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\controldates{15-DEC-1997,15-DEC-1997,15-DEC-1997,15-DEC-1997}
 
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\issueinfo{3}{21}{January}{1997}
\dateposted{December 17, 1997}
\pagespan{126}{130}
\PII{S 1079-6762(97)00037-1}
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\copyrightinfo{1997}%            % copyright year
  {American Mathematical Society}% copyright holder
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\begin{document}
\title{Invariants of twist-wise flow equivalence}
%    Information for first author
\author{Michael C. Sullivan}
%    Address of record for the research reported here
\address{Department of Mathematics (4408),
         Southern Illinois University,
Carbondale, IL 62901}
\email{msulliva@math.siu.edu}
%    General info
\subjclass{Primary 58F25, 58F13; Secondary 58F20, 58F03}
%\date{July 1, 1997}
\date{June 18, 1997}
\revdate{October 4, 1997}

\commby{Jeff Xia}

\keywords{Dynamical systems, flows, subshifts of finite type}
\begin{abstract}
Twist-wise flow equivalence is a natural generalization
of flow equivalence that takes account of twisting  in
the local stable manifold of the orbits of a flow. Here
we announce the discovery of two new invariants in this
category.
\end{abstract}
\maketitle

\section{Flow equivalence}
Square nonnegative integer matrices are used to describe
maps on Cantor sets known as subshifts of finite type.
Two such matrices are {\em flow equivalent} if their induced
subshifts of finite type give rise to topologically equivalent
suspension flows. Here {\em topologically equivalent} just means
there is a homeomorphism, taking orbits to orbits, while
preserving the flow direction. A matrix $A$ is {\em irreducible} if
for each
$(i,j)$ there is a power $n$ such that the $(i,j)$ entry of $A^n$ is
nonzero.  
%In terms of the corresponding subshift and suspension,
%irreducibility is equivalent to the existence of infinitely many
%periodic orbits.
%Permutation
%matrices, on the other hand, give flows with a single closed orbit and
%are thus said to form the {\em trivial flow equivalence
%class.}
In terms of the corresponding subshift and suspension,
irreducibility is equivalent to the existence of a dense orbit.
Irreducible permutation
matrices give rise to flows with a single closed orbit and
are thus said to form the {\em trivial flow equivalence
class.}
For nontrivial irreducible incidence matrices
John Franks has shown that flow equivalence
is completely determined by two invariants, the Parry-Sullivan
number and Bowen-Franks group. Let $A$ be an $n \times n$ incidence
matrix. Then
\[
PS(A) = \det (I-A) %\mbox
\text{\quad and\quad}
BF(A) = \frac{ %\Z
\bZ^n}{(I-A) %\Z
\bZ^n}
\]
are the Parry-Sullivan number and the Bowen-Franks group respectively.
See \cite{PS}, \cite{BF}, and \cite{F} or the recent text \cite{LM}.
Huang  has settled the difficult classification problem arising
when the assumption of irreducibility is dropped, \cite{H1},
\cite{H2},
\cite{H3}.

\section{Twist-wise flow equivalence}
Represent $ %\Z
\bZ_2$ by $\{ 1, t \}$, under multiplication with $t^2=1$.
Let $A(t)$ be an $n \times n$ matrix with entries of the form $a+bt$,
with $a$ and $b$ nonnegative integers. That is $A$ is a matrix
over the semigroup ring $ %\Z
\bZ^+ %\Z
\bZ_2$. Call such a matrix a {\em twist matrix}.
One interpretation of twist matrices is as follows. Suppose
the suspension flow for $A(1)$ is realized as a 1-dimensional
basic set, $\mathcal B$, of saddle type, of a flow on a 3-manifold.
For each orbit in $\mathcal B$ there is a 2-dimensional local stable
manifold, a {\em ribbon}, if you like. Call the union of such ribbons
the {\em ribbon set}, and denote it by $\mathcal R$. Each ribbon is
either an annulus, a \mobius band, or an infinity long strip.
Now, $A(1)$ is the incidence matrix for the first return map $\rho$
on the rectangles of a Markov partition, $\{R_1, \dots, R_n\}$, of
a cross section of a neighborhood of $\mathcal B$.
Thus, $A_{ij}(1)$ is the number of times $R_i$ passes through $R_j$.
If we orient the rectangles, then we can let $a_{ij}$ be the number
of components of $\rho(R_i) \cap R_j$ where  orientation is
preserved, and $b_{ij}$ be the number of components where orientation
is reversed by the action of $\rho$. Then $A_{ij}(t) = a_{ij} + b_{ij}t$.

