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\begin{document}


%\title{ \bf INERT ACTIONS ON \\ PERIODIC POINTS}
\title{Inert actions on periodic points}

%\author{\bf K.H. Kim \and \bf F.W. Roush \and \bf J.B. Wagoner}

\author{K. H. Kim}
\address{Department of Mathematics, Alabama State
University, Montgomery, Alabama 36101}
\email{kkim@@asu.alasu.edu}
\thanks{The first two authors were partially supported by  NSF Grants
DMS 8820201 and DMS 9405004. The last author was   partially supported
by NSF Grants DMS 8801333, DMS 9102959, and DMS 9322498.}

\author{F. W. Roush}
\address{Department of Mathematics, Alabama State
University, Montgomery, Alabama 36101}
\email{kkim@@asu.alasu.edu}

\author{J. B. Wagoner}
\address{Department of Mathematics, UCB, Berkeley, California 94720}
\email{wagoner@@math.berkeley.edu}

\commby{Douglas Lind}

\date{October 25, 1996}

\subjclass{Primary 54H20,  57S99,  20F99}

\issueinfo{3}{08}{January}{1997}
\dateposted{July 30, 1997}
\pagespan{55}{62}
\PII{S 1079-6762(97)00024-3}

\copyrightinfo{1997}{American Mathematical Society}


%\begin{center}
% {\large \bf Abstract}
%\end{center}

\begin{abstract}
      The action of  inert
automorphisms on finite sets of periodic points of mixing subshifts of
finite type
is  characterized in terms of  the sign-gyration-compatibility condition.
The main technique used is
variable length coding combined with a ``nonnegative  algebraic K-theory"
formulation of
state splitting and merging. One application  gives a counterexample to
 the  Finite Order Generation Conjecture by
 producing  examples of infinite order   inert automorphisms of mixing
subshifts of
finite type  which are not products of finite order automorphisms.%\\
\end{abstract}

\maketitle

%\setlength{\parskip}{.5cm}

%\noindent{\large \bf 
\section*{Introduction, main results, and applications }%%\\\bigskip

 Subshifts of finite type  $(X_A,\sigma_A)$ constructed
from a nonnegative integral matrices $A$ appear in a number of areas
 ranging from smooth dynamical systems to coding and information theory.
See \cite{LM} for a comprehensive introduction to these model
 dynamical systems and to
 the field of symbolic dynamics in general. The automorphism  group
$Aut(\sigma_A)$ of
 $(X_A,\sigma_A)$ consists  of those homeomorphisms of $X_A$ which commute with
the basic shift  homeomorphism $\sigma_A: X_A \longrightarrow X_A$. The
first systematic
study of
$Aut(\sigma_A)$ appeared  in \cite{He}, and it has subsequently been
studied in a
number of papers. For example, see  \cite{BF},
\cite{BK1},
\cite{BLR},
\cite{KR1},
\cite{KRW1}. In general,
 $Aut(\sigma_A)$ is a huge
%
%\vspace*{.25in}
%\noindent ------------------------------------------------------ \newline
% The first two authors were partially supported by  NSF Grants
% DMS 8820201 and DMS 9405004. The last author was   partially supported
%by NSF Grants DMS 8801333, DMS 9102959, and DMS 9322498
%
%
%
%\noindent
 countable group. It contains copies of the direct
 sum of any countable collection
of finite groups and is highly nonabelian. It contains a copy of the
free abelian group on  countably many generators.
It  contains copies of the fundamental groups of closed,
2-dimensional surfaces. It is residually finite, and therefore does not contain
 a divisible group. See \cite{BLR}. The group $Aut(\sigma_A)$
is interesting both in its own right and for its
relation to other questions in symbolic dynamics.
For example,
present understanding of the fundamental classification problem for
subshifts of finite type
is closely related to $Aut(\sigma_A)$. See \cite{B},
\cite{KRW1},
\cite{KR2},
\cite{W}.%\\



 One way to study  $Aut(\sigma_A)$ is through representations of it
to simpler groups,
and there  are currently two such representations which
play a key role; namely, the {\em periodic point} and
{\em periodic orbit  representations}
 and  the {\em dimension group representation}. %\\


