension of
the cocycle to $G$.
\end{prop}
\begin{pf} Let $p_0,\ldots,p_r$ be a finite set such that the
translates $\{\gamma(p_i); \gamma \in G'\}$ fill up $\Ll$. Choose an
upper bound $M$ for the finite sets of numbers $\{ |u(p_0)|,
\ldots,|u(p_r)|\}$ and $\{ |\sigma(\tau_{p_0})|, \ldots,
|\sigma(\tau_{p_r})|\}$. Then, for any $p\in \Ll$ we have:
$$|u(p)-\sigma(\tau_p)|=|u(\gamma(p_k))-\sigma(\tau_p)|=|u(p_k)-\sigma(\tau_{p_k})|\leq
2M.$$
\end{pf}
We will use another property of Birkhoff configurations that we now
explain.
\begin{prop} Suppose that $u$ is a Birkhoff configuration
that has integer valued rotation cocycle $\sigma:G\rightarrow \Bbb
Z$. Suppose that $u$ is periodic with respect to some finite index
subgroup $G'$ of $G$, that is,
$$u(\gamma(p))=u(p)+\sigma(\gamma)$$ for all $p$ in $\Ll$ and all
$\gamma$ in $G'$. Then $u$ is also $G$-periodic.
\end{prop}
The proof of this fact goes as follows. Let $\gamma$ be an element of
$G$. Then $\gamma^k \in G'$ for some integer $k$ because $G/G'$ is
finite. If $u$ is not $\gamma$-periodic, then let $p$ be such
$$u(\gamma(p)) > u(p)+\sigma(\gamma)$$ Using the Birkhoff property of
$u$ this implies that
$$u(\gamma^k(p))>u(p)+k\sigma(\gamma)$$ which contradicts the
$G'$-periodicity of $u$.
\section{Interaction potentials and the variational problem}
Let $\cal S$ denote the collection of finite nonempty subsets of
$\Ll$. An interaction potential is a collection of maps $H=\{H_B;
B\in {\cal S} \}$, where each $H_B:{\cal C}(\Ll) \rightarrow {\mathbb R}$, such
that $H_B(u) = H_B(v)$ whenever $u$ and $v$ agree on $B$ (so $H_B$ is
interpreted as a function $H_B:{\mathbb R}^B \rightarrow {\mathbb R}$), and for
any finite subset $X$ of $\Ll$, the series
$$ H(u,X)=\sum _{B\cap X\neq \emptyset} H_B(u)$$ converges. We call
$H(u,X)$ the (total) energy of $u$ in $X$.
A potential is absolutely summable if the series
$$\sum_{B,p\in B} |H_B|_\infty $$ converges for all $p\in
\Ll$. Denoting by $|H|_p$ be the value of the series above, we have a
family of semi-norms in the space ${\cal P}$ of absolutely summable
potentials, making it into a Frechet space.
A potential $H$ is said to be of finite range if for each $p\in\Ll$
there is some finite set $B_p \subset \Ll$ such that $H_B=0$ unless
$B\subset B_p$ for all $p\in B$. We say that $H$ is of bounded range
if there is a number $M$ such that $H_B=0$ if
$\text{diam}(B)>M$. Clearly, if $H$ is finite range and all $H_B$ are
bounded, then $H\in {\cal P}$. Moreover, absolutely summable finite
range potentials are dense in ${\cal P}$.
A configuration $u$ is a ground state for the interaction potential
$H=\{H_B\}$ if
$$ H(u,X)\leq H(v,X) $$ for any finite set $X$ and any configuration
$v$ such that $u=v$ on $\Ll \setminus X$. Thus $u$ is a ground state
if the energy of any finite perturbation of $u$ exceeds that of $u$.
Our goal is to seek minimal configurations which belong to the spaces
${\cal O}_\sigma$. Note that if $u$ is a minimal configuration for the
energy problem in ${\cal O}_\sigma$, it is also a minimal configuration
for the global problem in ${\cal C}(\Ll)$, simply because if $u\in \cal
O_\sigma$ and $u=v$ outside some finite set $X$, then also $v\in \cal
O_\sigma$. On the other hand, note that a configuration that minimizes
$H(\cdot,X)$ in ${\cal B}_\sigma$ may not necessarily be a ground state
configuration.