It is not necessary that the manifold be 3-dimensional or that
there be only one stable eigenvalue. We only need a means of assigning
orientations to rectangles of a Markov partition. We note that
$A(-1)$ is related to the {\em structure matrix} of \cite{BF}.
Two ribbon sets are {\em topologically equivalent} if there is a
homeomorphism between them that preserves the flow direction.
This leads us to define two twist matrices to be {\em twist-wise
flow equivalent} if they induce topologically equivalent ribbon sets.
\begin{theorem}
The numbers $PS(A(\pm 1))\!$ and the groups $BF(A(\pm 1))$ are invariants
of twist-wise flow equivalence.
\end{theorem}
It is clear that $PS(A(1))$ and $BF(A(1))$ are invariants in this
category. In \cite{S} it is shown that $PS(A(-1))$ is also invariant
and it can now be reported that $BF(A(-1))$ is too \cite{S2}.
We define an additional invariant in \S 4. However, we still do not
possess a complete set of invariants. See \S 6.


\section{Matrix moves}
Twist-wise flow equivalence (or {\em twist equivalence} for short) is
generated by three matrix moves \cite{S}. These are called, the {\em shift}
move, the {\em expansion} move, and the {\em twist} move, and are
denoted by $\eqs$, $\eqe$ and $\eqt$, respectively. The first two
generate flow equivalence \cite{PS}. We define them below.
{\bf Shift:} $A \eqs B$ if there exist rectangular matrices $R$ and $S$,
over $ %\Z
\bZ^+ %\Z
\bZ_2$, such that $A=RS$ and $B=SR$.
{\bf Expansion:} $A \eqe B$ if $A=[A_{ij}]$ and
\[
B    = \left[ \begin{array}{cccc}
        0 & A_{11} & \cdots & A_{1n} \\
        1 &    0   & \cdots &    0   \\
        0 & A_{21} & \cdots & A_{2n} \\
        \vdots & \vdots &       & \vdots \\
        0 & A_{n1} & \cdots & A_{nn}    \end{array} \right],
\]
or vice versa.
{\bf Twist:} $A \eqt B$ if $A=[A_{ij}]$ and
\[ B(t) = \left[ \begin{array}{cccc}
        A_{11} & tA_{12} & \cdots & tA_{1n} \\
        tA_{21}& A_{22}  & \cdots & A_{2n}  \\
        \vdots & \vdots  &        & \vdots  \\
        tA_{n1}& A_{2n}  & \cdots & A_{nn}  \end{array} \right]. \]
The shift move includes relabeling, so the expansion and twist moves
can be done on other ``locations'' in the matrix. See \cite{S} for
geometric motivations.

\section{The double cover flow}
We now consider another means of encoding the twisting of a ribbon
set. For a 2-dimensional ribbon set $\mathcal R$ place a flow on the
boundary with direction parallel to the flow on its core $\mathcal B$.
Call this the {\em double cover flow} of $\mathcal B$. An incidence
matrix $DA$ can be constructed from a twist matrix $A(t)$ by replacing
each entry $a+bt$ with $\left[ \begin{array}{cc} a & b \\ b & a
\end{array} \right]$. This amounts to using the matrix representation
\[
 %\Z
\bZ_2 \cong \left\{ 
\left[ \begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right],
\left[ \begin{array}{cc}
0 & 1 \\
1 & 0 
\end{array}
\right] \right\}.
\]
As this process is completely formal we need not be restricted to
2-dimensional ribbon sets. 
It is easy to show that $PS(DA)$ and $BF(DA)$ are invariants of twist
equivalence. However, $PS(DA) = PS(A(+1)) \times PS(A(-1))$, and thus
contains no new information. But $BF(DA)$ does distinguish twist
matrices that  the other invariants do not. Yet $PS(A(\pm1))$,
$BF(A(\pm1))$, and $BF(DA)$ are not complete. See \S 6.