Let  $P_k^{\bullet} = P_k^{\bullet}(A)$ denote the periodic points of
period exactly
$k$. Let $P_k = P_k(A)$ denote the disjoint union of
$P_l^{\bullet} = P_l^{\bullet}(A)$ for $1 \leq l \leq k$.
Let $Aut(\sigma_A|P_k^{\bullet})$  denote the group of
permutations of  $P_k^{\bullet}$ which commute with  $\sigma_A$.
Similarly, let  $Aut(\sigma_A|P_k)$  denote the group of permutations of
  $P_k$ which
 commute with  $\sigma_A$. Then as in \cite[Section 7]{BLR} we have the
  periodic point
 representations
\[ \pi_{A,k}^{\bullet}:Aut(\sigma_A)  \longrightarrow
Aut(\sigma_A|P_k^{\bullet}) \]
\noindent
and
\[ \pi_{A,k}:Aut(\sigma_A)  \longrightarrow Aut(\sigma_A|P_k). \]
%\noindent
Also discussed in \cite[Section 7]{BLR} are the closely related
 periodic orbit permutation representations
\[ \rho_{A,k,l}:Aut(\sigma_A)  \longrightarrow
\bigoplus_{r=k}^l Perm( %\mbox{
\text{$\sigma_A$ orbits of length $r$}) \]
%\noindent
%for $k \leq l$.
%\noindent
%and
%\[ \rho_{A;k}^{\bullet}:Aut(\sigma_A)  \longrightarrow
%Perm( %\mbox\text{$\sigma_A$ orbits of length k}) \]
%\vspace*{.15in}
%\noindent
for $k \leq l$. Computing the images of $\pi_{A,k}$ and  $\rho_{A,k,l}$ is
a basic problem.
If  $(X_A,\sigma_A)$ is a primitive subshift of finite type, it is  known
from
\cite[7.6]{BLR} that $\rho_{A,k,l}$ is onto for  $k$ sufficiently large.
On the other hand,
 an example where $(X_A,\sigma_A)$ is primitive and
$\pi_{A,1} = \pi_{A,1}^{\bullet} = \rho_{A,1,1} $ is not onto
is given in \cite[4.3]{KRW1}.
In Application 1 of our results below, we will show that in a number of  cases
the image of  $\pi_{A,k}$ can be effectively determined
 for $k$ sufficiently large. This is relevant to the problem of classifying
imbeddings of
one shift into another of larger entropy as discussed in \cite{BK2}.%\\


The periodic point  representations gave rise to   the  Boyle-Krieger sign
and gyration number homomorphisms
\[ OS_{A,k}:Aut(\sigma_A)  \longrightarrow
Aut(\sigma_A|P_k^{\bullet})  \longrightarrow Z/2, %\mbox
\qquad k\geq1,\]
and
\[GY_{A,k}: Aut(\sigma_A)  \longrightarrow
Aut(\sigma_A|P_k^{\bullet}) \longrightarrow Z/k,  %\mbox
 \qquad   k \geq 2,  \]
%\noindent
which were used in  \cite{BK1} to study involutions in
$Aut(\sigma_A)$. The orbit sign number homomorphism $OS_{A,k}$ is simply the
sign of the permutation induced on the orbits of length $k$ by an automorphism.
The gyration number homomorphism $GY_{A,k}$ is a measure of  how
 automorphisms move orbits of length $k$ parallel to  themselves.%\\

The dimension group representation
\[\delta_A:Aut(\sigma_A) \longrightarrow Aut(s_A)   \]
%\noindent
introduced by Krieger is essentially a matrix group representation
 which can be defined dynamically as in \cite{BLR} or more algebraically as in
\cite{W}. The group  $Aut(s_A)$ of order preserving
automorphisms of the dimension group is generally
 much simpler  than $Aut(\sigma_A)$.
It is often a torsion free, finitely generated abelian group. So typically,
much of the complexity of
$Aut(\sigma_A)$ lies in the subgroup $Inert(\sigma_A)$ of
{\em inert automorphisms} which is defined to be the kernel of $\delta_A$.
Application 2 below gives counterexamples to the FOG Conjecture
about $Inert(\sigma_A)$.%\\