Some other properties of ground state configurations that we will use
later are collected in the following proposition.
\begin{prop} The set of all ground state configurations for
a continuous potential $H$ is closed in ${\mathbb R}^\Ll$ with the product
topology.
If ${\cal O}$ denotes the space of configurations at bounded distance
from a fixed one, the space of ground states in ${\cal O}$ is closed in
the Banach space topology.
\end{prop}
\begin{pf} Suppose that a sequence $u_n$ of ground state
configurations converges point-wise to the configuration $u$. If $u$
is not a ground state configuration, then there is a finite set $X$
and a configuration $v$ which equals $0$ outside $X$ and such that
$$H(u,X)-H(u+v,X)\geq a >0.$$ On the finite set $X$, the convergence
$u_n \rightarrow u$ is uniform, and the continuity of the potential
implies that, for large $n$ we have
$$H(u_n,X)-H(u_n+v,X)>0$$ contradicting the minimality of the $u_n$'s.
The same argument also proves the second part, because if $u_n \in
{\cal O}$ converges to $u$ in the Banach space topology then it also
converges uniformly on each finite subset $X$ of $\Ll$.
\end{pf}
We introduce some more definitions in order to restrict the type of
potentials to be considered. An interaction potential $H=\{ H_B \}$
is $G$-invariant if
$$H_{\gamma B}(u)=H_B(T_\gamma u)$$ for all $\gamma \in G$, all $B$
and all configurations $u$ on $\Ll$.
If the interaction potential $H$ satisfies
$$ H_B({\cal R}_s u)=H_B(u)$$ for all configuration $u$, all $B$ and all
$s\in {\mathbb Z}^l$, then we say that $H$ has periodic phase. This
periodic phase property allows us to consider the $H_B$'s as maps
${\cal C}(\Ll)/{\cal R} \rightarrow {\mathbb R}$.
The Banach space structure of ${\cal O}_\sigma$ permits us to write down
the variational equations if the potential is differentiable. Thus,
with the assumption of differentiability of potentials, a necessary
condition for $u$ to be a minimal configuration is that it satisfies
the variational equations
$$ \nabla H(u,X)=\sum_{X\cap B\neq \emptyset, p\in B}
\frac{\partial}{\partial p} H_B(u) =0.$$
This sum converges for finite range interaction. In general, we have
to make some further restrictions.
Let ${\cal O}$ be a convex subset of the set of configurations. We say
that an interaction $H$ is $C^r$-bounded on ${\cal O}$ if
$$\|H\|_{r,{\cal O}} = \sup_{u\in {\cal O}} \sum _{|j|\leq r}|D^j H_B(u)
|<\infty $$
Note that the space ${\cal O}$ in the
definition above maybe a proper subset of one of the
${\cal O}_\sigma$'s. Examples show that interaction may not be bounded in
${\cal O}_\sigma$ but they are in subspaces of the form
${\cal O}=\{u; \, |u-u_\sigma|_\infty\leq K\}$.
If $H$ is a $C^1$-bounded potential on ${\cal O}$, we define a map
$u\mapsto V(u)$ on configurations $u\in {\cal O}$ with values in
$\ell^\infty(\Ll)$ by
$$V(u)(p) = -\sum_{B\ni p} \frac{\partial}{\partial p} H_B(u)$$ and
therefore the variational equations can be written as
$$V(u)=0$$
In the cases that we will consider ${\cal O}$ will be an affine space
over $\ell ^\infty(\Ll)$, in particular one of the ${\cal O}_\sigma$,
and the hypothesis of $C^r$ boundedness of $H$ will ensure that the
$r-1$ derivative of $V$ exists and is uniformly bounded in the sense
of derivatives of Banach spaces.
Since ${\cal O}_\sigma$ is a Banach space isomorphic to
$\ell^\infty(\Ll)$ the natural interpretation of $V$ is as a vector
field on ${\cal O}_\sigma$.
