%
%The original version of these macros is due to J.P. Eckmann
%
\magnification \magstep1
\vsize=22 truecm
\hsize=16 truecm
\hoffset=0.8 truecm
\normalbaselineskip=5.25mm
\baselineskip=5.25mm
\parskip=10pt
\immediate\openout1=key
\font\titlefont=cmbx10 scaled\magstep1
\font\authorfont=cmcsc10
\font\footfont=cmr7
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
\font\smaller=cmr5
%%%%%constant subscript positions%%%%%
\fontdimen16\tensy=2.7pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=2.7pt
%%%%%%%%%%%%%%%%%%%%%%
%%% macros %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\def\dowrite #1{\immediate\write16 {#1} \immediate\write1 {#1} }
%\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
\newcount\EQNcount \EQNcount=1
\newcount\SECTIONcount \SECTIONcount=0
\newcount\APPENDIXcount \APPENDIXcount=0
\newcount\CLAIMcount \CLAIMcount=1
\newcount\SUBSECTIONcount \SUBSECTIONcount=1
\def\SECTIONHEAD{X}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\QED{\hfill\smallskip
\line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
\ \ \ \ \ \ }
\bigskip}
% These ones cannot be used in amstex
%
\def\real{{\bf R}}
\def\rational{{\bf Q}}
\def\natural{{\bf N}}
\def\complex{{\bf C}}
\def\integer{{\bf Z}}
\def\torus{{\bf T}}
%
% These ones can only be used in amstex
%
%\def\real{{\Bbb R}}
%\def\rational{{\Bbb Q}}
%\def\natural{{\Bbb N}}
%\def\complex{{\Bbb C}}
%\def\integer{{\Bbb Z}}
%\def\torus{{\Bbb T}}
%
%
%
\def\Re{{\rm Re\,}}
\def\Im{{\rm Im\,}}
\def\PROOF{\medskip\noindent{\bf Proof.\ }}
\def\REMARK{\medskip\noindent{\bf Remark.\ }}
\def\NOTATION{\medskip\noindent{\bf Notation.\ }}
\def\PRUEBA{\medskip\noindent{\bf Demostraci\'on.\ }}
\def\NOTA{\medskip\noindent{\bf Nota.\ }}
\def\NOTACION{\medskip\noindent{\bf Notaci\'on.\ }}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1 \dowrite{undefined equation #1}
\else\csname e#1\endcsname\fi}
\def\clm(#1){\ifundefined{c#1}$\clubsuit$#1 \dowrite{undefined claim #1}
\else\csname c#1\endcsname\fi}
\def\EQ(#1){\leqno\JPtag(#1)}
\def\NR(#1){&\JPtag(#1)\cr} %the same as &\tag(xx)\cr in eqalignno
\def\JPtag(#1){(\SECTIONHEAD.
\number\EQNcount)
\expandafter\xdef\csname
e#1\endcsname{(\SECTIONHEAD.\number\EQNcount)}
\dowrite{ EQ \equ(#1):#1 }
\global\advance\EQNcount by 1
}
\def\CLAIM #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~\SECTIONHEAD.\number\CLAIMcount.} {\sl #3}\par
\expandafter\xdef\csname c#2\endcsname{#1\
\SECTIONHEAD.\number\CLAIMcount}
%\immediate \write16{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
%\immediate \write1{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
\dowrite{ CLAIM #1 (\SECTIONHEAD.\number\CLAIMcount) :#2}
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\CLAIMNONR #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~#2} {\sl #3}\par
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\SECTION#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\def\SECTIONHEAD{\number\SECTIONcount}
\immediate\dowrite{ SECTION \SECTIONHEAD:#1}\leftline
{\sectionfont \SECTIONHEAD.\ #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\APPENDIX#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\def\SECTIONHEAD{\ifcase \number\APPENDIXcount X\or A\or B\or C\or D\or E\or F \fi}
\global\advance\APPENDIXcount by 1
\vfill \eject
\immediate\dowrite{ APPENDIX \SECTIONHEAD:#1}\leftline
{\titlefont APPENDIX \SECTIONHEAD: }
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SECTIONNONR#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\immediate\dowrite{SECTION:#1}\leftline
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SUBSECTION#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\def\SUBSECTIONHEAD{\number\SUBSECTIONcount}
\immediate\dowrite{ SUBSECTION \SECTIONHEAD.\SUBSECTIONHEAD :#1}\leftline
{\subsectionfont
\SECTIONHEAD.\number\SUBSECTIONcount.\ #1}
\global\advance\SUBSECTIONcount by 1
\nobreak\smallskip\noindent}
\def\SUBSECTIONNONR#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\immediate\dowrite{SUBSECTION:#1}\leftline{\subsectionfont
#1}
\nobreak\smallskip\noindent}
%%%%%%%%%%%%%TITLE PAGE%%%%%%%%%%%%%%%%%%%%
\let\endarg=\par
\def\finish{\def\endarg{\par\endgroup}}
\def\start{\endarg\begingroup}
\def\getNORMAL#1{{#1}}
\def\TITLE{\beginTITLE\getTITLE}
\def\beginTITLE{\start
\titlefont\baselineskip=1.728
\normalbaselineskip\rightskip=0pt plus1fil
\noindent
\def\endarg{\par\vskip.35in\endgroup}}
\def\getTITLE{\getNORMAL}
\def\AUTHOR{\beginAUTHOR\getAUTHOR}
\def\beginAUTHOR{\start
\vskip .25in\rm\noindent\finish}
\def\getAUTHOR{\getNORMAL}
\def\FROM{\beginFROM\getFROM}
\def\beginFROM{\start\baselineskip=3.0mm\normalbaselineskip=3.0mm
\obeylines\sl\finish}
\def\getFROM{\getNORMAL}
\def\ENDTITLE{\endarg}
\def\ABSTRACT#1\par{
\vskip 1in {\noindent\sectionfont Abstract.} #1 \par}
\def\ENDABSTRACT{\vfill\break}
\def\TODAY{\number\day~\ifcase\month\or January \or February \or March \or
April \or May \or June
\or July \or August \or September \or October \or November \or December \fi
\number\year}
\newcount\timecount
\timecount=\number\time
\divide\timecount by 60
\def\DRAFT{\font\footfont=cmti7
\footline={{\footfont \hfil File:\jobname, \TODAY, \number\timecount h}}
}
%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\period{\unskip.\spacefactor3000 { }}
%
% ...invisible stuff
%
\newbox\noboxJPE
\newbox\byboxJPE
\newbox\paperboxJPE
\newbox\yrboxJPE
\newbox\jourboxJPE
\newbox\pagesboxJPE
\newbox\volboxJPE
\newbox\preprintboxJPE
\newbox\toappearboxJPE
\newbox\bookboxJPE
\newbox\bybookboxJPE
\newbox\publisherboxJPE
\def\refclearJPE{
\setbox\noboxJPE=\null \gdef\isnoJPE{F}
\setbox\byboxJPE=\null \gdef\isbyJPE{F}
\setbox\paperboxJPE=\null \gdef\ispaperJPE{F}
\setbox\yrboxJPE=\null \gdef\isyrJPE{F}
\setbox\jourboxJPE=\null \gdef\isjourJPE{F}
\setbox\pagesboxJPE=\null \gdef\ispagesJPE{F}
\setbox\volboxJPE=\null \gdef\isvolJPE{F}
\setbox\preprintboxJPE=\null \gdef\ispreprintJPE{F}
\setbox\toappearboxJPE=\null \gdef\istoappearJPE{F}
\setbox\bookboxJPE=\null \gdef\isbookJPE{F} \gdef\isinbookJPE{F}
\setbox\bybookboxJPE=\null \gdef\isbybookJPE{F}
\setbox\publisherboxJPE=\null \gdef\ispublisherJPE{F}
}
\def\ref{\refclearJPE\bgroup}
\def\no {\egroup\gdef\isnoJPE{T}\setbox\noboxJPE=\hbox\bgroup}
\def\by {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
\def\paper{\egroup\gdef\ispaperJPE{T}\setbox\paperboxJPE=\hbox\bgroup}
\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
\def\vol{\egroup\gdef\isvolJPE{T}\setbox\volboxJPE=\hbox\bgroup\bf}
\def\preprint{\egroup\gdef
\ispreprintJPE{T}\setbox\preprintboxJPE=\hbox\bgroup}
\def\toappear{\egroup\gdef
\istoappearJPE{T}\setbox\toappearboxJPE=\hbox\bgroup}
\def\book{\egroup\gdef\isbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\publisher{\egroup\gdef
\ispublisherJPE{T}\setbox\publisherboxJPE=\hbox\bgroup}
\def\inbook{\egroup\gdef\isinbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\bybook{\egroup\gdef\isbybookJPE{T}\setbox\bybookboxJPE=\hbox\bgroup}
\def\endref{\egroup \sfcode`.=1000
\if T\isnoJPE \item{[\unhbox\noboxJPE\unskip]}
\else \item{} \fi
\if T\isbyJPE \unhbox\byboxJPE\unskip: \fi
\if T\ispaperJPE \unhbox\paperboxJPE\unskip\period \fi
\if T\isbookJPE ``\unhbox\bookboxJPE\unskip''\if T\ispublisherJPE, \else.