It turns out that the sign and gyration number homomorphisms are related
 to the dimension group representation through the so-called
{\em SGCC homomorphisms}  named after the {\em
sign-gyration-compatibility-condition}
which arose in the fundamental paper \cite{BK1}.
 For $k \geq 2$,  we have the SGCC homomorphisms
\[  SGCC_{A,k}:Aut(\sigma_A)  \longrightarrow Aut(\sigma_A|P_k)
\longrightarrow Z/k \]
%\noindent
given in \cite{KRW1},
\cite{W} by the formula
\[ SGCC_{A,k} =  GY_{A,k} + \sum_{i>0}OS_{A,k/2^i}\]   % \label{SGCC}
%\noindent
where we take $OS_{A,k/2^i} = 0 $ whenever $k/2^i$ is not an integer and
where, for $k$
even, we  consider  $Z/2$  as the subgroup $ \{0,k/2\}$ of $Z/k$. In
particular,
$SGCC_{A,k} = GY_{A,k}$ whenever $k$ is odd, and
\[ SGCC_{A,2} =  GY_{A,2} + OS_{A,1}   .\]
For simplicity of notation we shall usually just write $GY_k = GY_{A,k}$ ,
$OS_k = OS_{A,k}$, and $SGCC_k = SGCC_{A,k}$. The relationship between
the sign
 and gyration homomorphisms $OS_k$ and $GY_k$ and
the dimension group homomorphism $\delta_A$ is that $SGCC_k$ factors through
$\delta_A$. See \cite{KRW1}. We therefore know that
\[ SGCC_k(\alpha) = 0 %\mbox
\text{\hspace*{.2in}whenever\hspace*{.05in}} \alpha
       %\mbox
\text{\hspace*{.05in}lies in $Inert(\sigma_A)$. }\]
%\noindent
This paper announces the  characterization of the action of inert automorphisms
on the  periodic points $P_k(A)$ for sufficiently large $k$,
thereby  computing the image of $\pi_{A,k}$
restricted to the  inert subgroup $Inert(\sigma_A)$. Details are available
in the paper
\cite{KRW2}.
The principal result is %\\

%\noindent
%{\bf Main Theorem}
\begin{mthm}
%\begin{it} 
Assume that A is primitive and $k \geq 2$ is a
 positive integer such that
for every $n \geq k $ there is at least one periodic point of period
exactly $n$.
Let $\alpha$ be an element of $Aut(\sigma_A|P_k)$. Then there is an
automorphism
 $\beta$ in
$Inert(\sigma_A)$ satisfying $\alpha =  \beta|P_k$  iff  $SGCC_r(\alpha) = 0$
for every $2 \leq r \leq k$.  \label{mainth}
%\end{it}%\\
\end{mthm}

This is a corollary of the
following three theorems.%\\

%\noindent
%{\bf Theorem A}\begin{it} 
\begin{thma}\label{gyth} Assume that A is primitive and $2 \leq n \leq p$.
Let $\alpha$ be an
element of $Aut(\sigma_A|P_n^{\bullet})$ such that  $ GY_n(\alpha) = 0 $
and such
that $\alpha$ induces the identity permutation on orbits of length  n. Then
there is
an  automorphism $\beta$ in $Inert(\sigma_A)$ satisfying
\[\begin{array}{lll}
  \beta|P_n^{\bullet}& = &\alpha,  \\
  \beta|P_k^{\bullet}& = & %\mbox
\text{Identity, for
            $1 \leq k \leq p$  and $k \neq n$.}
\end{array}\] %\label{gyth}
%\end{it}%\\
\end{thma}

%\newpage\noindent{\bf Theorem B}\begin{it} 
\begin{thmb}\label{osth} 
Assume that $A$ is primitive and $1 \leq n \leq p$
are positive
integers. Let $\alpha$  be an even permutation of the orbits of length n.
Then there
is an  automorphism $\beta$ in $Inert(\sigma_A)$  satisfying
\[\begin{array}{cll}
  \alpha & = &%\mbox
\text{the permutation induced on orbits of length  n  by
$\beta$, }  \\
  GY_n(\beta)& = & 0,  \\
  \beta|P_k^{\bullet}& = & %\mbox
\text{Identity,  for
            $1 \leq k \leq p$  and $k \neq n$.}
\end{array}\] %\label{osth}
%\end{it}%\\
\end{thmb}
%\noindent
%{\bf Theorem C}\begin{it} 
\begin{thmc}\label{thc}
Assume that A is primitive and $k \geq 2$ is a
positive integer such that
for every $n \geq k $ there is at least one periodic point of period
exactly $n$.
Let $\alpha$ be in $Aut(\sigma_A|P_k)$. Let $gy_r = GY_r(\alpha)$  for $ 2
\leq r \leq k $
and  let $os_r = OS_r(\alpha)$   for $ 1 \leq r \leq k $. Assume that the
 sign-gyration-compatibility condition
\[  gy_r + \sum_{p>0}os_{r/2^p} = 0\]
holds for  $ 2 \leq r \leq k $. Then there is an  element $\beta$ in
$Inert(\sigma_A)$  such that
\[ GY_r(\beta) = gy_r %\mbox
\text{\hspace{.2in}for\hspace{.2in}} 2 \leq r \leq
k, \]
\[ OS_r(\beta) = os_r %\mbox
\text{\hspace{.2in}for\hspace{.2in}} 1 \leq r \leq
k. \] %\label{thc}
%\end{it}%\\
\end{thmc}