In any case, the above definitions show that the function $V$
associated to a $C^r$-bounded potential $H$ on ${\cal O}$ is globally
Lipschitz, that is:
\begin{cor} If $V$ is defined as above, then
$$ |V(u)-V(v)|_\infty \leq \|H\|_2 |u-v|_\infty.$$
\end{cor}
Before we state the main theorem on the existence of minimal
configurations we still need new definition concerning the potentials
to be considered. This is the twist condition of Hamiltonian
mechanics or the ferromagnetic property in statistical mechanics. More
precisely, we say that an interaction potential $H$ which is
$C^2$-bounded on ${\cal O}$ satisfies the twist condition if
$$\sum_{B\ni q}\frac{\partial^2}{\partial p\partial q}H_B(u) \leq 0$$
for all configurations $u$ in ${\cal O}$ and all $p$, $q$ in $\Ll$ with
$p\neq q$.
This property is obviously implied by the strong twist condition, that
is
$$ \frac{\partial^2}{\partial p \partial q }H_B(u)\leq 0$$ for all $u$
and all $p\neq q$.
\section{Existence of solutions to variational equations}
In this section we prove the existence of solution to the variational
equation of a twist potential. The idea is inspired by \cite{4} and
\cite{6}. There the strategy of the proof is to consider the flow
$\phi_t:{\cal O}_\sigma \rightarrow {\cal O}_\sigma$ defined by the
differential equation
$$\frac{d}{dt}\phi_t (u) = -V(\phi_t(u)).$$ Since the right-hand side
is globally Lipschitz, solutions exist for all times. Then we will
argue that this flow preserves the Birkhoff condition on
configurations, so it is actually a flow on $\cal
B_\sigma$. Furthermore, it commutes with the operators ${\cal R}$ so
that the end result is a flow on ${\cal B}_\sigma/{\cal R}$. Once here we
will be able to use compactness and therefore fixed points for the
flow.
Unfortunately, for this outline to work we need to add a further
condition on the space $\Ll$, namely that it has polynomial growth
function with respect to the action of $G$. In general this case is
the exception rather than the rule. To overcome this difficulty we
consider a sequence of gradient flows that converge to the one defined
by the variational equations.
\begin{theorem} Let $G$ be a finitely generated group acting on
$\Ll$ with finitely many orbits, and with small stabilizers. Let
$\sigma$ be a cocycle on $G$ and let $H$ be an interaction potential
which is $C^2$-bounded on ${\cal O}_\sigma$ and which is $G$-invariant
and has periodic phase. Assume that either $H$ satisfies the strong
twist condition or that $H$ satisfies the twist condition and is of
finite range. Then there is a solution of the variational equations
which lies in ${\cal B}_\sigma$.
\end{theorem}
Suppose that $H$ is of finite range. For any finite subset $F$ in
$\Ll$ consider the function $H_F$ on configuration space defined by
$$H_F(u)=\sum_{B\cap F\neq \emptyset} H_B(u).$$
Let $-\nabla H_F$ be the gradient vector field defined by this
function and let $\phi_F$ be the flow defined by the differential
equation
$$\frac{d}{dt}\phi_F(u,t) = -\nabla H_F(\phi_F(u,t)).$$
\begin{prop} The flow $\phi_F$ is defined on ${\cal O}_\sigma$
for all times.
\end{prop}
Indeed, the gradient $\nabla H_F$ is a Lipschitz vector field in
${\cal O}_\sigma$ with the Banach space topology.
\begin{prop} The flow $\phi_F(t,u)$ defined by the
differential equation
$$\frac{d}{dt} \phi_F(t,u) = -\nabla H_F(\phi_F(t,u))$$
preserves the
order structure on the space of configurations.
\end{prop}
The proof of this proposition is along the lines of the one in \cite{6}. Briefly, the
idea is that the twist condition implies that the corresponding
linearized equation is given by a matrix with positive entries off the
diagonal. This is shown to imply the monotonicity of the gradient
flow.