\fi\fi
\if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE,
\else\period \fi\fi
\if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi
\if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if
T\isyrJPE \ \else\period \fi\fi\fi
\if T\istoappearJPE (To appear)\period \fi
\if T\ispreprintJPE Preprint\period \fi
\if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi
\if T\isvolJPE \unhbox\volboxJPE\unskip, \fi
\if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi
\if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def\cite#1{ [#1]}
\def\bfx{{\bf x}}
\def\bfn{{\bf n}}
\def\bfe{{\bf e}}
\def\bfu{{\bf u}}
\def\SS{{\cal S}}
\def\RR{{\cal R}}
\def\OO{{\cal O}}
\def\BB{{\cal B}}
\def\CC{{\cal C}}
\def\Tau{{\cal T}}
\def\FF{{\cal F}}
\def\tG{{\tilde G}}
\TITLE AUBRY--MATHER THEORY FOR FUNCTIONS ON LATTICES.
\footnote{${}^{\rm (1)}$ } {\rm \baselineskip = 12 pt
This preprint is available from the
math-physics electronic preprints archive.
Send e-mail to {\tt mp\_arc@math.utexas.edu} for instructions}
\AUTHOR Hans Koch
\footnote{${}^2$}{ e-mail address {\tt koch@math.utexas.edu}},
Rafael de la Llave
\footnote{${}^3$}{ e-mail address {\tt llave@math.utexas.edu}},
Charles Radin
\footnote{${}^4$}{ e-mail address {\tt radin@math.utexas.edu} }
\FROM Mathematics Department
The University of Texas at Austin
Austin TX\ \ 78712
\ENDTITLE
\ABSTRACT
We generalize the Aubry-Mather theorem
on the existence of quasi-periodic
solutions of one dimensional
difference equations
to situations in which the
independent variable ranges
over more complicated lattices.
This is a natural generalization
of Frenkel-Kontorovna models to
physical situations in a
higher dimensional space.
We also consider generalizations
in which the interactions among the
particles are not just nearest neighbor,
and indeed do not have
finite range.
\ENDABSTRACT
\SECTION Introduction.
The goal of this paper is to generalize the Aubry-Mather theorem
on the existence of quasi-periodic solutions of
certain difference equations to cases in
which the unknown depends not just on one, but
on several variables by modifying a method used
in [Go] in the one-dimensional case.
Such generalizations are
natural from the point of view of the
solid state motivations of the theorem.
For example, the Frenkel-Kontorovna model considered by Aubry and Mather
describes configurations which are one-dimensional
chains of particles
in a periodic external potential
and interacting with their nearest neighbors
via a harmonic potential.
Such models
are one-dimensional caricatures
of the physical situation of a
layer of a material over a substratum of other material.
If we denote by $V(x) = V(x+1)$ the periodic potential
and by $u_i$ the displacement of the $i^{\rm th}$
particle from the its equilibrium position,
we seek
configurations that are critical points of the
energy:
$$
\SS(u) = \sum_{i\in\integer} {1 \over 2} (u_i -u_{i+1} -a)^2 + V(u_i)
\EQ(ActionFK)
$$
(We also point out that there are other
physical interpretations. For example,
we could have atoms whose state is described
by an internal variable $u_i$.
The internal energy is periodic but there is a
coupling between nearest variable. For example,
the model \equ(ActionFK) has been used as models
of spin waves. In that interpretation,
$V(u_i)$ would be the magnetic energy of
the $i^{\rm th}$ atom and the term
${1 \over 2} (u_i -u_{i+1})^2$ -- or a modification --
would be the exchange interaction between neighboring spins.)
Even if the sum on the R.~H.~S. of \equ(ActionFK)
is only formal, the variational
equations are quite well defined, namely:
$$
u_{i+1} + u_{i-1} - 2 u_i + V'(u_i) = 0
\EQ(variationFK)
$$
which are equivalent to the well known standard-like mappings
of Hamiltonian mechanics.
The Aubry-Mather theorem establishes, among other things, that
for every $\omega \in \real$ the variational
equations \equ(variationFK) admit a solution
$u$ such that $\sup_i |u_i - \omega i| < \infty$.
In this paper we will be concerned with
generalizing this theorem to situations in which:
\item{$\bullet$} The physical space is higher dimensional.
\item{$\bullet$} The interactions are not necessarily just nearest neighbor.
\item{$\bullet$} The interactions are invariant under a smaller symmetry group than the
full lattice.
An example to keep in mind of a model to which our results apply
is the $n$-dimensional analogue
of the Frenkel-Kontorovna model,
described by the action
$$
\SS(u) = \sum_{i \in \integer^n} {1 \over 2 n}\sum_{j:|j-i| =1}|u_i -u_j|^2
+V(u_i)
$$
which leads to the variational equations:
$$
(\Delta u)_i + V'(u_i) =0
$$
where $\Delta$ denotes the discrete Laplacian.
Such models for $n=2$ have been considered in [V],
where one can also find an extensive discussion
of properties of solutions,
and physical consequences.
Another generalization which we will be able to deal
with is
$$
\SS(u) = \sum_{i \in \integer^n} {1 \over 2 n}\sum_{j:|j-i| =1}|u_i -u_j|^2
+V(u_i,i)
$$
where $V(x,i)$ is periodic in $i$ and similar
generalizations. In the case $n = 1$, this admits the
dynamic interpretation of a composition of finite number of
standard maps or, alternatively, as a map in a higher
dimensional space. These generalizations, however do
not seem to include higher dimensional
standard maps.
The method of proof we will use is motivated by a recent
paper of Gol\'e [Go]
which presents a new proof of the classical
Aubry-Mather theorem for one dimensional Frenkel-Kontorovna models.
We point out that it is also possible using the methods
employed here to give a simple proof
of generalizations of a theorem of
Moser [Mo] on partial differential
equations. Since the latter results
seem to require other ingredients from
P.D.E., and have different motivation, we will
report on them elsewhere.
The use of heat flow methods in Aubry-Mather theory
was introduced in the paper \cite{An}, but the methods
we use here are more elementary.
We also point out that results very related to those
of this paper, can be found in [Bl1], [Bl2] by using
quite different methods.
We also note that the papers
[Bl1] and [Bl2] look for minimal solutions.
Following the lead of [Mo], [An] and [Go], we look for solutions
of the variational problem that are well ordered.
This makes the estimates at infinity much simpler.
In this respect, it is interesting to point out that
in higher dimensional ambient spaces
there are examples in [Bl2] of
minimal solutions which are not well ordered.
These phenomena have been observed numerically in [OV].
\SECTION Acknowledgments
We thank C. Gol\'e for acquainting us with [Go]
prior to publication, and for his encouragement.
He and F. Tangerman made very useful comments on
a preliminary version of this paper.
We also thank L. M. Floria for making us aware of
[V] and providing us with a copy and J. Moser for
several discussions and making us aware of [Bl1], [Bl2].
The authors have been supported by NSF grants.
R.L. also acknowledges the support of the
AMS Centennial Fellowship and a URI from
U. Texas during the last revision of this paper.
\SECTION Notation and statement of results.
We will be concerned with functions defined on
lattices (i.e. discrete subgroups of $\real^d$ with its additive structure).
We will find it useful to distinguish between the
lattices as subsets of $\real^d$ and as
groups of motions.
(For example, we will find it useful to consider
smaller groups of motions that the full lattice.)
Hence, we will use the
word crystals when we want to emphasize the
fact that the group structure is irrelevant
We will also refer to functions on
lattices as configurations.
This is motivated by the physical interpretation of
the models in \equ(ActionFK) for which
the $i$ represent labels of
atoms, $u_i$ represent characteristics of each atom.
The function $u_i$ represents the state -- or configuration
of the crystal --
and the action in \equ(ActionFK) is the energy associated
to the configuration.
As it is well known, discrete subgroups of
$\real^d$ can be identified with
sets of the form
$\{ x \in \real^d\ \big| \ \ x = \sum_{i = 1}^n k_i v_i, k_i \in \integer\}$
for some linearly independent vectors $v_i$.
(Without loss of generality we will assume that $n = d$)
Hence, all crystals can be identified with $\integer^d$.
The identification is not unique, but we will assume it is
done and fixed through the proof. This allows us to define
the distance between two points in the crystal
using the absolute value of points in $\integer^d$
(The sum of the absolute value of the coordinates)
Again, this is a non-canonical choice and many
others would have worked, but we fix it for the sake
of definiteness.
We will denote by $\Gamma$ the points of a lattice
and denote
by $G$ a group of translations that
leaves invariant. Clearly, $G$ is a subgroup of the
lattice and indeed we will fins it useful to
consider the case where it is a non-trivial
subgroup.
If $k$ is an element of the lattice, we denote by
$T_k$ the translation by $k$ on the crystal.
That is $T_k x = x + k$.
For each subset $B$ of $\Gamma$
we denote by $|B|$ the number of elements of
$B$, and by $\partial_n B$ the set:
$$ \partial_n B \equiv \{ x \in \Gamma - B\ :\ {\rm dist}(x,B) \le n\} \cup
\{ x \in B\ :\ {\rm dist}(x,\Gamma - B) \le n\}$$
We write that $ \lim_n \Lambda_n = \Gamma$
if for any site $j \in \Gamma$
there is an $N(j)$ such that
$n > N(j)$ implies $j \in \Lambda_n$.
A particularly important role in the argument will
be played by collections of sets that cover the whole space
but are hierarchically ordered in sizes of different scales.