%\noindent{\bf 
\subsection*{Application 1.  Computing the image of $\pi_{A,k}$ for $k$ large} %.}
Our three theorems above show how to find generators
$\{\alpha_1, \ldots ,\alpha_p\}$ for  the subgroup
\[ \pi_{A,k}(Inert(\sigma_A))\subset Aut(\sigma_A|P_k)\]
%\noindent
for large enough $k$. If we can find a set of elements
 $\{\beta_1, \ldots ,\beta_q\}$ in
$Aut(\sigma_A)$ so that the set
$\{\delta_A(\beta_1), \ldots ,\delta_A(\beta_q)\}$ generates the image of
$Aut(\sigma_A)$ under the dimension group representation $\delta_A$, then
$\{\alpha_1, \ldots ,\alpha_p,\pi_{A,k}(\beta_1), \ldots ,\pi_{A,k}(\beta_q)\}$
is a set of generators for
the subgroup
\[ \pi_{A,k}(Aut(\sigma_A))\subset Aut(\sigma_A|P_k).\]
%\noindent
In other words, if we can find automorphisms which generate  the image of
the dimension
 group representation, then we can find generators for  the image
of  $\pi_{A,k}$ for $k$ large. For example, this applies to the full Bernoulli
$n$-shift $(X_n,\sigma_n)$. A set of generators for the image of $\delta_n$
consists of the automorphisms
\[  \sigma_r \times Id_{n/r}: X_r\times X_{n/r}\longrightarrow X_r\times
X_{n/r}\]
%\noindent
where $r$ is a prime divisor of $n$. %\\
%\noindent{\bf 
\subsection*{Application 2.  FOG  Counterexamples} %. } 
Boyle and Fiebig in \cite{BF}
characterized
the action of products of finite order inert automorphisms on periodic points
of mixing  SFT's. Together with the theorems above,
 this produces many
subshifts of finite type $(X_A,\sigma_A)$ having inert  automorphisms which
are not
products of finite order inert automorphisms. One of the simplest examples
is where
\[ A = \left(  \begin{array}{cccc}
                  1 & 1 & 0 & 0 \\
                  0 & 1 & 1 & 0 \\
                  0 & 0 & 1 & 1 \\
                  1 & 0 & 0 & 0
                 \end{array}      \right). \]     \label{A}
%\noindent 
According to Theorem B, the cyclic permutation (123) of the three
fixed points is the restriction of an inert automorphism $\beta$ to the set of
 fixed points.
 On the other hand, since $(X_A,\sigma_A)$  has no points of period 2 or
period 3, \cite{BF} says
that  any product of finite order inert automorphisms must be the identity on
the fixed points. Incidentally, for this example it can be verified as in
\cite[6.4]{BLR} that the automorphism group $Aut(s_A)$ of the dimension
 group is a
torsion free, finitely generated abelian group. Hence, every finite order
 element of  $Aut(\sigma_A)$ is inert. So
 $\beta$ is not even a product of finite order  automorphisms.%\\

Heretofore, the big subgroups of infinite order elements in $Inert(\sigma_A)$
were produced
using very ingenious variations on the well-known ``marker method"  which
gives rise to  products of  finite order elements.  An interesting problem
is to find natural
 representations of $Inert(\sigma_A)$ into, say, a vector space or a 
torsion-free group which would
 detect the new infinite order inert automorphisms.%\\