A consequence of this and the fact that the interaction potential has
periodic phase is that $\phi_t$ induces a flow on
${\cal B}_\sigma /{\cal R}$.
\begin{prop} If the interaction potential $H$ is
ferromagnetic, then $H_F$ is non-increasing along the flow, that is
$$ \frac{d}{dt}H_F(u) \leq 0$$
\end{prop}
We compute the derivative along the flow $\phi:{\cal O}\times \Bbb
R\rightarrow {\cal O}$ defined by the differential equation. By the
chain rule we have that:
\begin{eqnarray*}
\frac{d}{dt} H_F(\phi_F(u,t)) & = & -\nabla H_F (u)\frac{d}{dt}\phi_F(u,t)\\
&=& -\left( \frac{\partial}{\partial p} \sum_B H_B(u)\right)_{p\in BF}
\cdot \left( \frac{\partial}{\partial p} \sum _B H_B(u)\right)_{p\in BF} \\
&=& \sum_{p\in BF}\left(\frac{\partial}{\partial p}H_B(u)\right)^2 \leq 0.
\end{eqnarray*}
By the order preserving property, we can consider the flow on $\cal
B_\sigma/{\cal R}$. The last detail we need is:
\begin{prop} The space ${\cal B}_\sigma/ {\cal R}$ is compact
(for the pointwise convergence metric).
\end{prop}
\begin{pf} Let $F$ be a fundamental domain for the action of $G$ on
$\Ll$. If $u$ is a configuration, we can apply one of the operators
${\cal R}$ so that $u(p)\in [0,1]$ for $p\in F$. If furthermore $u$ is a
Birkhoff configuration, then
$$\sigma(\gamma) \leq u(\gamma p)-u(p) \leq \sigma(\gamma) +1$$ Hence
we can view ${\cal B}_\sigma/{\cal R}$ as a subspace of an infinite
product of circles. It is also closed, hence compact.
\end{pf}
Note that the potential $H_F$ induces a map on this quotient space
because of the invariance of the interactions $H_B$ under the
transformations ${\cal R}$.
Since $H_F$ is bounded below in this compact space, and it is
non-increasing along the orbits of the flows, there must be a sequence
$t_n\rightarrow \infty$ for which
$$\frac{d}{dt}\phi_F(u,t_n) \rightarrow 0.$$ Therefore, if we start
with a Birkhoff configuration $u$, we obtain a sequence of Birkhoff
configurations $\phi(u,t_n)$ which is compact in the space $\cal
B_\sigma/{\cal R}$. Thus there is a convergent subsequence whose limit
is a critical point $u_F$.
Consider now an increasing sequence $F_0\subset F_1 \subset \cdots$ of
finite sets exhausting $\Ll$. For each of then we have a critical
point $u_n$ in ${\cal B}_\sigma$ for the gradient flow $\nabla
H_n$. After modification by the operators ${\cal R}$, which preserves
the property of being a critical point, we see that these sequence of
points has a subsequence which converges point-wise to $u$, a
configuration in ${\cal B}_\sigma$.
To finish the proof of the theorem we only need the following:
\begin{prop} This configuration $u$
satisfies the variational equations $V(u)=0$.
\end{prop}
\begin{pf} We have to check that $V(u)(p)=0$ for any $p$ in
$\Ll$. Fix $p$ and choose $n$ large enough so that if $B$ contains $p$
then $B\subset F_n$. The convergence $u_n \to u$ is uniform on $F_n$,
and $\nabla H_n(w)(p)=V(w)(p)$. It follows that $u$ satisfies the
variational equations.
\end{pf}
\section{Properties of ground state configurations}
For a configuration $u$ and a point $p$ in $\Ll$ define the energy of
$u$ due to $p$ as
$$ E_p(u)= \sum_{B\ni p}\frac{1}{|B|} H_B(u).$$ For $X$ a finite
subset of $\Ll$ let
$$h_X(u) = \frac{1}{|X|} \sum_{p\in X}E_p(u).$$ We would like to take
the limit as $X$ approaches $\Ll$, but there is no canonical way to do
so. We will regard it as a function on finite parts of $\Ll$,
$X\in{\cal F}(\Ll) \mapsto h_X(u)$.