The sets of one scale can be decomposed into pieces
which are sets of the smaller scale.
(This is quite similar to the usual Calder\'on-Zygmund
decompositions of harmonic analysis)
For example, using the representation of
the crystal as $\integer^d$, we can consider
the collection $\CC$ of
finite subsets $\Lambda$ which are of the
form:
$$
\Lambda = [ k_1 -l_m, k_1 + l_m] \times
[ k_2 -l_m, k_2 + l_m] \times \cdots
[ k_n -l_m, k_n + l_m]
$$
for some $k_1,\ldots,k_n \in \integer$,
$l_m = (3^{m+1} -1)/2$.
In other words, cubes of sizes $l_m$ growing exponentially
and centered at any point.
It is important to note that since
$[k -l_m, k+l_m] =
[ (k -l_m +l_{m-1}) -l_{m-1}, (k -l_m +l_{m-1}) + l_{m-1}] \cup
[ (k -l_m + 3 l_{m-1} +1) -l_{m-1}, (k -l_m + 3 l_{m-1} +1 ) + l_{m-1}] \cup
[ (k -l_m + 5 l_{m-1} +2) -l_{m-1}, (k -l_m + 5 l_{m-1} +2 ) + l_{m-1}]
$ (Recall that $-l_m + 6 l_{m-1} +2 = l_m$.) we conclude that
the cubes of size $l_m$ can be divided into
$3^n$ disjoint cubes of size $l_{m-1}$.
We also note that as the sizes of the cubes grow, their
boundaries are negligible with respect to their volume.
This property will also play an important role.
We will record these in
a definition to make clear what
properties of the collection $\CC$ will be needed later. This may lead to
developing an Aubry-Mather theory in a more abstract setting.
We also note that very similar properties play a role in the
theory of the thermodynamic limit (See e.g \cite{Ru}).
\CLAIM Definition (inflation)
We say that a collection
$\CC$ of finite subsets of
a lattice $\Gamma$ admits
an inflation rule if:
\item{$i)$} If $\{ \Lambda_n\}_{n \in \natural} \subset \CC$
and $|\Lambda_n| \to \infty$, then there are translations
$k_n \in G$ such that
$\lim_n T_{k_n}(\Lambda_n) = \Gamma$.
\item{$ii)$} There exist $N, L \in \natural$ and $M \in \real$
such that
for any set $\Lambda \in \CC$ with $|\Lambda| > L$,
we can find $N$ mutually disjoint sets $\Lambda_1,\cdots,\Lambda_N \in \CC$,
$|\Lambda_i| > M$, such that
$\Lambda = \Lambda_1 \cup \Lambda_2,\cup \cdots, \cup \Lambda_N.$
\item{$iii)$} For every $n \in \natural$,
if we define
$\phi_n(m) \equiv \max\{|\partial_n \Lambda|\ :\ |\Lambda| \le m,\
\Lambda \in \CC\}$,
we have
$ \lim_{m\to \infty } \phi_n(m)/m^\alpha = 0$,
for any $\alpha > \log(M) / \log(N)$ for the
$N,M$ in $ii)$.
It is obvious that for the $M,N$ in $ii)$
we should have
$M \ge N$. The difference between $M$ and $N$ is
a measure of the difficulty of fitting sets of the same size in
$\CC$ together.
For example, if we consider the sets of cubes
with all possible translations, we can take $M=N$.
For the set consisting of up and down triangles
of size at least $S$
on a two dimensional hexagonal lattice we can take
$N = 4$ and $M = 4 + O(1/S)$.
For the above two cases it is easy to verify
condition $ii)$ with
$\alpha =1$, but nevertheless for
fixed $n$ and sufficiently large
sets, $|\partial_n \Lambda| \approx n |\delta_n \Lambda| \approx
n |\Lambda|^{1 - 1/\nu}|$
where $\nu$ is the dimension of the lattice.
Aubry-Mather theory will assert the existence of
real valued functions on the lattice satisfying certain
properties. Again, we point out that we
will use the physically motivated name
``configurations" often.
\CLAIM Definition(configuration)
A configuration is a map $u:\Gamma \mapsto \real$.
We define sums of configurations and their
multiplication by numbers in the usual way for maps.
Our next goal is to find a definition, analogous to that of Birkhoff orbits,
that
generalizes to higher dimensional situations. We observe that the
essential part of the definition of Birkhoff orbits
is that translations preserve order properties.
In what follows, we note that the group of translations that we consider
may be a subgroup of finite index of the full group of
translations of the lattice. To emphasize that, we will denote it by
$\tG$.
\CLAIM Definition (actions)
If we have a crystal $\Gamma$,
a subgroup $\tG$ -- of finite index --
of the group $G$ of translations of the lattice.
we will denote the action of $\tG$ acting on configurations
by $(\Tau_k u)_i = u_{i+k}$.
Similarly, for $\ell \in \integer$
we denote by $\RR_\ell$ the mapping
defined by $(\RR_\ell u)_k = u_k + \ell$.
Note that $\Tau_k$ are linear operators and the
$\RR_\ell$ are affine.
An immediate consequence of these definitions is:
\CLAIM Proposition (commute)
$\Tau_k \Tau_{k'} = \Tau_{k'}\Tau_k = \Tau_{k + k'}$,
$\RR_\ell \RR_{\ell'} = \RR_{\ell'}\RR_\ell = \RR_{\ell + \ell'}$,
and, for $k$ in ${\tilde G} $,
$\Tau_k \RR_\ell = \RR_\ell \Tau_k$.
\CLAIM Definition (order)
Given two configurations $u, u'$, we write
$u \le u'$ if $u_i \le u'_i \hbox{ for all } i \in \Gamma$, with
analogous definitions for any of the other comparison symbols $\ge,\
<$ or $>$.
Notice that these comparisons among configurations
are not total orders (except in the trivial case where $\Gamma$
consists of one element). That is, there are pairs
of configurations
which do not satisfy any of the comparisons.
This order works very nicely with all the
other elements of structure that we have introduced so
far (actions of translations, addition of configurations)
\CLAIM Proposition (orderpreserved)
If $u \le u'$, then for every $k \in \tG$ and
every $\ell \in \integer$ and for any
configuration $u''$
we have: $\Tau_k u \le \Tau_k u'$,
$\RR_\ell u \le \RR_\ell u'$, and
$u + u'' \le u' +u''$.
\CLAIM Definition (uniform)
Given a vector $\omega\in\real^n$,
we say that a configuration $u$ is
of type $\omega$
if
$$
\sup_{j \in \Gamma}|| u_j - \omega \cdot j|| < \infty
\EQ(newuniform)
$$
where we denote by $\omega \cdot j$ the usual scalar product.
We denote by $\OO_\omega$ the set of configurations of type
$\omega$.
We refer to the configuration $u^\omega$ defined
by $u^\omega_k = \omega \cdot k$ as the plane wave of
frequency $\omega$
We observe that $\Tau_k \RR_\ell (\OO_\omega) = \OO_\omega$.
Notice also that $\OO_\omega$ is an affine space modeled on
$
\ell^{\infty}( \Gamma, \real) \equiv \{ u:\Gamma \to \real\
: \ || u || \equiv \sup_{j \in \integer^n} |u_j| < \infty .\}
$
Part of the difficulties of the theory arise from the fact that
$\ell^\infty$ is a space ill suited for many
constructions in the calculus of variations.
Nevertheless, there is a natural way to restore some control at
$\infty$. The following is a definition that has played an important
role in dynamical systems.
\CLAIM Definition (Birkhofforbit)
Given a discrete subgroup $\tG$ of trasnlations
leving invariant a crystal $\Gamma$,
we say that a configuration
$u \in \OO_{\omega}$
is a Birkhoff configuration if
for every $k \in \tG$, every $\ell \in \integer$
we have either $ \Tau_k \RR_\ell u \ge u$
or
$ \Tau_k \RR_\ell u \le u$
We denote by $\BB_\omega$ the set of Birkhoff configurations
in $\OO_\omega$. We note that this set is not
empty (e.g. the plane wave $u^\omega$ belongs to it).
Note that the concept of Birkhoff depends on the
subgroup $\tG$ that we are considering.
We will omit this from the notation when it does not
lead to confusion. Usually in Aubry-Mather theory, one
just takes $\tG$ = $G$.
consider other models.
\CLAIM Proposition (orders)
Let $u \in \BB_\omega$. If for
some $k \in \tG$, $\ell \in \integer$,
$\omega \cdot k + \ell < 0$. Then
$\Tau_k \RR_\ell u \ge u$.
If
$\omega \cdot k + \ell > 0$ then
$\Tau_k \RR_\ell u \le u$.
(In other words, the choice of
sign in the Birkhoff definition is the same
as for the plane wave of frequency $\omega$)
\PROOF
We will only prove the first case.
Assume the hypothesis was true and the conclusion false.
By the Birkhoff property,
if $\Tau_k \RR_\ell u \ge u$ is false
then, we should have
$\Tau_k \RR_\ell u \le u$.
If $\Tau_k \RR_\ell u \le u$ then, for any
natural number $n$ we also
have
$
(\Tau_k \RR_\ell)^n u \le u
$.