%\newpage\bigskip\noindent{\large \bf 
\section*{Polynomial matrices}%\\



\ip The key technique we use is to present SFT's in terms graphs arising from
square matrices with nonnegative polynomial entries (variable length coding)
 and to construct
conjugacies between SFT's from row and column operations on these matrices
(as in algebraic K-theory)
but with certain natural positivity conditions brought in.
 A {\em nonnegative polynomial matrix}
 is one with entries in the
set $tZ^+[t]$ of polynomials in the variable $ t$ having nonnegative integer
coefficients and zero
constant term. If $ A = (A(i,j))$ is a nonnegative polynomial matrix,
construct the
directed graph, or equivalently, the zero-one matrix $A^{\#}$ as follows: The
indices $i$ and $j$ will be called the {\em primary vertices} of $A^{\#}$.
Suppose
\[A(i,j) = a_1t + a_2t^2 + \dots  + a_nt^n. \]
Corresponding to the term $a_kt^k$ in $A(i,j)$ draw $a_k$ simple paths of
length $k$ from  $i$ to $j$, each having $k$ edges and $k-1$ {\em secondary
vertices}. By definition, we will let
\[ (X_A,\sigma_A) = (X_{A^{\#}},\sigma_{A^{\#}}).  \]
%\noindent
The usual SFT associated to a nonnegative integral matrix $A$ is obtained by the
 above construction from the nonnegative polynomial matrix $tA$.%\\


\ip Polynomial matrices turn out to be a very compact and efficacious way of
representing SFT's and their automorphisms. For example, a special case of
\cite[Theorem 1.7]{BGMY} shows that
\[  det(I-A)  = det(I-tA^{\#}).        \label{zeta}\]
%\noindent 
So the Bowen-Lanford formula \cite{Sm} for the zeta function yields
\[ \zeta_A(t) =\zeta_{A^{\#}}(t) = \frac{1}{det(I-A)}.\]
%\noindent
There is a corresponding formula for dimension groups. Namely,
there is an isomorphism of $Z[t,t^{-1}]$-modules
\[
  \frac{Z[t,t^{-1}]^n}{ %\mbox
\text{Image}(I-A)} \cong
            \frac{Z[t,t^{-1}]^{n^{\#}}}{%\mbox
\text{Image}(I-tA^{\#})}.
\]
%\noindent
This result has also been obtained by M. Boyle.%\\

The polynomial matrix setting provides useful approximation lemmas.  Suppose
we have a sequence of nonnegative polynomial matrices
\[A_m = B + t^mP_m \]
%\noindent
 where the entries $ p(t) = p_1t + \dots + p_rt^r$ of $P_m$ satisfy the
conditions
$r \leq Dm$ and $p_i\leq Cm^e$ for  constants $e,C,D$ which are independent
of $m$. %\\
%\noindent{\bf Entropy Limit Lemma}\begin{it} 
\begin{ella}$\lim_{m \rightarrow \infty} h(A_m) = h(B). $
\end{ella} %\end{it}%\\

%\noindent{\bf Periodic Point Lemma}\begin{it} 
\begin{ppl}
Let $A$ be a nonnegative  polynomial
matrix such that $A^{\#}$ is primitive. Assume
\[\lim_{m \rightarrow \infty} h(A_m) = L  < h(A).\]  Then   there
is an integer $K$  such that whenever  $m \geq K $  we have
\[ |P_q^{\bullet}(A_m)| \leq |P_q^{\bullet}(A)|   %\mbox
\text{\hspace{.3in}for $ q
                      \geq m. $}     \] \label{perA}
\end{ppl} %\end{it}%\\

