\magnification 1200
\hsize=5.8truein \hoffset=.25truein %was \hoffset=1.2truein
\vsize=8.8truein %\voffset=1truein
\pageno=1 \baselineskip=12pt
\parskip=0 pt \parindent=20pt
\overfullrule=0pt \lineskip=0pt \lineskiplimit=0pt
\hbadness=10000 \vbadness=10000 % REPORT ONLY BEYOND THIS BADNESS
\let\nd\noindent % NOINDENT
\def\NL{\hfill\break}% NEWLINE
\def\qed{\hbox{\hskip 6pt\vrule width6pt height7pt depth1pt \hskip1pt}}
\parindent=40pt
\pageno=1
\footline{\ifnum\pageno=1\hss\else\hss\tenrm\folio\hss\fi}
\hbox{}
\vskip 1truein\centerline{{\bf Z$^n$ Versus Z Actions for Systems of
Finite Type}\footnote*{Research supported in part by NSF Grant No. DMS-
9001475\hfil}}
\vskip .7truein\centerline{by}
\centerline{Charles Radin}
\vskip .2truein\centerline{Mathematics Department}
\centerline{University of Texas}
\centerline{Austin, TX\ \ 78712}
\centerline{USA}
\vskip 1truein\centerline{Conference talk at ``Symbolic Dynamics and its
Applications''}
\vskip0truein
\centerline{July 28 - August 2, 1991 at Yale University}
\vskip .5truein\centerline{ABSTRACT}
{\narrower\vskip .2truein\noindent We consider dynamical systems of finite
type with {\bf Z$^n$} \hbox{actions,}
and discuss the differences between the cases $n=1$ and \hbox{$n\ge 2$.} For
the latter we examine the degree of ``order'' which is \hbox{possible} when the
system is uniquely ergodic.\smallskip}
\hfil\eject
\parindent=20pt
\leftline{{\bf Systems of finite type.}}
\vskip .05truein
We begin with some notation. Let {\it A} be a finite alphabet, and
consider the ``infinite arrays'' ${\it A}^{{\bf Z}^n}$ as functions on
${\bf Z}^n$. We say that the dynamical system which consists of the
natural action of ${\bf Z}^n$ on $X \subseteq {\it A}^{{\bf Z}^n}$ is
``of finite type'' if there is some finite $C \subset {\bf Z}^n$ and
finite set of finite arrays $F \subset {\it A}^C$ (thought of as
restrictions $x\vert _C$ to $C$ of functions $x$ in ${\it A}^{{\bf
Z}^n}$) such that $$ X = X_F \equiv \{x \in {\it A}^{{\bf
Z}^n}\,:\,\hbox{for all }t\in {\bf Z}^n,\, x_t\vert _C \notin F\}
\eqno 1)$$
\noindent where $x_t(j) \equiv x(j-t)$ for $t,j\in {\bf Z}^n$.
It is easy to see that certain choices of $F$ will lead to an empty
$X_F$. For this and other reasons it is useful to ``redefine'' $X_F$,
whereby instead of {\bf forbidding} restrictions in $F$ from appearing in
the arrays we just {\bf minimize their appearance}. We do this
using the ``energy function'' $E\,:\,{\it A}^C
\to {\bf R}$ defined to be the characteristic function of $F$.
(That is, $E(f) = 1\hbox{ for }f\in F,\ E(f) = 0\hbox{ for } f\notin F$.)
We then define:
$$\eqalign{\tilde X_E \equiv \big\{x \in {\it A}^{{\bf Z}^n}\,:\,\hbox{for every }
&\hbox{finite }B\subset {\bf Z}^n,\cr &E^B(x) = \inf\{E^B(y)\,:\,y = x\hbox{
outside }B\}\big\}\cr}\eqno 2)$$
\vskip .1truein
\noindent where $E^B(z) \equiv \sum\{E(z_t\vert _C)\,:\,t\in Z^n,\,B_{-t}\cap
C\neq\emptyset\}$.