If we compare with the plane wave solution, using that
$
(\Tau_k \RR_\ell)^n u^\omega = u + ( k \cdot \omega +\ell)n
$, we have using that $\Tau_k$ is linear and $\RR_\ell$
affine
$$
(\Tau_k \RR_\ell)^n u - (\Tau_k \RR_\ell)^n u^\omega =
(\Tau_k \RR_\ell)^n u - u + ( u - u^\omega) - n(k \cdot \omega + \ell)
\le (u -u^\omega) - n(k \cdot \omega + \ell).
$$
On the other hand
$
\inf_j
[(\Tau_k \RR_\ell)^n u]_j - [(\Tau_k \RR_\ell)^n u^\omega]_j =
\inf_j u_j - u^\omega_j
$
This is the desired contradiction.
\QED
\CLAIM Proposition (bounds)
For every
$k \in \tG$,
there exist $\ell^{-}, \ell^{+} $ such that
for every $u \in \OO_\omega$,
$ \RR_{\ell^{-}} u \le \Tau_k u \le \RR_{\ell^{-}} u $.
Actually, this is the main property of the set
of Birkhoff orbits that
we will use. Notice that its strength comes from the fact that
$\ell^{-}, \ell^{+}$ depend only on $k$ and not on the element of the set.
If $\omega \cdot k$ is not an integer, we can
get $\ell^{+}$ and $\ell^{-}$ to differ by $1$.
If $\omega \cdot k $ happens to be an integer,
we can get them to differ by $2$.
Another property of Birkhoff sets we will use
is:
\CLAIM Proposition (Rpreserves)
$\Tau_k \RR_\ell \BB_\omega$ $= \BB_\omega$, for $k \in \tG$.
From this we deduce that if we define an equivalence relation
among configurations
by $u \RR u' $ if and only if $ u = \RR_\ell u'$
for some $\ell$, it is possible to restrict this relation
to $\BB_\omega$ and hence speak
of $\BB_\omega/\RR$.
This set will play an important role.
The previous definitions involve only the geometry
of configurations. To obtain meaningful physical models
it is necessary to define interactions among the
particles.
The following definition is standard in statistical mechanics.
See e.g [Ru].
\CLAIM Definition (interaction)
An interaction is a collection of maps
$\{ H_B\ :\ B \subset \Gamma,\ B\ {\rm finite}\}$
that to each configuration $u$ associate a number $H_B(u)$
which
depends only on the restriction of $u$ to $B$.
\CLAIM Definition (invariances)
We will say that an interaction is invariant
under the group $\tG$ of translations
if for all the translations $k \in \tG$,
all $B$, and all configurations $u$,
$H_{T_k B}(\Tau_k u ) = H_B(u)$.
We will say that an interaction has periodic phase space if
for all $\ell \in \integer$, all $B$, and all
configurations $u$,
$$
H_B( \RR_\ell u) = H_B(u).
\EQ(perphase)
$$
One consequence of \equ(perphase) is that the
maps $H_B$ can be considered not just as maps
$\BB_\omega \to \real$
but rather as $\BB_\omega/\RR \to \real$.
Notice that the Frenkel-Kontorovna models correspond to
taking $H_{\{i\}}(u) = V(u_i)$,
$H_{\{i,j\}}(u) = {1 \over 2n } (u_i - u_j)^2 $ when $|i -j | = 1$, and all other
$H_B(u)$ equal to zero. They are
translation invariant and have periodic phase space.
The variational principle we will consider comes from the
formal action:
$$
\SS(u) = \sum_{B \subset \Gamma \atop B {\rm finite}} H_B(u)
\EQ(actionstatmech)
$$
The variational equations
corresponding to the formal variational principle above are:
$$
\sum_{B \ni i} {\partial \over \partial u_i } H_B( u ) = 0, \quad \hbox{for all } i \in \Gamma
\EQ(variationsstatmech)
$$
The equations \equ(variationsstatmech)
are well defined whenever the sums converge in a
sufficiently strong sense.
This motivates the following definitions.
\CLAIM Definition (rbounded)
We say that an interaction is
$r$-bounded on the configurations in the convex set $\OO$
of configurations
if:
$$
|| H||_{r} \equiv \sup_{u\in \OO} \sup_{j_1}\sum_{j_2, \cdots,j_r} \big| \sum_{B \ni j_1,\cdots j_r}
{\partial \over \partial u_{j_1} }\cdots {\partial \over \partial u_{j_r}} H_B(u)\big|
< \infty \EQ(rbounded) $$
\REMARK
Note that we have chosen the definition in such a way that
$||H||_r$ depends only on the derivatives of
$H$ of order exactly $r$. In particular, assuming that
a function is $r$-bounded does not afford any control
on derivatives of order less than $r$.
\REMARK
Note that the equations \equ(variationsstatmech) make perfectly
good sense when the interaction is $1$ bounded.
\REMARK
Notice that the property of being $1$ bounded depends on the set
of configurations we are considering. For example, if we
consider the Frenkel-Kontorovna model, it is
$1$ bounded on sets of the form
$\OO^K_\omega = \{ u\ :\ \sup_i | u_i - \omega \cdot i | \le K < \infty \}$
(Hence, it is $1$ bounded in $\BB_\omega$)
Nevertheless, it is not $1$ bounded on $\OO_\omega$.
We will still omit the dependence on the set
when it is obvious to which set we are referring.
Notice that by \clm(bounds) the set $\BB_\omega$ is
contained in $\OO^2_\omega$.
The reason for introducing these semi-norms
is that the variational equations can
be written as
$
\FF_i(u) = 0
$
where $ \FF_i(u) = \sum_{B \in i} {\partial \over \partial u_i} H_B(u)$.
In the cases that we will be interested in,
$\OO$ will be an affine space over $\ell^\infty$
and the conditions \equ(rbounded) are conditions that
ensure that the $r-1$ derivative of $\FF$ exists and is uniformly bounded
in the sense of derivatives in the Banach space $\ell^\infty$.
In particular we have:
\CLAIM Corollary (interpolation)
If we define $\FF$ as above,
we have $||\FF(u) - \FF(\tilde u)||_{\ell^\infty} \le ||H||_{2} ||u - \tilde u||_{\ell^\infty}$.
Finally, we will need an extra hypothesis
which is analogous to the twist condition in
Hamiltonian mechanics and to ferromagnetism in statistical mechanics.
\CLAIM Definition (twist)
We will say that an interaction which is $2$ bounded on $\OO$ satisfies
the twist condition -- or is ferromagnetic --
if for all configurations $u\in \OO$,
and all $j \ne j' \in \Gamma$,
$$
\sum_{B \ni j}
{\partial^2 \over {\partial u_j \partial u_{j'}}}
H_B(u) \ge 0
\EQ(twist)
$$
The property \equ(twist) is obviously implied by the stronger one:
$$
{\partial^2 \over \partial u_j \partial u_{
j'}} H_B(u) \ge 0
\EQ(strongtwist)
$$
for all $j \ne j' \in \Gamma$, $B \subset \Gamma$.
We refer to \equ(strongtwist) as the
strong twist condition or the strong ferromagnetic condition.
(Notice that the Frenkel-Kontorovna models satisfy \equ(strongtwist).)
\CLAIM Theorem (main)
Let $\Gamma$ be a crystal with group of translations $G$.
Let $H$ be an interaction invariant under the
group $\tG$ of finite index in $G$,
$2$ bounded on $\OO^3_\omega$.
Assume furthermore either:
\item{$a)$} $H$ satisfies the twist condition and is finite range.
\item{$b)$}$H$ satisfies the strong twist condition.
\vskip 0em
Then, there is a solution of \equ(variationsstatmech)
which lies in $\BB_\omega$.
The crux of the proof is to verify
the second part. Then, the first part is just a
very easy approximation argument.
We also note that the hypothesis of boundedness
of the flow can be weakened considerably.
See the remarks at the end of the proof for the details.
\SECTION Proof of \clm(main)
Our first task will be to prove \clm(main)
under the extra assumption that the interaction is of
finite range
and that it is $3$-bounded.
Later, we will use this result to prove the full
result by approximating our original problem
by $3$-bounded finite range models.
The proof of this weak version of \clm(main)
will follow, roughly, the scheme in [Go].
We will derive a contradiction
to the assumption that there are no
critical points of \equ(variationsstatmech) of the desired type;
namely, we will show that if there is no critical point of
\equ(variationsstatmech) in $\BB_\omega$, then we can
find a map $T: \BB_\omega \to \BB_\omega$,
and a continuous function
$G:\BB_\omega \to \real$, such that
$G(T(u)) \ge G(u) + \delta$ for some $\delta>0$,
and this will lead to a contradiction.
The argument in [Go] uses the fact that
if a configuration $u_0$ is not
a critical point of \equ(variationFK) we can find a $G$
such that $G(T(u) ) \ge G(u)$ with the inequality being
strict for $u$ in a neighborhood of $u_0$.
Unfortunately, some steps of the argument of
[Go] do not generalize when the dimension of ``time''
is greater than one
and we have to use a more direct argument.
\CLAIM Lemma (compact)
$\BB_\omega/\RR \subset \OO_\omega/\RR$ is compact when $\OO_\omega/\RR$ is
given the topology generated by the following basis:
$\{O_{B,A}\ :\ B \hbox{ a finite subset of }\Gamma, A \hbox{ an open subset of }
\real^d\}$, where
$\{O_{B,A}\} \equiv \{\hat u\ :\ (u_j - \omega \cdot j) \subset A\hbox{ for } j\in B,
\hbox{ for some } u \in \hat u\}$.