     We now discuss how the polynomial matrix setting can be used to
produce strong
shift equivalences and topological conjugacies between SFT's. If $a(t) =
\sum_r a_r
t^r$ and  $b(t) = \sum_r b_r t^r$ are polynomials, we define $a \leq b $
iff $a_r
\leq b_r $ for each $r$. Let  $A$  be an $n \times n$ nonnegative polynomial
matrix.
Fix an entry $A(k,l)$ of $A$ where $k \neq l $, and let $b$ denote a polynomial in
$tZ[t]$. Let $E_{kl}(b)$  denote the elementary matrix which is the identity on
the diagonal and where the only nonzero entry off the diagonal is $b$ in the
$k$th row and $l$th column.  $E_{kl}(b)$
will be  called a {\em positive shear} if $b\geq 0$. Let the matrix $B$ be
defined by
one of the following equations:
\[ \begin{array}{lll}
  I - B & = & E_{kl}(b)(I-A),  \\
  I - B & = & (I-A)E_{kl}(b).
 \end{array} \label{elem}\]
%\noindent
%{\bf Positive Shear Lemma}\begin{it} 
\begin{psl} Assume that $0 \leq b
\leq A(k,l)$.
Then $B$ is a
nonnegative polynomial matrix and there are topological conjugacies
\[R_{kl}(b):(X_{A^{\#}},\sigma_{A^{\#}}) \longrightarrow
(X_{B^{\#}},\sigma_{B^{\#}}), \]
\[C_{kl}(b):(X_{A^{\#}},\sigma_{A^{\#}}) \longrightarrow
(X_{B^{\#}},\sigma_{B^{\#}}) \]
corresponding respectively to the first and second equations above.
\label{elemc}
\end{psl} %\end{it}%\\

This lemma is proved  using state splitting and merging \cite{LM}. The
relation to
algebraic K-theory is that, roughly speaking, state splitting and merging
corresponds to elementary
 row and column operations but with an extra positivity condition imposed.
 %Mike Boyle has shown  that 
Conjugacies between subshifts
 of finite type
are generated by the positive row and column  type conjugacies $R_{kl}(b)$
and $C_{kl}(b)$
together with isomorphisms of graphs. This gives an algebraic formulation
to the well-known classification  theorem that conjugacies between SFT's
are generated by state splitting and merging together with graph
isomorphisms. %\\
%\bigskip\noindent{\large \bf Idea of Proof}%\\
\section*{Idea of proof}

\ip  The purpose of this section is to indicate the proof of a special case of
 Theorem~B. Namely, assume that $(X_A,\sigma_A)$ is primitive and 
$ P_1^{\bullet}(A) = \{p_1,\ldots,p_r\} $ for $r\geq3$. Then the  cyclic
permutation  $(123)$ on
 fixed points may be
 realized by the action of an inert automorphism.
The strategy is to  construct certain inert automorphisms
of  model subshifts of finite type $(X_C,\sigma_C)$  corresponding to
polynomial
matrices $C = C_m$ such that
\begin{itemize}
\item $\sigma_{C}$ has  exactly three fixed points and no other
 periodic points of period $q$ for $2\leq q \leq m$.
\item For $m$ large,  $ h(C) < h(A)$ and
$|P_s^{\bullet}(C)| \leq |P_s^{\bullet}(A)|$
    for $s \geq m$.
\item   $(X_C,\sigma_C)$ has an inert automorphism $\beta$
inducing $(123)$ on fixed points.
\end{itemize}

\noindent Now let $X_B = X_C \cup \{p_4,\ldots,p_r\}$
and extend $\beta$ by letting it be the identity on  $ \{p_4,\ldots,p_r\}$.
 We then apply
the Imbedding Theorem \cite{K} and the
 Inert Extension Theorem  \cite{KR1} to imbed $(X_B,\sigma_B)$ into
$(X_A,\sigma_A)$
and to extend $\beta$ to an inert automorphism of $(X_A,\sigma_A)$ as
required. %\\


  To construct the model $(X_C,\sigma_C)$, we  find $6\times 6$ nonnegative
polynomial matrices
$C = C_m$ and $D = D_m$  such that

\begin{itemize}
\item $\sigma_{C}$ has  exactly three fixed points $\{\mu_1,\mu_2,\mu_3\}$
and no other
 periodic points of period $q$ for $2\leq q \leq m$.
\item Let $\theta = (12)$. Then $C\theta = \theta C$,  $\theta(\mu_1) = \mu_2$,
and $\theta(\mu_2) = \mu_1$.
\item $\delta_C(\theta)$ is in the center of $Aut(s_C)$.
\item $\sigma_{D}$ has  exactly three fixed points $\{\nu_1,\nu_2,\nu_3\}$
and no other
 periodic points of period $q$ for $2\leq q \leq m$.
\item Let $\phi = (23)$. Then $D\phi = \phi D$,  $\phi(\nu_2) = \nu_3$,
and $\phi(\nu_3) = \nu_2$.
\item There is a conjugacy $\gamma$ between $(X_C,\sigma_C)$
and $(X_D,\sigma_D)$ satisfying
$\gamma(\mu_i) = \nu_i$ for $i = 1,2,3 $.
\end{itemize}