Clearly $X_F \subseteq \tilde X_E$. Furthermore it is easy to prove
using the compactness of ${\it A}^{{\bf Z}^n}$ that $\tilde X_E$ is
always nonempty, in fact for an arbitrary function \hbox{$E\,:\,{\it A}^C\to
{\bf R}$}, not just characteristic functions. We therefore define a
``zero temperature'' dynamical system as one defined by 2) for any
fixed function $E$.
\vskip .2truein
\leftline{{\bf Unique ergodicity.}}
\vskip .05truein
We say $X \subset {\it A}^{{\bf Z}^n}$ is ``uniquely ergodic'' if
there is one and only one Borel probability measure on $X$ invariant
under the natural action of ${\bf Z}^n$. Consider the following three
classes of uniquely ergodic systems.
\vskip .1truein\noindent
i)\ All uniquely ergodic $X \subset {\it A}^{{\bf Z}^n}$
\vskip .05truein\noindent
ii)\ All uniquely ergodic zero temperature $X \subset {\it A}^{{\bf Z}^n}$
\vskip .05truein\noindent
iii)\ All uniquely ergodic $X \subset {\it A}^{{\bf Z}^n}$ of finite type
\vskip .1truein\noindent
It is clear by construction that there is containment as one descends
the list, but proper containment is not obvious. Our first result
along this line is the following (known as the Third Law of
Thermodynamics).
\vskip .1truein\noindent
Theorem (J.~Mi\c ekisz and C.~R.~[7]).\ All uniquely ergodic zero
temperature
\vskip 0truein\noindent$X \subset {\it A}^{{\bf Z}^n}$ have zero
topological entropy.
\vskip .1truein\noindent
This together with the theorem of Jewett-Kreiger-Weiss [11] shows that
the first containment is proper. We do not know a proof that the
second containment is proper, but there is a preprint by Mi\c ekisz
[3] going part way.
\vskip .2truein
\leftline{{\bf ${\bf Z}^n$ Versus {\bf Z} Actions.}}
\vskip .05truein\noindent
To say that a symbolic system $X \subset {\it A}^{{\bf Z}^n}$ is of
finite type implies that each variable, with values in {\it A},
corresponding to a point of ${\bf Z}^n$ can only directly affect
nearby variables. This is reminiscent of the differential equations of
the natural sciences. One of the main points we wish to make follows
by analyzing such nonmathematical applications of ${\bf Z}^n$ and {\bf
Z} actions. We envision {\bf Z} actions as typically modeling {\bf
evolution} problems; that is, {\bf Z} represents time. We reformulated
the condition of finite type above as an optimization condition, for
reasons we will soon discuss. With this in mind, we note that
evolution problems can also sometimes be reformulated as optimization
problems -- think of the least action principle for Hamiltonian
systems for example. However in such a reformulation it is typical
that one seeks as solutions critical points, not global optima, and
that such critical points can represent a wide variety of curves. On
the other hand, the $n$ translation variables of ${\bf Z}^n$ actions
often represent spatial translations, and, as in the crystal problem
of condensed matter physics, or the sphere packing problem, or the
problems of space tiling, what one seeks is typically (generically
[5,6,7]) a {\bf unique} solution (a well defined ``structure'', so to
speak) to a global optimization problem of the general form of our
zero temperature condition [8]. When properly formulated, the
solution is sought in a space of invariant probability measures, and
the uniqueness of the solution translates into the property of unique
ergodicity.
Thus {\bf Z} actions and ${\bf Z}^n$ actions naturally represent very
different situations, the former accomomodating a very flexible class
of arrays (in particular they are highly nonunique), and the latter
representing some unique structure such as a crystal or quasicrystal.
This is ``why'' unique ergodicity is not as natural for {\bf Z}
actions as it is for ${\bf Z}^n$ actions.
Our interest is primarily with ${\bf Z}^n$ actions, and more
specifically in the degree to which uniquely ergodic zero temperature
systems (or systems of finite type) tend to be ``ordered''. (Why does
low temperature matter tend to be crystalline, why do there always
seem to be periodic examples among the densest sphere packings in any
dimension, why is it hard to find tiles which can only tile space
nonperiodically? See [8].) Consider the following extreme cases of
``order'' for a zero temperature system $X$ with unique invariant
measure $\rho$.