(We will refer to this topology as the topology of
component-wise convergence since convergence in it is just convergence
component by component.)
\PROOF
If we pick $p$ representatives $\{i_1,\cdots,i_p\}$ of the
classes $\Gamma/ {\tilde G}$,
-- here we use that $\tG$ is of finite index in $G$ --,
given a configuration $u$ we can apply
one and only one $\RR_\ell$
in such a way that
$0 < u_{i_\alpha} <= 1$.
This means that we can identify
$\BB_\omega/\RR$ as the set of
configurations $u$ in $\BB_\omega$ which
satisfy $u_{i_\alpha} \in (0,1]$
if we identify the ends of $(0,1]$.
If a configuration $u \in \OO_\omega$ is Birkhoff
by \clm(bounds)
$$
[\omega \cdot k ] + u_{i_\alpha} \le u_{i_\alpha+ k} \le [\omega \cdot k ] +1 + u_{i_\alpha}
\EQ(coordinatebounds)
$$
for every $\alpha = 1,\cdots, p$ and every $k \in {\tilde G}$,
where $[\quad]$ denotes integer part.
Since, by assumption, the $i_1,\cdots,i_p$ cover all the conjugacy classes of
$\Gamma/ {\tilde G}$, we see that if
an orbit belongs to $\BB_\omega$ and the $u_{i_\alpha}$
remain bounded, then the values at each site have a bounded range.
Applying Tychonov's theorem,
we can conclude that
$\BB_\omega / \RR$ is precompact when we give it the topology of
point-wise convergence. By the definition of
Birkhoff orbit, it is clear that $\BB_\omega$ is closed in this topology
since it is expressed as the intersection of conditions on each coordinate
that are preserved under point-wise limits. \QED
\CLAIM Lemma (flow)
Consider the map
$\FF$
defined on $\OO^3_\omega$ by:
$$
\left[ \FF(u)\right]_j =
- \sum_{B \ni j } {\partial \over \partial u_j} H_B(u);
\EQ(flowequations)
$$
If $H$ is $1$-bounded on bounded sets of $\OO_\omega$, then
$\FF$ maps $\OO_\omega$ into $\ell^{\infty}$.
Moreover if
$H$ is $r$--differentiably bounded, then $\FF^i$
is a ${r-1}$ map from $\OO_\omega$ to
$\ell^{\infty}$,
and the norm of the derivatives
of order $r-1$ is uniformly bounded.
\PROOF
The proof just consists of restating
the definition of $1$ bounded
and noting that if an
interaction is $2$ bounded the
derivative of
the right hand side of \equ(flowequations)
can be obtained by taking derivatives term by term in the sum. \QED
\REMARK
Notice that $\FF$ is formally the derivative
of the variational principle
\equ(actionstatmech).
Notice also that in the Frenkel-Kontorovna case it reduces to
an analogue of the heat equation in which the Laplacian
is discrete.
If we remember that $\OO_\omega$ is an affine
space modeled on $\ell^\infty$,
the usual existence theorem of differential equations
implies that we can define
a local flow $\Phi_t: \OO_\omega \to \OO_\omega$
satisfying
$$
{d \over dt } \Phi_t(u) = \FF(\Phi_t(u)).
\EQ(floweq)
$$
From the fact that the right hand side of \equ(flowequations),
is uniformly Lipschitz, we conclude that the
solutions are defined for all time.
Using the hypothesis about the
symmetry of the coefficients of the
interaction we obtain that for
$k \in {\tilde G}$, $q \in \integer$.
$$
\Tau_k R_q \FF = \FF \Tau_k.
\EQ(symmetries)
$$
\noindent Hence, using the uniqueness of solutions of the differential equations
defining the flow, we also have for $k \in {\tilde G}$
and $q \in \integer$:
$$
\Tau_k \RR_q \Phi_t = \Phi_t \Tau_k \RR_q
\EQ(symmetryflow)
$$
We show next that the partial orders of
\clm(order) are also preserved by the flow
when the interaction is ferromagnetic.
\CLAIM Lemma (comparison)
Let $\FF$ be defined as before and let
$H$ be a ferromagnetic interaction.
Then
if $u \le {\tilde u}$ (resp. $u \ge {\tilde u}$)
we have $\Phi_t(u ) \le \Phi_t({\tilde u})$
(resp.$\Phi_t(u) \ge \Phi_t( {\tilde u})$).
\PROOF
It suffices to prove the result in the case
$u \le {\tilde u}$.
Define
$u_\lambda = \lambda u + (1 - \lambda){\tilde u}$.
By the theorem on smooth dependence on parameters of solutions of
O.D.E.'s in Banach space, we have that
$\Phi_t( u_\lambda)$ is $C^{r-2}$ with respect to parameters
if $H$ is $r$-bounded. (In finite dimensions, one does not
need an extra derivative since,
in that case,
continuous functions on closed bounded sets
are uniformly continuous.)
Hence, if $r \ge 3$ we have that $\Phi_t( u_\lambda)$
is $C^1$ and, hence, to prove the conclusion it
suffices to show
${ d \over d \lambda} \Phi_t( u_\lambda) \le 0$.
Recall that if we consider ${ d \over d \lambda} \Phi_t( u_\lambda)$
as a function of $t$ it satisfies the
variational equations:
$$
\eqalign{
& {d \over dt } \left[ {d \over d \lambda} \Phi_t( u_\lambda)\right] =
D\FF( \Phi_t(u_\lambda) \left[ {d \over d \lambda} \Phi_t( u_\lambda)\right] \cr
& {d \over d \lambda} \Phi_t( u_\lambda)\big|_{t = 0} = u - {\tilde u}
}
\EQ(variation)
$$
\noindent Observe that, for fixed $t$ and $\lambda$, $D\FF(\Phi_t(u))$
is a bounded linear operator
on
$\ell^\infty( \Gamma, \real).$
And note that in this representation the matrix elements of
$D \FF( \Phi_t( u_\lambda)) $ are positive if they are off the
diagonal, and are bounded on the diagonal.
Let $M(t) = D\FF(\Phi_t(u_\lambda))$, $D_t$ be the diagonal part of
$M(t)$ and $N(t)$ be the non-diagonal part.
If $H$ is $C^3$ it follows that the mappings $t \mapsto M(t), D(t), N(t)$
are $C^1$ and the derivatives are uniformly bounded.
Following the method of variation of constants, we try to write
the solution of \equ(variation) as
$\exp\left[ \int_0^t D(s) ds \right] C(t)$.
Since all the diagonal elements commute among themselves,
we have:
$$
{d \over dt } \exp\left[ \int_0^t D(s) ds \right] =
\exp\left[ \int_0^t D(s) ds \right] D(t) =
D(t) \exp\left[ \int_0^t D(s) ds \right].$$
Hence equation \equ(variation) becomes:
$$
\eqalign{
&{d \over dt} C(t) = \left( \exp\left[ -\int_0^t D(s) ds \right] N(t)
\exp\left[ -\int_0^t D(s) ds \right] \right) C(t) \cr
& C(0) = u - {\tilde u} \cr
}
\EQ(Cequation)
$$
Noting that
$\exp\left[ -\int_0^t D(s) ds \right]$,
$ N(t)$, $\exp\left[ -\int_0 ^t D(s) ds \right]$
all have non-negative matrix elements, it follows that
the initial value problem \equ(Cequation)
is equivalent to:
$$
C(t) = ( u - {\tilde u} ) + \int_0^t R(s) C(s) ds
\EQ(integeq)
$$
where $R(s)$ is a matrix that has non-negative entries.
As is standard, we obtain that the R.H.S. of
\equ(integeq) can be considered an operator acting
on continuous functions defined on an interval $[0,T]$.
It is a contraction if this space of continuous functions
has the metric
$ d( C, \tilde C) = \sup_{t \in [0,T]} || C(t) - {\tilde C} (t) || e^At,$
where $A$ is sufficiently large.
From the fact that $R(s)$ has positive entries and that
$u - \tilde u \le 0$ it follows that
the operator defined by the R.H.S. of \equ(integeq)
preserves the set of $C$'s such that
$C(t) \ge 0$ for all $t$.
And finally, since the solution can be obtained by iterating the
R.H.S. of \equ(integeq) with starting point
$u - \tilde u$,
it follows that the solution is negative and the proof
of \clm(comparison) is complete. \QED
Since the fact that a configuration is Birkhoff
can be expressed in terms of
the translations that commute with the heat flow, and
of the order (which we have shown is preserved),
we obtain:
\CLAIM Corollary (birkhoffpreserved)
Under the above conditions,
$$
\Phi_t( \BB_\omega) \subset \BB_\omega.
$$
Moreover, the flow $\Phi_t$ can be defined on
$\BB_\omega/ \RR$.