\section{Other Abelian groups}
If we replace $ %\Z
\bZ_2$ with an arbitrary
Abelian group $G$, the only major change is that the twist move must
be replaced by a series of $g$-moves for each $g \in G$ defined by
multiplying the first row by $g$ and the first column by $g^{-1}$.
The Parry-Sullivan invariant becomes an element of the group
ring $ %\Z
\bZ G$. In \cite{BF} the Bowen-Franks group was defined for
arbitrary Abelian groups, so there should be no trouble there.
If one has a matrix representation of $G$, then even an analog of the 
double cover operation should go through. 
Suppose one has a finite directed graph and an associated 
subshift of finite type. Let the edges be labeled with elements
of $G$ and suppose we seek conjugacies that preserve this additional
information. Each closed orbit is paired with the product of
its edge labels and this pairing is to be preserved under
conjugacies. Then the analogs of the Parry-Sullivan and Bowen-Franks
invariants should be useful. We are hopeful that researchers in
other areas of dynamics or coding theory will find that they can
exploit these ideas.

\section{Examples}
Consider $2 \times 2$ matrices with entries 0, 1, or $t$, but which
are irreducible and nontrivial. For brevity
we shall denote $\left[ \begin{array}{cc} a & b \\ c& d \end{array}
\right]$ by $abcd$. We will divide these matrices into 6 classes:
\begin{itemize}
\item $A=\{ 1111, 1110, 1101, 1011, 1tt1, 1tt0, 0tt1 \}$,
\item $B=\{ t111, 111t, ttt1, 1ttt \}$,
\item $C_1=\{ 1t11, 11t1  \}$,
\item $C_2=\{ tt11, t1t1, 1t1t, 11tt, 1t10, 11t0, 0t11, 01t1, t110,
ttt0, 011t, 0ttt \}$,
\item $D=\{ t11t, tttt, tt10, t1t0, 01tt, 0t1t \}$,
\item $E=\{ t1tt, tt1t \}$.
\end{itemize}
Within each of these classes we have shown that the matrices are twist
equivalent by constructing the necessary matrix moves. In Table 1
we list the invariants for each class. Classes $A$ and $B$ cannot
be distinguished, yet their ribbon sets cannot be homeomorphic since
the ribbon set for $B$ contains \mobius bands whereas the closed ribbons
for $A$ are all annuli. Hence our invariants are not complete. Of
course, one can simply take orientability as an additional invariant.
The frustrating point is that the motivation behind all the new invariants
was precisely to capture orientation data. The classes $C_1$ and $C_2$
also have the same set of invariants. However, we have not been able
to tell if they form a single twist class or not.

\begin{table}[ht]
\caption{}
\[ \begin{array}{|c|r|c|r|c|c|c|} \hline
%\mbox
\text{Class} & PS^+ & BF^+ & PS^- & BF^- & BF^D \\  \hline \hline
   A    &  -1  &  0   &  -1  &  0   &  0   \\ \hline
   B &  -1  &  0   &  -1  &  0   &  0   \\ \hline
  C_1   &  -1  &  0   &   1  &  0   &  0   \\ \hline
  C_2   &  -1  &  0   &   1  &  0   &  0   \\ \hline
   D    &  -1  &  0   &   3  &  %\Z
\bZ_3 &  %\Z
\bZ_3 \\ \hline
   E    &  -1  &  0   &   5  &  %\Z
\bZ_5 &  %\Z
\bZ_5 \\ \hline
\end{array} \]
\end{table}
In Table 2 we have listed the invariants for a sampling of $3 \times
3$ matrices. Several features stand out. Any finitely generated
Abelian group can be realized as a $BF(A(1))$ group \cite{F}, and
we think this is
likely true for $BF(A(-1))$ as well. However, certain groups do not
seem to show up as Bowen-Franks groups of double cover flows.
For example, we have not found $ %\Z
\bZ_2 \oplus  %\Z
\bZ_4$ or $ %\Z
\bZ_{12}$,
though our work here is still very preliminary. In some cases we have
$BF^D = BF^+ \oplus BF^-$ (we use a more condensed notation here and
in the tables). This never seems to happen if $BF^D$ has infinite
order. Does this say anything interesting about the flows?