One possibility would be to take a fundamental domain $X$ for $G$,
list the elements of $G$ as $\{e,\gamma_1,\gamma_2,\ldots\}$ and set
$X_n=X\cup\ldots\cup \gamma_nX$. Then, if $u$ is a configuration which
has integer periods with respect to a finite index subgroup $G'$ of $G$, we
see that the limit
$$\lim_{n\to\infty} h_{X_n}(u)$$ exists and is equal to the specific
energy of $u$ in a fundamental domain for $G'$.
We want to mention one property about the specific energy of
configurations. In \cite{2} this property is proved under the
assumption that the configuration is minimal and satisfies the
Birkhoff condition. It appears that considerably less is needed. We
include a proof for completeness.
\begin{prop} Let the interaction potential $H$ be of finite
range, $G$-invariant and satisfy the strong ferromagnetic
condition. Let $\sigma$ be a cocycle on $G$. If $u$ is a configuration
which belongs to ${\cal O}_\sigma$ then there is a constant
$K(u,\sigma)$ such that
$$ H(u,X)\leq K(\sigma,u)\,\text{\rm Card}(X) ,$$ for any finite
subset $X$ of $\Ll$.
\end{prop}
\begin{pf} Let $u$ be a configuration in ${\cal O}_\sigma$. Let
$u_\sigma$ be a configuration defined by the cocycle
$\sigma$. Applying the ferromagnetic hypothesis in the development
provided by Taylors theorem we have that for any $B$ with $H_B\neq 0$
there are positive constants $\alpha(B)$, $\beta(B)$ such that
$$H_B(u)\leq \alpha(B)\sum_{p,q\in B} (u(p)-u(q))^2 +\beta(B).$$ This
follows from Taylor's theorem. The constants $\alpha$ and $\beta$
depend on $\sigma$, on the distance between $u$ and $u_\sigma$, and on
$B$. There are only finitely many $B$'s up to translation, so the
invariance of $H$ allows us to choose $\alpha$ and $\beta$
independently of $B$.
Next note that for any $p$, $q$ and for $u$ in ${\cal O}_\sigma$ we have
previously shown that
$$|u(p)-u(q)|\leq C+|\sigma(\gamma_{pq})|,$$ where $C$ is a constant
and $\gamma$ is any transformation that takes $p$ to $q$. Therefore,
if the interaction potential is of bounded range, there is a constant
$C$ such that
$$|u(p)-u(q)|\leq C$$ for any $p,q$ belonging to the same set $B$ with
$H_B\neq 0$.
We now put all this information together. Let $X$ be a finite set and
$u\in {\cal O}_\sigma$. We have
\begin{eqnarray*}
H(u,X) & =& \sum_{B\cap X \neq\emptyset} H_B(u)\\
& \leq & \sum_B \sum_{p,q\in B}\alpha (u(p)-u(q))^2 +\beta \\ & \leq & \sum_B
n_B^2( C^2 +\beta)\\
& \leq& K(u,\sigma) \text{Card}(X)
\end{eqnarray*}
\end{pf}
Aubry's fundamental lemma for the Frenkel-Kontorova model shows the
non-intersecting property of ground state configurations. The analysis
of Aubry and Mather of configurations on the line which are minimal
for the Lagrangian shows that they satisfy a non-intersecting
property, namely, if $u:{\mathbb Z} \rightarrow {\mathbb R}$ is a ground state
configuration, then it is Birkhoff.
In higher dimensions this need not be true, and Blank \cite{2}
provides the following example:
$$H(u_1,u_2,u_3)= (u_1-u_2+b_1)^2 +(u_1-u_3+b_2)^2$$ which admits
$f(x_1,x_2)=x_1x_2$ as a solution to the variational equations.
Blank showed that a similar property holds for minimum energy
configurations for the model he studies in \cite{2}. The following
proposition exhibits a similar property for the models we study. From
a geometric point of view, it is essentially the maximum principle
for minimal surfaces.