Given any finite $\Lambda \subset \Gamma$ we define:
$$\SS_\Lambda (u) = \sum_{B\cap \Lambda \ne \emptyset} H_B (u ).$$
\noindent A simple calculation shows:
$$\eqalign{
{d\over dt} S_\Lambda \bigl(\Phi_t (u)\bigr)\Big|_{t=0}
&= \sum_{B\cap \Lambda\ne \emptyset} \sum_{j\in B;i}
{\partial H_B (u) \over\partial u^i_j} \FF(u)_{j} (u)\cr
& = \sum_{B\cap \Lambda \ne\emptyset} \sum_{j\in B\cap\Lambda;i}
{\partial H_B (u) \over \partial u^i_j} \FF(u)_{j} (u)
+ \sum_{B\cap \Lambda \ne\emptyset} \sum_{j\in B/\Lambda;i}
{\partial H_B (u)\over \partial u^i_j} \FF(u)_{j} (u) \cr
&= \sum_{j\in\Lambda;i} \FF(u)_{j} \sum_{B\owns j}
{\partial H_B(u)\over\partial u^i_j}
+ \sum_{B\cap \Lambda \ne \emptyset}
\sum_{j\in B/\Lambda;i}
{\partial H_B(u)\over \partial u^i_j} \FF(u)_{j} (u)=\cr
&= -\sum_{j\in\Lambda;i} |\FF(u)_{j}(u)|^2 +
\sum_{B\cap \Lambda\ne\emptyset} \sum_{j\in B/\Lambda;i}
{\partial H_B(u)\over\partial u^i_j} \FF(u)_{j} (u)\ .\cr}$$
\noindent
Notice that on the last line the first term is obviously non positive and that the
second one only involves boundary terms.
Intuitively, the first term should dominate
since it is a term that depends on the bulk
while the other depends only on the boundary. In the case $m=1$
and nearest neighborhood interactions this was proved in [Go].
Nevertheless, that proof does not generalize to higher dimensions since
it relies on the fact that there are only two boundary terms
independently of the size of the cube.
Fortunately, for our purposes considerably less is needed.
\CLAIM Lemma(bulkbounds)
Let $\CC$ be a collection of
subsets of $\Gamma$ as in \clm(inflation).
Assume that there is no critical point in
$\BB_\omega$.
Then we can find $\epsilon_0 > 0$ and $L_0 \in \natural$
such that for all configurations $u \in \BB_\omega$,
all sets $\Lambda \in \CC$, and $|\Lambda| > L_0$,
we have:
$$\sum_{j \in \Lambda} |\FF(u)_{j}|^2 \ge \epsilon_0.$$
\PROOF
Let $E_{\Lambda}(u) \equiv \sum_{j \in \Lambda;i} |\FF(u)_{j}|^2$.
Notice that $E_{\Lambda'}(u) \le E_{\Lambda}(u)$
whenever $\Lambda' \subseteq \Lambda$.
We now proceed by contradiction
and assume, contrary to the conclusion, that we can
find a sequence $u^{(n)}\in {\BB _{\omega}}$
and a sequence $\Lambda_n$ of sets in $\CC$
with $|\Lambda_n| \to \infty$ such that
$$E_{\Lambda_n} (u^{(n)}) \longrightarrow \ 0
\qquad \hbox{ as }n\to \infty.
$$
\noindent By assumption $i)$ in \clm(inflation),
we can find ${k_n} \in {\tilde G}$
such that $T_{k_n} \Lambda_n$ converges to $\Gamma$.
Since $E_{\Lambda}$ is invariant under translations
in ${\tilde G}$, (i.e. $E_{\Lambda} (u)=E_{T_k \Lambda} ( \Tau_k u)$
whenever $k \in {\tilde G}$),
if we consider $\Lambda'_n = T_{k_n} \Lambda_n$
and $u'^{(n)} = \Tau_{k_n} u^{(n)}$
we have obtained a sequence of sets ${\Lambda'}_n$ converging to
$\Gamma$ and a sequence of configurations $u'^{(n)} \in \BB_\omega$
such that $E(\Lambda'_n)( {u'}^{(n)} )$ converges to zero.
By the compactness of $\BB_{\omega}/\RR$ we can find a subsequence
$u'^{(n_k)}$ that converges (in $\BB_{\omega}/\RR$) to $u^{(\infty)}.$
We want to conclude that $u^{(\infty)}$ is
a critical point. (This would contradict the assumption that there is
no critical point and hence prove
\clm(bulkbounds).
In effect, if we fix $j \in \Gamma$
we have
$$
|\FF (u^{(\infty)})_j|^2 = \lim_k |\FF(u'^{n_k})_j|^2
\le \lim_k E(\Lambda'_{n_k})( u'^{(n_k)})
$$
The first equality is true because
$\FF_j$ depends only on
the value of the configuration on a finite number of sites
and we have point-wise convergence.
The second holds because $E$ is a sum
of non-negative terms one of which is eventually
$|\FF_j|^2$; since the limit of the R.H.S. is zero, we
conclude that $\FF_j = 0$ for any $i,j$.
Hence, $u^{(\infty)}$ is a critical point and our contradiction
is established. \QED
\CLAIM Corollary(bulkboundsimproved)
Assume there are no critical points in ${\BB}_\omega$. Then
we can find $\epsilon^* >0$ and $L^* \in \natural $ such that for all
configurations $u\in {\BB}_\omega,$ and all sets in $\CC$
of size longer than $L^*,$
we have
$$\sum_{j \in \Lambda;i} |\FF(u)_j|^2 \ge \epsilon^* |\Lambda|^\alpha$$
where $\alpha$ is as in \clm(inflation).
\PROOF
Let $L^*$ be as in \clm(bulkbounds).
Applying repeatedly part $iii)$ of
\clm(inflation),
we obtain that all sets $\Lambda$ of sufficiently large
size can be broken into $N^k$ disjoint
pieces contained in $\CC,$
each of which has size
larger than $|\Lambda|/M^k$.
If we choose $k = [ \log( | \Lambda|/N*)/ \log(N) ] $
we conclude that we can divide the set into
at least $K |\Lambda|^\alpha$ disjoint pieces each of which
is of size at least $N^*$.
Since the sum can be divided into the sum of
each of these pieces, and \clm(bulkbounds)
tells us that they are bounded uniformly away
from zero, the claim is established.\QED
Continuing the argument to prove \clm(main) for
finite range systems, we have on the other hand
that if the range of the interaction is $r$
and $\Lambda$ is a set in $\CC$:
$$
\left| \sum_{B\cap \Lambda\ne\emptyset} \sum_{j\in B/\Lambda;i}
{\partial H_B (u) \over \partial u^i_j} \FF(u)_j\right| \le
\sum_{B\cap \Lambda\ne\emptyset} \sum_{j\in B/\Lambda;i}
\left| {\partial H_B (u) \over \partial ui_j}\right| M_1
\le K \phi(r,|\Lambda|)
$$
\noindent where $\phi$ has the same meaning as in \clm(inflation).
Since by assumption $iii)$
of \clm(inflation) this is negligible compared with the
lower bounds in \clm(bulkboundsimproved),
we obtain:
\CLAIM Corollary(derivative)
If there are no critical points in ${\BB}_\omega$, then
for all large enough sets $\Lambda$ of $\CC$ we have:
$${d\over dt} S_\Lambda (u)\big|_{t=0} \ge K(|\Lambda|) >0$$
where $K(|\Lambda|)$ does not depend on $u$.
Furthermore, since ${d^2\over dt^2} \Phi_t(u)$ is a continuous function
on ${\BB}_\omega/\RR$, which is a compact set, we can bound it uniformly
in $u$.
It follows that if there is no critical point in $\BB_\omega$ we can find
$t_0 >0$ and a cube $\Lambda\subset \integer^m$ such that
$$S_\Lambda (\Phi_{t_0}(u)) \ge S_\Lambda (u) +\delta$$
with $\delta >0$. This is impossible since $\SS_\Lambda$ is a continuous
function on the compact set.
From this contradiction
we conclude there is a critical point in $\BB_\omega$ and
\clm(main)
is proved under the assumption that $H$ is of finite range
and $3$ bounded.
To conclude the proof of \clm(main) as stated,
we just have to show that a $2$ bounded interaction can be approximated
by a sequence of interactions for which the theorem
as proved so far applies in such a way that the
critical points thus produced converge to a
critical point of the interaction.
\CLAIM Lemma (approximation)
Let $\FF^n$ be a sequence of
vector fields derived as in \equ(flowequations)
from an interaction $H^n$ satisfying
condition \equ(perphase) (periodic phase space).
Assume that each of the
$\FF$'s admits a critical point in $\BB_\omega$.
Assume furthermore that there is another vector field $\FF^\infty$
coming from an interaction $H^\infty$
satisfying condition \equ(perphase).
Finally, assume:
\item{$i)$} For every $i \in \Gamma$, every $u \in \BB_\omega$,
$\FF^n(u)_i \to \FF^\infty(u)_i$
\item{$ii)$}
$\sum_l
\sup_n
\sum_{|B| = l}
\sup_{u \in \OO_\omega}
\sum_{i \in \Gamma}
| {\partial^2 \over \partial u_i \partial u_j } H^n_B(u)|
$
\vskip 0em
Then $\FF^\infty$ has a critical point
in $\BB_\omega$.
\REMARK
The interpretation of
condition $ii)$ is that if we truncate
any of the interactions $H^n$
to those interaction
terms which correspond to sets of size $L$,
the error incurred can be made arbitrarily
small uniformly in $n$,
in the sense of the $||\quad||_{2}$ semi-norm.
This is a technical condition that
will be easy to verify for the cases that we have in mind.
The main one is when we consider a
$2$ bounded interaction and approximate it
by finite range ones which are obtained by
ignoring the interaction terms that
have size larger than a certain number.