\noindent
Let  $\beta = [\theta, \gamma\phi\gamma^{-1}]$. Then $\beta$ is inert and is the
 cycle (123) on  $\{\mu_1,\mu_2,\mu_3\}$. %\\


Here is the algebraic formulation. We  find $6\times 6$ nonnegative polynomial 
matrices $C$ and $D$ so that
%\begin{eqn}
\refstepcounter{theorem}
\begin{equation}
D -I = S(C-I)T \label{jjj}
\end{equation}
%\end{eqn}\noindent
 where $S$ and  $T = S^t$ are certain  products of positive shears
 which give rise to the desired conjugacy $\gamma$. $S$ is fixed
 in advance, and
 we show that $C$ can be chosen so that  $C\theta = \theta C$ and
so that
the $D$ defined by (\ref{jjj}) is nonnegative and $D\phi = \phi D$.
 This involves
solving a set of homogeneous linear equations satisfied by the unknown
$X = C-I$ with coefficients which are products of the entries of $S$.
To control the number of low order periodic points, there is
 a congruence
condition  which must be satisfied by $X$ modulo $t^m$. We also require
that $X$
satisfies a
certain congruence condition  modulo $\Delta^2$, where $\Delta = t-1-t^m$,
as well as  an additional
nonhomogeneous  linear equation
%\begin{eqn}
\refstepcounter{theorem}
\begin{equation}
X_{11}- X_{12} = \Delta.
\end{equation}
%\end{eqn}\noindent
 This will insure that there is a decomposition of the dimension group
%\begin{eqn}
\refstepcounter{theorem}
\begin{equation}
 Coker(I-C) \otimes Q = K_- \oplus K_+  \label{kkk}
\end{equation}
%\end{eqn}\noindent 
as a $Q[t,t^{-1}]$-module where $K_-$ and $K_+$ are the
 $-1$ and $+1$ eigenspaces of
$\theta$  and where  $K_-$ and $K_+$ have relatively prime annihilators.
 So any $Q[t,t^{-1}]$-module isomorphism of  $Coker(I-C) \otimes Q$ must
preserve
(\ref{kkk}), and consequently $\delta_C(\theta)$ will be  in
the center of $Aut(s_C)$.%\\

   The proof of Theorem A is very similar to that of Theorem 
B. In both cases, the key step is to construct
inert automorphisms of model SFT's
 with low entropy and good
control over the growth of  periodic points. This is done by fixing {\it a
priori}
 certain products of positive shears $S$ and $T= S^t$, and then showing how
to choose
$C$  so that the matrix $D$ defined by
\[D -I = S(C-I)T  \label{CD} \]
%\noindent 
satisfies a symmetry condition. The proof of Theorem C
is somewhat different. We first construct
 {\it a priori}  nonnegative polynomial matrix models $C_1$, $D_1$
and products of shears $S_1$, $T_1$ so that
\[ D_1-I = S_1(C_1-I)T_1 + Y(t^m\Delta^2)^r\]
%\noindent
for a certain polynomial  $\Delta $ which is relatively prime to $t^m$.
Then in several steps we enlarge $C_1$
to another  matrix $C$ satisfing  an involution symmetry condition
 and an appropriate  eigenspace determinant condition,
and we multiply  $S_1$ and $T_1$  on the left and right respectively by
 additional products of shears  depending
on the error term $Y(t^m\Delta^2)^r$ to obtain  new products of shears
 $S$ and $T$  so that the matrix
$D$ defined by
\[D -I = S(C-I)T\]
%\noindent
 satisfies  another  involution symmetry condition and
  the same  eigenspace determinant condition and so that
$S$ and $T$  satisfy certain positive shear conditions.

%\vspace*{.25in}\noindent{\large\bf Acknowledgement}
\section*{Acknowledgement}
\ip We would like to thank Mike Boyle and Doug Lind for  conversations
about this material. %\\


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\end{thebibliography}

%\setlength{\parskip}{0pt}


%\noindent K.H.Kim and F.R.Roush, Department of Mathematics, Alabama State
%University, Montgomery, Alabama 36101 \\
%kkim@@asu.alasu.edu\\

%\noindent
% J.B.Wagoner, Department of Mathematics, UCB, Berkeley, California
%  94720\\
%wagoner@@math.berkeley.edu


\end{document}

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