%The $BF$ groups are presented in Smith Normal Form \cite{New} and many of
%the
%calculations were done with the $\tt ismith$ command in Maple's
%$\tt linalg$ package.
\begin{table}[ht]
\caption{}
\[ \begin{array}{|c|r|c|r|c|c|c|} \hline
%\mbox
\text{Matrix} & PS^+ & BF^+ & PS^- & BF^- & BF^D \\  \hline \hline
t11111111   & -2  &  %\Z
\bZ_2  & -4  &  %\Z
\bZ_4  &  %\Z
\bZ_8             \\ \hline
1t1111111   & -2  &  %\Z
\bZ_2  &  0  &  %\Z
\bZ    &  %\Z
\bZ               \\ \hline
tt1111111   & -2  &  %\Z
\bZ_2  & -2  &  %\Z
\bZ_2  &  %\Z
\bZ_2^2           \\ \hline
t111t111t   & -2  &  %\Z
\bZ_2  &  0  &  %\Z
\bZ \oplus  %\Z
\bZ_3  &  %\Z
\bZ \oplus  %\Z
\bZ_3 \\ \hline
tt11t1111   & -2  &  %\Z
\bZ_2  & -4  &  %\Z
\bZ_4  &  %\Z
\bZ_8             \\ \hline
1t1t11111   & -2  &  %\Z
\bZ_2  &  2  &  %\Z
\bZ_2  &  %\Z
\bZ_2^2           \\ \hline
ttt111111   & -2  &  %\Z
\bZ_2  &  0  &  %\Z
\bZ    &  %\Z
\bZ               \\ \hline
t11t11111   & -2  &  %\Z
\bZ_2  & -2  &  %\Z
\bZ_2  &  %\Z
\bZ_2^2           \\ \hline
tt1tt1111   & -2  &  %\Z
\bZ_2  & -2  &  %\Z
\bZ_2  &  %\Z
\bZ_2^2           \\ \hline
t111t1111   & -2  &  %\Z
\bZ_2  & -6  &  %\Z
\bZ_6  &  %\Z
\bZ_2 \oplus  %\Z
\bZ_6 \\ \hline
ttttttttt   & -2  &  %\Z
\bZ_2  &  4  &  %\Z
\bZ_4  &  %\Z
\bZ_8 \\ \hline
tttttt1tt   & -2  &  %\Z
\bZ_2  &  6  &  %\Z
\bZ_6  &  %\Z
\bZ_2 \oplus  %\Z
\bZ_6 \\ \hline
tt1ttt1tt   & -2  &  %\Z
\bZ_2  &  0  &  %\Z
\bZ \oplus  %\Z
\bZ_3  &  %\Z
\bZ \oplus  %\Z
\bZ_3 \\ \hline
0t1111111   & -3  &  %\Z
\bZ_3  & -1  &  0    &  %\Z
\bZ_3             \\ \hline
0111t1111   & -3  &  %\Z
\bZ_3  & -5  &  %\Z
\bZ_5  &  %\Z
\bZ_{15}          \\ \hline
0t11t1111   & -3  &  %\Z
\bZ_3  & -3  &  %\Z
\bZ_3  &  %\Z
\bZ_3^2           \\ \hline
01t1011t1   & -4  &  %\Z
\bZ_4  &  0  &  %\Z
\bZ    &  %\Z
\bZ \oplus  %\Z
\bZ_2   \\ \hline
01t10111t   & -4  &  %\Z
\bZ_4  &  0  &  %\Z
\bZ    &  %\Z
\bZ \oplus  %\Z
\bZ_2   \\ \hline
011t011t1   & -4  &  %\Z
\bZ_4  & -2  &  %\Z
\bZ_2  &  %\Z
\bZ_8 \\ \hline
011t0111t   & -4  &  %\Z
\bZ_4  &  2  &  %\Z
\bZ_2  &  %\Z
\bZ_8 \\ \hline
01tttt110   & -4  &  %\Z
\bZ_4  &  6  &  %\Z
\bZ_6  &  %\Z
\bZ_{24}          \\ \hline
011t01110   & -4  & %\Z
\bZ_2^2 &  0  &  %\Z
\bZ \oplus  %\Z
\bZ_2  &  %\Z
\bZ \oplus  %\Z
\bZ_2^2 \\
\hline
\end{array} \]
\end{table}
The $BF$ groups are presented in Smith Normal Form \cite{New} and many of
the
calculations were done with the $\tt ismith$ command in Maple's
$\tt linalg$ package.
Finally, we present two twist matrices where the Bowen-Franks groups of the
double covers is the only distinguishing invariant. Let
\[ A(t) =
\left[ \begin{array}{ccc}
  3  & 1+t &  2  \\
1+t &  7  & 1+t \\
1+t & 1+t & 31
\end{array} \right]
\,\, %\mbox
\text{and}\,\,\, B(t) =
\left[ \begin{array}{ccc}
  3  & 1+t & 1+t  \\
1+t &  7  & 1+t \\
1+t & 1+t & 31
\end{array} \right].
\]
Then we get
\[ PS^+ = -224,\,\, BF^+ =  %\Z
\bZ_2 \oplus  %\Z
\bZ_4 \oplus  %\Z
\bZ_{28}, \]
\[ PS^- = -360,\,\, BF^- =  %\Z
\bZ_2 \oplus  %\Z
\bZ_6 \oplus  %\Z
\bZ_{30}, \]
for both $A(t)$ and $B(t)$, but
\[ BF(DA) =  %\Z
\bZ_4 \oplus  %\Z
\bZ_{24} \oplus  %\Z
\bZ_{840}, \]
while
\[ BF(DB) =  %\Z
\bZ_2^2 \oplus  %\Z
\bZ_{24} \oplus  %\Z
\bZ_{840}. \]