\PROOF
Denote by $\FF_n$ the vector field in $\OO_\omega$
corresponding to the interaction $H^n$.
Denote by $\FF_\infty$ the
vector field corresponding to the
limiting interaction.
If $u^n$ are the critical points corresponding to
$H^n$, we have
$\FF_n(u^n) = 0$.
Given the invariance under $\RR$
of the interaction,
by substituting $\RR_{q_n} u^n$ for our original
configurations
and using \equ(coordinatebounds),
we can assume
that for all $n,n'\in \natural$ we have $||u^n - u^{n'}||_{\ell^\infty} \le 2 p$, so
$$
||u^n - u^{\infty}||_{\ell^\infty} \le 2 p.
\EQ(uniformdistance)
$$
By the compactness of $\BB_\omega/\RR$ given a sequence
as above, we can obtain a subsequence
that converges to a limiting configuration in
the sense of point-wise convergence.
So that, by passing to a subsequence,
we can assume without loss of generality that
$u^n \to u^\infty$ in the point-wise sense
and $\FF^n(u^n) = 0$.
If we fix $i \in \Gamma$,
we have for any $n, L \in \natural$:
$$
\eqalign{
|\FF^\infty( u^\infty )_i | =
&|\FF^\infty(u^\infty)_i - \FF^n(u^n)_i| \cr
&\le | \FF^\infty(u^\infty)_i - \FF^n(u^\infty)_i| +
| \FF^n(u^\infty)_i - \FF^n(u^n)_i| \cr
&\le |\FF^\infty(u^\infty)_i - \FF^n(u^\infty)_i | \cr
&\ \ \ + \big|\sum_{B\ni i \atop |B| \le L}
{\partial \over \partial u_i} H^n_B( u^\infty) -
\sum_{B\ni i \atop |B| \le L}
{\partial \over \partial u_i} H^n_B( u^n)\big| \cr
&\ \ \ +\sum_{B\ni i \atop |B| > L}
\left| {\partial \over \partial u_i} H^n_B( u^\infty) -
{\partial \over \partial u_i} H^n_B( u^n)\right|\cr
}
\EQ(splitting)
$$
Given any $\epsilon > 0$,
by condition $ii)$
$
\sup_{u \in \OO_\omega} \sum_{|B| > L} \sum_{i'}
|{\partial^2 \over \partial u_i \partial u_{i'}} H^n_B(u^n) | \le \epsilon/6p
$.
Using \equ(uniformdistance) and \clm(interpolation), this implies that
$| {\partial \over \partial u_i} H^n_B( u^\infty) -
{\partial \over \partial u_i} H^n_B( u^n)| \le \epsilon/3$
independently of $n$.
Once $L$ is chosen, we can find
$n$ large enough so that the first term in the sum in \equ(splitting)
is smaller that $\epsilon/3$,
and given the uniform convergence assumption
in \clm(approximation) we can make arbitrarily small
the last term in \equ(splitting). \QED
To conclude the proof of
\clm(main), we observe that a $2$ bounded function of
finite range can be approximated,
in the sense required
\clm(approximation), by $C^3$ functions of finite range
by using just the usual smoothing of finite dimensional variables
for each of the $H_B$'s.
The twist condition is preserved since it only depends on the
second derivatives of sums
of of finitely many functions of finitely many variables,
which we are approximating.
Hence, we can use the results we have already proved to
prove $a)$ of \clm(main).
In the case where we have an interaction satisfying $b)$
of \clm(main), we can just approximate the interaction
by the cut-off interactions which agree with the original interaction
on sets of size up to $n$ and are zero for larger sets.
It is easy to check that under the hypothesis of
the strong twist condition all these interactions
will also satisfy the strong twist condition
and also converge in a sense strong enough for
\clm(approximation) to apply.
This concludes the proof of \clm(main). \QED
\REMARK
We note that the regularity hypothesis in \clm(main) can be
slightly weakened. The most delicate analytical point of the proof is that the
heat flow can be defined on $\BB_\omega$ for all time. Then, one
can use the argument starting in \clm(bulkbounds).
We concluded the existence of the flow for all time by making assumptions that
implied that $\FF$ was uniformly Lipschitz and then showed that the
comparison principle and symmetries lead to the preservation of
the Birkhoff character.
Note that since we know that $\BB_\omega \subset \OO^2_\omega$,
it would have sufficed to show that there is a flow defined
defined for a positive time for conditions starting in
$\OO^3_\omega$ and that comparison held. Then, the same argument
we have detailed shows that the $\BB_\omega$ is preserved, hence,
that the flow exits for all times when the conditions are Birkhoff.
There are indeed some models for which this generality leads
to new results, but we do not know of any physically interesting one.
Nevertheless, analogues of these arguments are needed when
generalizing this argument to PDE's.
\SECTION Applications
In this section, we discuss some models
that can be reduced to situations
in which $G \ne \tG$.
That is, the group that leaves invariant the interaction
-- and with respect to which we define Birkhoff orbits -- is
smaller than the group that leaves invariant the lattice.
Besides the conceptual simplification of not identifying
objects that play different roles, having these models
was an important motivation.
As a first example we show how finite compositions of
twist mappings can be considered in this framework.
(This application had been considered by other methods in
\cite{Ma3}) This addresses the fact that even if the
conclusions in the dynamical version of Aubry-Mather theory
are invariant under changes of coordinates, the twist hypothesis is
not. Relatedly, even if the conclusions of
Aubry-Mather theory for a map
$f$ imply the same conclusions for $f^n$, it is not true in general that
$f^n$ satisfies the twist hypothesis.
We recall -- we refer to \cite{Ma1} for
further details and for some precisions
about uniformity assumptions that
one has to do to deal with the
the absence of -- that, given a diffeomorphisms
$M$ of $\torus^n\times \real^n$
that preserves the symplectic form
$\omega = \sum_i d p_i \wedge d q_i$ and
is exact ($ M^* ( \sum_i p_i d q_i ) = \sum_i p_i d q_i + d S$)
where $S: \torus^n\times\real^n \to \real$.
If we denote by $(P',Q')$ the image
of $(P,Q)$ under the map
and assume that $(Q,Q')$ is a
good system of coordinates.
(This happens if $ |det(\partial Q' \partial P) | \ge a > 0$.)
Then, we can find a generating function $h$ such that
the statement $M(P,Q) = (P',Q')$ is
equivalent to $P'= \partial_2 h(Q,Q')$,
$P = - \partial_1 h(Q,Q')$.
In that case, a sequence $Q_n$ is the projection
of an orbit if and only if
it is a critical point of the
formal action
$ \sum_{n \in \integer} h(Q_n,Q_{n+1})$
If $h^\alpha$, $\alpha = 1,\cdots,p,$
is a finite set of these generating functions,
corresponding to twist maps $M_\alpha$,
we see that
if we consider the formal
action principle:
$$
\SS( Q) = \sum_n h^\alpha( Q_{n p + \alpha}, Q_{np +\alpha +1})
\EQ(actioncomposition)
$$
the variational equations are:
$$
\partial_1 h^\alpha( Q_{n p + \alpha}, Q_{n p + \alpha +1})
+
\partial_2 h^{\alpha -1} ( Q_{n p + \alpha -\alpha}, Q_{n p + \alpha }).
\EQ(hamiltoniancritical)
$$
That is, if we set
$$
P_{n p + \alpha} =
-\partial_1 h^\alpha( Q_{n p + \alpha}, Q_{n p + \alpha +1}),
\EQ(Pdefinition)
$$
the equations
\equ(hamiltoniancritical)
imply that:
$$
P_{n p + \alpha +1} =
\partial_2 h^\alpha( Q_{n p + \alpha}, Q_{n p + \alpha +1}).
\EQ(derivedcritical)
$$
Taking together \equ(Pdefinition)
and \equ(derivedcritical), we obtain:
$$( Q_{n p + \alpha +1}, P_{n p + \alpha +1} ) =
M^\alpha( Q_{n p + \alpha}, P_{n p + \alpha} ).$$
\noindent Hence $ \{ (Q_{n p}, P_{n p}) \}$ is an orbit for
$M^p \circ M^{p-1} \circ M^1$.
The variational principle
for the action \equ(actioncomposition)
can be fitted into
those considered in \clm(main)
if we consider a one-dimensional
lattice $\Gamma = \integer$
with group of translations $\integer$
and reduced group $p \integer$.
Clearly, the action is invariant under these reduced groups.
The only partial derivatives that are not zero are those
corresponding to neighboring points, and in this case
they reduce to the twist condition for the individual maps.
Notice that this problem is equivalent to the problem of considering
the map obtained by composing the $p$
twist maps. Nevertheless, in general
the map will not satisfy the twist condition.
This construction has an analogue in higher dimensions
if we associate to a site in a lattice $\integer^d$
a value in $\real^{l^{(d-1)2d}}$.
Notice that there is an identification
$\sigma$ between $[1, {l^{(d-1)2d}}]$
and the boundary of $\Lambda_l$, the cube of size $l$.
Given a state $u$ we can associate to it the
energy $V(u) = \min\{E({\tilde u})\ :\
{\tilde u}: \Lambda_l \to \real, {\tilde u}|_{\partial \Lambda_l}
= u\circ \sigma\},$
where $E({\tilde u})$ satisfies the twist conditions.