The space $\Ll$ has no structure, but given an interaction potential
$H=\{H_B\}$, those sets $B$ for which $H_B\neq 0$ define a sort of
topology on $\Ll$. So if $X$ is a finite subset of $\Ll$, we will
denote by $N(X)$ the union of those sets $B$ which intersect $X$ and
with $H_B\neq 0$, and by $X^0$ the subset of those elements $p$ of $X$
for which there is some $B$ which meets $X$ in some other points
different from $p$. Although these definitions may look strange, they
are simply verified for models whose supporting set $\Ll$ is
corresponds to the vertices of a graph and the interaction potential
is defined in terms of the connecting edges.
First we need some new terminology. Say that a subset $X$ of $\Ll$ is
connected if for any pair of elements $p$, $q$ of it, there is a
sequence $p_0,\ldots,p_k$ in $X$ with $p_0=p$, $p_k=q$, and sets
$B_i$, each containing a pair $p_i,p_{i+1}$, and for which
$H_{B_i}\neq 0$.
The meaning of this condition is obvious. Each connected component of
$\Lambda$ behaves in its own independent way, and there is no
restriction in assuming our models have connected state space.
\begin{prop} Let $u$ and $v$ be ground state configurations
for the ferromagnetic potential $H$ on the configuration space of
$\Ll$. Suppose that there is a finite connected set $X$ such that
$u\geq v$ on $N(X)$. Then, either $u=v$ or $u>v$ in all of $X$.
\end{prop}
\begin{pf} We use the standard technique of the Hilbert integral in
calculus of variations. Any ground state configuration must satisfy
the variational equations. Thus, if $p\in X$ we get
$$\sum_{B\ni p} \frac{\partial}{\partial p}[H_B(u)-H_B(v)]=0$$ and
replacing the terms in the sum by their integrals,
$$\sum_{B\ni p} \int_0^1 \frac{d}{dt}\frac{\partial}{\partial
p}H_B(tu+(1-t)v)dt=0.$$ This expression may be written as:
$$\sum_{B\ni p}\sum_{q\in B}[u(q)-v(q)] \int_0^1
\frac{\partial^2}{\partial q\partial p}H_B(tu+(1-t)v) dt=0.$$ Let
$p\in X^0$ be such that $u(p)=v(p)$. For $q\neq p$ in $X^0$, the term
$u(q)-v(q)$ in the sum above is $\geq 0$ by hypothesis, while the
integral term is strictly negative due to the ferromagnetic
condition. This forces $u=v$ through all of $N(p)$. Now the fact that
$X$ is connected allows us to extend the equality $u=v$ to all $X$.
\end{pf}
\begin{cor} Let $u$, $v$ be two critical configurations on
$\Ll$, which is assumed to be connected. If $u\leq v$, then either
$u=v$ or $uv$.
\end{prop}
\begin{pf} The proof is essentially the same as the one in the
previous section. First start with a point in $F_n$ which is
equivalent to no point of $N(F_n)$. If $u=v$ at this point, the proof
in the previous section shows the $u=v$ in the whole neighborhood of
the point.
For points of $F_n$ that have some equivalent in $N(F_n)$, the last
equation we have written allows us to use again the Hilbert integral
technique.
In conclusion, if $u(p)=v(p)$ at some point of $F_n$, we obtain that
$u=v$ through all of $F_n$, as we are assuming the fundamental domains
to be connected.
\end{pf}
Hence, if $u$, $v$ are periodic and minimize $H_n$ we can consider the
configurations $u\lor v=\max\{u,v\}$ and $u\land v=\min\{u,v\}$ which
are also periodic, and the twist condition implies
$$H_n(u)+H_n(v) \geq H_n(u\lor v)+H_n(u\land v)$$ Hence these
configurations also minimize $H_n$. By applying the argument of the
previous paragraph to the pairs $u$, $u\lor v$, etcetera, we conclude
that either $u>v$ or $v>u$ or they are equal.
Before proceeding we note the following fact.
\begin{lemma} If $u$
is a $G_n$-periodic configuration, so is ${\cal T}_\gamma u$ for any
$\gamma$ in $G$.