The author would like to thank John Franks for several helpful
conversations.
%\pagebreak
%\vskip -1cm
\bibliographystyle{amsplain}
\begin{thebibliography}{100}

\bibitem{BF}
R. Bowen and J. Franks, {\em
Homology for zero-dimensional nonwandering sets},
Ann. of Math. {\bf 106} (1977), 73--92.
\MR{56:16692}
\bibitem{F}
J. Franks, {\em
Flow equivalence of subshifts of finite type},
Ergod. Th. Dynam. Sys. {\bf 4} (1984), 53--66.
\MR{86j:58078}
\bibitem{H1}
D.~Huang, {\em
Flow equivalence of reducible shifts of finite type},
Ergod. Th. Dynam. Sys. {\bf 14} (1994), 695--720. \MR{95k:46110}
\bibitem{H2}
D.~Huang, {\em
Flow equivalence of reducible shifts of finite type and Cuntz-Krieger
algebras}, J. Reine Angew. Math. {\bf 462} (1995), 185--217.
\MR{96m:46123}
\bibitem{H3}
D.~Huang, {\em
Flow equivalence of reducible shifts of finite type and non-simple
Cuntz-Krieger algebras II: Complete classifications}, Preprint.

\bibitem{LM}
Douglas Lind and Brian Marcus,
{\em An introduction to symbolic dynamics and coding},
Cambridge University Press, 1995.
\MR{97a:58050}
\bibitem{New}
M.~Newman,
{\em Integral matrices},
Academic Press, 1972. \MR{49:5038}
\bibitem{PS}
B. Parry and D. Sullivan, {\em
A topological invariant of flows on 1-dimensional spaces},
Topology {\bf 14} (1975),  297--299.
\MR{53:9179}
\bibitem{S}
Michael C.~Sullivan, {\em
An invariant for basic sets of Smale flows},
To appear in Ergod. Th. Dynam. Sys.,
Preprint on http://nkrs465.math.siu.edu/$\sim$mike/Preprints.
\bibitem{S2}
Michael C.~Sullivan, {\em
Invariants of twist-wise flow equivalence},
Preprint on \newline
http://nkrs465.math.siu.edu/$\sim$mike/Preprints.
\end{thebibliography}
\end{document}
%---------------------------------------------------------------------------
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%% Modified December 2, 1997 by A. Morgoulis


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