\vskip .1truein\noindent
\vbox{\noindent a)\ Periodic; (corresponding to a finite set $X$, the orbit of a
periodic array)
\vskip .05truein\noindent
b)\ Quasiperiodic; (corresponding to $X$ with purely discrete
dense spectrum)
\vskip .05truein\noindent
c)\ Weakly mixing $\rho$; (corresponding to purely singular continuous
spectrum)
\vskip .05truein\noindent
d)\ Strongly mixing $\rho$; (corresponding to purely absolutely continuous
spectrum)}
\vskip .1truein\noindent
(Note that as in probability theory we are using measure theoretic --
chiefly spectral -- properties to analyze the order of the dynamical
system.)
\vskip .1truein
Examples of a) are easy to obtain. Examples of b) were first obtained
by R.~Berger in 1966 [1], with nicer examples by R.~Robinson [9] and
others. Examples of c) are due to S.~Mozes in 1989 [4,7]. It is unknown
if there are examples of d), and this is an important open problem.
There are several reasons for our introduction of the class of zero
temperature systems. They constitute a natural generalization of
systems of finite type with the conditions defining the system still
strictly local, and there are real parameters for the class so that
one can address questions of genericity. Furthermore, ever since the work of
G.~Toulouse [10,2] it has been commonly felt by condensed matter
theorists that energy functions $E$ as above which are ``frustrated''
(that is, cannot be reformulated as a characteristic function as is
the case for systems of finite type), are more likely to lead to
``disorder'' -- or smooth spectrum, as in models of spin
glasses. In other words, physical intuition suggests that the
class of zero temperature systems is broader than, and should contain
more disordered examples than (perhaps of type d) above), the class of
systems of finite type. Needless to say, it would be most interesting
if this could be proved true.
\hfil\eject
\centerline{{\bf REFERENCES}}
\vskip .2truein\noindent
[1] R.\ Berger, The undecidability of the domino problem, {\it
Mem.\ Amer.\ Math.\ Soc. no.\ 66}\ (1966).
\vskip.1truein\noindent
[2] J.\ Mi\c ekisz, Frustration without competing interactions, {\it
J.\ Stat.\ Phys.}\ 55 (1989), 351-355.
\vskip.1truein\noindent
[3] J.\ Mi\c ekisz, ``The global minimum of energy is not always a sum
of local minima -- a note on frustration'', preprint, University of
Louvain-la-Neuve, June, 1991
\vskip.1truein\noindent
[4] S.\ Mozes, Tilings, substitution systems and dynamical systems
generated by them, {\it J. d'Analyse Math.}\ 53 (1989), 139-186.
\vskip.1truein\noindent
[5] C.\ Radin, Correlations in classical ground states, {\it J. Stat.
Phys.}\ 43 (1986), 707-712.
\vskip.1truein\noindent
[6] C.\ Radin, Low temperature and the origin of crystalline symmetry,
{\it Int.\ J.\ Mod.\ Phys.}\ 1 (1987), 1157-1191.
\vskip.1truein\noindent
[7] C.\ Radin, Disordered ground states of classical lattice models,
{\it Revs.\ Math.\ Phys}\ 3 (1991) 125-135.
\vskip.1truein\noindent
[8] C.\ Radin, ``Global order from local sources'', to appear in October,
1991 issue of {\it Bull.\ Amer.\ Math.\ Soc}.
\vskip.1truein\noindent
[9] R.M.\ Robinson, Undecidability and nonperiodicity for tilings of the
plane, {\it Invent.\ Math.}\ 12 (1971), 177-209.
\vskip.1truein\noindent
[10] G.\ Toulouse, Theory of frustration effect in spin glasses, I, {\it
Commun.\ Phys.\ (G.B)}\ 2 (1977), 115-119.
\vskip.1truein\noindent
[11] B.\ Weiss, Strictly ergodic models for dynamical systems, {\it Bull.
Amer. Math. Soc.}\ 13 (1985), 143-146.
\end