\end{lemma}
\begin{pf} If $\gamma_1 \in G_n$, then by normality of $G_n$ in $G$,
there is $\gamma_2$ in $G_n$ such that $\gamma \gamma_1=\gamma_2
\gamma$. Hence $\sigma(\gamma_1)=\sigma(\gamma_2)$ and
$${\cal T}_\gamma u(\gamma_1 p) = u(\gamma \gamma_1 p)=u(\gamma_2 \gamma
p)= u(\gamma p)+\sigma(\gamma_2).$$
\end{pf}
%>From this the Birkhoff property follows as we now show.
>From this the Birkhoff property follows as we now show.
\begin{prop} Suppose that $u$ is $G_n$-periodic and
minimizes $H_n$. Then it is a Birkhoff configuration.
\end{prop}
\begin{pf} We show that for any $\gamma$ in $G$ we have
$$|{\cal T}_\gamma u -u -\sigma(\gamma)|\leq 1.$$ If $[ \sigma(\gamma)]$
denotes the integer part of $\sigma(\gamma)$, it follows from the
previous lemma that the configurations $\overline{u}=T_\gamma
u-[\sigma(\gamma)]$ and $\underline{u}=\overline{u}-1$ are also
$G_n$-periodic and minimize $H_n$. Hence the differences $\overline{u}
-u$ and $\underline{u} -u$ have constant sign. Clearly these
differences are invariant by $G_n$, and so we can determine their sign
by adding over a fundamental domain for $G_n$.
To do this, observe that for a $G_n$-invariant configuration $v$ we
have
$$\sum_{F_n} {\cal T}_\gamma v - v = \sigma(\gamma) |F_n|$$ and in fact
the sum can be taken over any fundamental domain for $G_n$. Hence
$$\sum_{F_n} \overline{u}- u = (\sigma(\gamma) -[\sigma(\gamma)]
)|F_n|\geq 0,$$ and similarly
$$\sum_{F_n} \underline{u} - u = (\sigma(\gamma) -[\sigma(\gamma)] -1)|F_n|\leq 0.$$
It follows that $0\leq {\cal T}_\gamma u -u -[\sigma(\gamma)]\leq 1$,
which is equivalent to the Birkhoff property.
\end{pf}
This discussion has the following consequence. We start with a tower
of connected fundamental domains $F=F_0 \subset \cdots\subset F_n
\subset \cdots$ corresponding to normal finite index subgroups of
$G_0\supset G_1\supset \ldots$ of $G$, then we obtain a sequence of configurations $u_0,
u_1, \ldots$, where $u_n$ is periodic with respect to $G_n$ and
minimizes $H_n$. Furthermore,
since these configurations satisfy the Birkhoff property and
the cocycle is integer valued on $G_0$, they are all
periodic with respect to $G_0$. By invoking the operators ${\cal R}$, we
may assume that this sequence $\{u_n\}$ has a subsequence which
converges pointwise to a configuration $u$. Hence $u$ is a Birkhoff
configuration. Furthermore,
\begin{prop} The configuration $u$ is a ground state for
$H$.
\end{prop}
Indeed, let $v$ be a configuration that agrees with $u$ outside a
finite subset $X$ of $\Ll$. Then $X\subset F_n$ for some large $n$. We
can find a larger $m$ such that every set $B$ which meets $F_n$ is
inside $F_m$. Now restrict $v$ to $F_m$ and extend to $\Ll$ by
periodicity with respect to $G_m$. This does not change the value of
$H(v,F_n)$. Then, for $k\geq n$,
$$H(v,F_n) \geq H(u_k,F_n).$$ Since $u_n$ converges uniformly to $u$
on $F_n$, we take the limit with respect $k$ and the ground state
property of $u$ is verified.
The case in which the cocycle $\sigma$ is arbitrary is now
immediate. We approximate $\sigma$ by rational cocycles. For each one
the argument just described produces a Birkhoff ground state. By using
the operators ${\cal R}$, we may assume this sequence converges. The
limit is then a Birkhoff ground state for $\sigma$.