The Frenkel-Kontorovna model for this interaction can be identified
with a one-component Frenkel-Kontorovna model in which each of the
sites is blown up to a cube $\Lambda_l$ and the interaction between the new sites
is given by the expression $E$ if they are in the same block, and
by the original Frenkel-Kontorovna if they have adjacent boundaries.
In general this process leads to models in which the potential
cannot be expressed as a sum of functions of the one-dimensional variables.
Hence the multidimensional analogue of the twist condition is
violated.
This is very similar
to the process that is called {\sl ``conjunction''} in
[Ma3]. It can also be considered an analogue of
the block spin renormalization of
statistical mechanics, and indeed the renormalization group
picture of [McK] can also be formulated in this language.
Notice that if the dimension of the {\sl `` time'' }
is larger than one the renormalization
process increases the number of variables.
In the one-dimensional time case -- the one most
interesting for dynamics -- the number of
variables does not increase.
We also point out that an statistical mechanics interpretation of these
models is molecules laid in a linear substratum.
The atoms in the molecule interact with their nearest neighbors
and with the substratum
in a way that depends on the chemical element. Since the molecules
are arranged periodically, the interactions are periodic
with a period equal to the number of atoms in the molecule.
We also point out that this formalisms also allows to deal with some
restricted classes of several component models.
In the statistical mechanics interpretation,
this would correspond to situations where the internal
state of an atom is describe by several parameters.
Frenkel-Kontorovna models with $p$ components
have configurations which are maps
$u: \integer^n \to \real^p$.
We will denote the components by superscripts
and, as before, the site of the lattice
by a subindex.
So for a vector $i \in \integer^n$,
$u_i$ denotes the vector in $\real^p$ which is the value
taken by the configuration at the site $i$.
For an integer $j \in \{1,\cdots,p\}$,
$u_i^j$ will denote component $j$ of
the $\real^p$ vector $u_i$.
Similarly, we will denote by $u^j$ the mapping
$\integer^n \mapsto \real$ that to each
site associates the $j^{th}$ component of the configuration at the site.
The interaction of such a multicomponent
Frenkel-Kontorovna model is given by
$H_{\{i\}} = V(u_i)$,
where $V(x + e) = V(x)$ if $e$ is a
vector with integer components of length $1$,
$H_{\{i,j\}}(u) = 1/2n\ |u_i - u_j|^2$ if $|i -j| = 1$,
and $H_B(u)=0$ for any other set $B$.
To reduce such models to the situation discussed in
\clm(main), we introduce an auxiliary one-component model
in which $ \hat \Gamma = \integer^n \times \integer$.
The group $G$ of the new model
will be $\integer^n \times \integer$
acting in the obvious way, and
${\tilde G} = \integer^n \times p \integer$.
The configurations $\hat u$ in the auxiliary model can be
obtained from those in the old one by
setting
${\hat u}_{(i_1,\cdots,i_p,i_{p+1}) } =
u_{(i_1,\cdots,i_p)}^{i_{p+1} \bmod p}$.
The interactions are given by:
$$
\hat H_{\{(i_1,\cdots,i_p, pn +1), \cdots, (i_1,\cdots,i_p, pn +p)\}} =
V( {\hat u}_{(i_1,\cdots,i_p,p n +1)},\cdots,{\hat u}_{(i_1,\cdots,i_p,p n +p)}
$$
If $\sum_{\alpha = 1,\ldots,p} |i_\alpha - j_\alpha| = 1$,
we define:
$$
{\hat H}_{ \{(i_1,\ldots,i_p, \ell),(j_1,\ldots,j_p,\ell)\}} =
{1 \over 2}\left( {\hat u}_{(i_1,\cdots,i_p,\ell)} -
{\hat u}_{(j_1,\cdots,j_p,\ell)}\right)^2
$$
This identification can be interpreted as saying that we
lump together segments of $p$ elements in the vertical direction
and consider them as a site in the original problem.
If we have a configuration $\hat u$ of the extended system
satisfying
$$
{\hat u}_{(i_1,\ldots,i_p,i_{p+1})}
= {\hat u}_{(i_1,\ldots,i_p, i_{p+1} +p )}
\EQ(periodicity)
$$
and it satisfies the equations
\equ(variationsstatmech)
the $p$ component configuration $u$
satisfies the equation
\equ(variationsstatmech) for the
reduced model.
Given a frequency $\omega =(\omega_1,\ldots,\omega_p)$
we can consider an extended frequency
$\hat \omega = (\omega_1,\ldots,\omega_p, 1/p)$.
The Birkhoff configurations of this extended frequency
have to satisfy the periodicity condition \equ(periodicity).
Unfortunately, these models,
even if they have a natural interpretation in
Statistical Mechanics, they do not have such a
nice interpretation as twist maps in
the annulus. (They are dynamical systems in $\real^{2d}$.)
For the
Frenkel Kontorovna model to verify the twist condition of
\clm(main) we have to assume
that ${\partial^2 \over \partial u_{i} \partial u_{i'} } V(u) \ge 0$
when $i' \ne i$. Unfortunately,
if we want that they are
functions in a torus
$ V(u_1 +1, u_2,\cdots,u_p) = V( u_1, u_2 +1,\cdots,u_p)= \cdots =
V(u_1,u_2,\cdots,u_p+1) = V(u_1,u_2,\cdots, u_p)$
we have considered this implies that $V(u) = V_1(u_1)+\cdots+V_p(u_p)$,
so the model reduces to
uncoupled one-dimensional cases.
\SECTION References
\ref \no{AD} \by{S. Aubry, P. Y. Le Daeron} \paper{ The discrete Frenkel Kontorovna model and its extensions: Exact results for the ground state} \jour{Physica} \vol{8D} \pages{240-258} \yr{1983} \endref
\ref \no{An} \by{S. Angenent} \paper{ Monotone recurrence relations, their Birkhoff orbits and topological entropy} \jour{Ergod. Th. and Dyn. Syst.} \vol{10} \pages{15-41} \yr{1990} \endref
\ref
\no{BK} \by{D. Bernstein, A. Katok} \paper{ Birkhoff periodic orbits for small perturbations of completely integrable systems with convex Hamiltonians} \jour{Inv. Mat.} \vol{88} \pages{225-241} \yr{1987} \endref
\ref\no{Ba} \by{V. Bangert} \paper{Mather sets for twist maps and geodesics on tori} \jour{Dynamics Reported} \vol{1} \pages{1-54} \yr{1988} \endref
\ref\no{B1}\by{M.L. Blank}\paper{Metric properties of minimal solutions of discrete periodical variational problems} \jour{Nonlinearity}\vol{2}\pages{1-22}\yr{1989}\endref
\ref\no{B2}\by{M.L. Blank}\paper{Chaos and order in the multidimensional Frenkel-Kontorovna model} \jour{Sov. Phy. JEPT}\vol{85}\pages{1255-1268}\yr{1989}\endref
\ref \no{He} \by{M. Herman} \paper{In\'egalit\'es `` a priori'' pour des tores Lagrangiens invariantes par des diff'eomorphismes symplectiques} \jour{Pub. Mat. I.H.E.S} \endref
\ref\no{Go}\by{C. Gol\'e}\paper{A new proof of Aubry--Mather's theorem}\jour{E.T.H. preprint}\endref
\ref \no{Gr} \by{R. Griffiths} \paper{Frenkel-Kontorovna models of commensurate-incommensurate transitions} \jour{preprint} \yr{1991} \endref
\ref\no{McK}\by{R.S. McKay}\paper{Scaling exponents at the transition by breaking of analyticity for incommensurate structures}\jour{Physica}\vol{50D}\pages{71-79}\yr{1991}\endref
\ref\no{Ma1}\by{J. Mather}\paper{A criterion for non existence of invariant circles}\jour{Pub. Mat. I.H.E.S.}\vol{63}\pages{153 --204}\yr{1987}\endref
\ref\no{Ma2}\by J. Mather\paper{Existence of quasiperiodic orbits for twist homeomorphisms of the annulus}\jour{Topology}\vol{21} \pages{457-467}\yr{1982}\endref
\ref \no{Ma3} \by{J. Mather} \paper{Variational construction of orbits of twist diffeomorphisms} \jour{Jour. Am. Math. Soc.} \vol{4} \pages{207-263} \yr{1991} \endref
\ref \no{Mo} \by{J. Moser} \paper{Minimal solutions of variational problems on a torus} \jour{Ann. Inst. H. Poinc. An. non lin.} \vol{3} \pages{229-272} \yr{1986} \endref
\ref \no{Mo2} \by{J. Moser} \paper{A stability theorem for minimal foliations on a torus} \jour{Ergod. Theo. Dyn. Syst.} \vol{8} \pages{251-281} \yr{1988} \endref
\ref\no{OV}\by{A. Olvera, C. Vargas}\paper{A continuation method to study periodic orbits in the Froeschl\'e map}\jour{Physica}\vol{72 D} \pages{351-371}\yr{1994}\yr{1992}\endref
\ref \no{Ru} \by{D. Ruelle} \book{Statistical Mechanics} \publisher{Benjamin, Reading MA} \yr{1969} \endref
\ref \no{V} \by{F. Vallet} \paper{Thermodynamique unidimensionelle, et structures bidimensionelles de quelques mod\`eles pour des syst\`emes incommensurables} \jour{ U. Paris VI thesis} \yr{1986} \endref
\end