\section{Examples and problems}
The typical example is the following. Let $G$ be a group with a finite
generating set $S=S^{-1}$. With it we construct a graph whose vertices
are the elements of $G$ and whose edges are labeled by the elements of
$S$, that is, there is an edge from $g$ to $h$ if $h=gs$, and we
denote it by $(g,s,gs)$.
This graph is made into a metric space by taking each edge to be
isometric to the unit interval in the real line.
Furthermore, there is a natural left action of $G$ on $\Gamma$, namely, $
g(v,s,vs)=(gv,s,gvs)$, which is effective and transitive.
There is a natural potential $H$ defined by
$$H_B(u)=(u(p)-u(q))^2$$ if $p$ and $q$ are connected by an edge, and
trivial otherwise. The ground state configurations of this model are
harmonic functions on the group $G$, where the discrete Laplacian is
defined by
$$\Delta u(p) =\frac{1}{n(p)} \sum_{q\sim p}u(q)-u(p),$$ where $n(p)$
is the number of edges emanating from $p$, and $q\sim p$ means that
$q$ is connected to $p$ by an edge.
It is then elementary that any cocycle on $G$ defines a harmonic
function. This corresponds to the integrable case.
The Frenkel-Kontorova model in the group $G$ involves also the site
potentials
$$H_p(u)=V(u(p))$$ where $V$ is a periodic function satisfying
$V''<0$. The variational equations are
$$ n(g)u(g)-\sum_{s\in S} u(gs) +V'(u(g))=0.$$ Most of what was done
for ${\mathbb Z}$ carries over to this general situation.
In physics there are two types of lattices which are frequently
used. One is the Euclidean lattice ${\mathbb Z}^d$, which is amenable, and
the other goes by the name of Bethe lattice, non-amenable. This one is
typically used to exemplify strange phenomena in comparison with the
Euclidean lattices (and non-amenability is responsible for this), and
also because computations for models on it are simplified by the fact
that Bethe lattices are trees.
%>
>From our point of view, these lattices are the graphs of free products
of groups of the form $G_1\ast\cdots\ast G_n$, where each $G_i$ is a
copy of the infinite cyclic group ${\mathbb Z}$ or of the two element group
${\mathbb Z}/2{\mathbb Z}$. They are finitely generated groups and it is
elementary how to compute their first cohomology group. Therefore our
theory applies to them.
To conclude, we mention some problems that appear to have some
interest. Of course, the theorem of
existence of quasi-periodic solutions is
only the first step in the very rich
Aubry-Mather theory and it wold be interesting to
see how much generalizes to the present setting.
On the other hand, there are problems that seem to be suggested
by the present formulation.
The first concerns Gibbs states for the model $\Ll$, $H$,
$G$. It would be interesting to know what type of ground states are
contained in the support of a Gibbs state. For instance, under what
conditions is it true that the Dirac mass supported on a Birkhoff
ground state is a Gibbs state? Examples related to this question are
the Shlosman staircases (see \cite{3}).
In a more geometrical vein, it would be interesting to develop a
theory of minimal surfaces in Riemannian manifolds corresponding to
the relation between the Frenkel-Kontorova model and geodesics on a
torus. For manifolds of dimension three the theory of minimal surfaces
admits a simplicial version (Jaco and Rubinstein). Given a compact
three manifold $M$ with a triangulation, a simplicial minimal surface
is described by how is should intersect the simplices of the
triangulation. Passing to the universal cover of $M$ we obtain a space
$\Ll$, namely, the set of simplices, as well as an action of the
fundamental group of $M$ on $\Ll$. An interaction potential can be
written down in terms of the gluing data, so that solutions to the
corresponding variational equations are simplicial minimal surfaces.
We also note that, given the interpretation of $S$ as energy and
of the rotations numbers as density, it is interesting physically
to study the function that, to a rotation number associates
the energy per particle of the minimal configuration.
This has been studied when $\Ll = {{\mathbb Z}}^d$
numerically in \cite{9} and rigorously in \cite{11}.
We also call attention to the models of quasi-crystals based on
aperiodic tilings \cite{10}. These have actions of semi-groups that
correspond to dilations. It would be interesting to know whether
some version of Aubry-Mather theory could be developed for them.
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\end{document}