POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
M. BOYLE AND J.B.WAGONER
Dedicated to Anatole Katok on the occassion of his 60th birthday.
A BSTRACT. This paper discusses classification of shifts of finite type using positive algebraic
K-theory.
1. I NTRODUCTION
Since R.F. Williams’ work in the early 1970’s, one of the main themes in studying shifts of
finite type and their isomorphisms has been strong shift equivalence theory. See the overviews
[B1,KR,W5]. On the other hand, an important technique in coding theorems and concrete applications involving shifts of finite type has been state splitting and merging. See [LM, M]. The purpose
of this article is to place state splitting and merging into an algebraic setting directly related to algebraic K-theory and to show how this is related to strong shift equivalence theory. We discuss how to
associate a shift of finite type (X(A), σ(A)) to a matrix A which has nonnegative integral polynomial entries and which satisfies the no Z-cycles condition (NZC) in (2.1). In addition, we show how
positive row and column operations over Z + [t] on I − A as in (3.2) give rise to conjugacies of shifts
of finite type. Fix a pair of indices (k, l) where k 6= l. Let b be an integral polynomial satisfying
0 ≤ b ≤ Akl . A positive row operation adds b times the lth row to the k th row of I − A, and a positive column operation adds b times the k th column to the lth column of I − A. The resulting matrix
is of the form I − B where B has nonnegative integral polynomial entries and satisfies NZC too.
Corresponding respectively to these row and column operations there are topological conjugacies
Lkl (b) and Rkl (b) from (X(A), σ(A)) and (X(B), σ(B)). This generalizes the material in [KRW]
and gives geometric content to the polynomial strong shift equivalence equations (PSSE) in Section
4. The construction of shifts of finite type in the presence of NZC is also known independently to
K.H.Kim and F.W.Roush.
Assume A and B are nonnegative polynomial matrices satisfying NZC. The main results are
Classification Theorem (X(A), σ(A)) and (X(B), σ(B)) are topologically conjugate iff there is
a sequence of positive row and column operations over Z + [t] connecting I − A and I − B.
Conjugacy Theorem Every topological conjugacy from (X(A), σ(A)) to (X(B), σ(B)) arises
from some sequence of positive row and column operations over Z + [t] connecting I − A and I − B.
The matrices I − A and I − B are the same in the algebraic K-theory group K1 (Q(t)) iff they are
connected by a sequence of row and column operations over the field of rational functions Q(t). The
Classification Theorem gives geometric meaning to positive row and column operations over Z + [t].
The Conjugacy Theorem is really part of the Classification Theorem. But we state it explicitly, because it is the first step in describing the group of automorphisms Aut(σ(A)) of (X(A), σ(A)) as
some kind of positive algebraic K-theory group K2 (Z + [t]). A sequence of row and column operations over Q(t) from I − A to itself gives rise to an element in K2 (Q(t)). Analogously, the
Conjugacy Theorem says that any element of Aut(σ(A)) arises from positive row and column operations over Z + [t] from I −A to itself. To construct the positive algebraic K-theory group K2 (Z + [t]),
it remains to specify what natural relations are satisfied.
Key words and phrases. positive algebraic K-theory, positive row and column operations, shifts of finite type.
The first author was supported in part by NSF Grant DMS 9971501
The second was supported in part by NSF Grant DMS 9706852.
1
2
M. BOYLE AND J.B.WAGONER
Section 2 describes the basic construction of the shift of finite type (X(A), σ(A)). Section 3
discusses positive row and column operations Lkl (b) and Rkl (b) and shows how they are related
to the conjugacies c(R, S) arising from strong shift equivalence theory. Section 4 proves the Conjugacy Theorem. Section 5 discusses zeta functions and dimension modules from the nonnegative
polynomial matrix viewpoint. Section 6 proves the Classification Theorem. Finally, the Appendix
gives background on strong shift equivalence theory necessary for this paper.
2. N ONNEGATIVE P OLYNOMIAL M ATRICES
Matrices with nonnegative polynomial entries provide a very compact, efficient, and powerful
way of representing shifts of finite type. Perhaps the first appearance of this idea is in Shannon’s
work [Sh] on information theory. In [KRW] and [KOR] the polynomial matrix technique is indispensible in studying automorphisms of shift spaces on the one hand and characterizing the spectra of
primitive, nonnegative integral matrices on the other. Polynomial matrix methods are used in [BL]
to get small presentations of shifts of finite type.
Consider a nonnegative polynomial matrix A = {Aij } where Aij = Aij (t) lies in Z + [t], the set
of polynomials in t with nonnegative integer coefficients. The indices i and j will range through the
positive integers, and we will assume that A has finite support, i.e., Aij 6= 0 for at most finitely many
pairs of indices (i, j). We will generally let I denote the identity matrix of infinite size. For matrices
appearing in the form I ± U , we will always assume U has finite support. This section describes
how to construct a shift of finite type (X(A), σ(A)) under the no Z-cycles condition (NZC)
(2.1)
The nonnegative integer matrix A(0) = {Aij (0)} has no periodic cycles
This construction generalizes the one in [KRW] where it was assumed that A(0) = 0, i.e., each
polynomial Aij is divisible by t. From the viewpoint of this paper, a compelling reason to construct
(X(A), σ(A)) in the presence of NZC is to give geometric meaning in the context of symbolic dynamics to the Polynomial Strong Shift Equivalence equations (PSSE) in Section 4. These are similar
to equations found in algebraic K-theory.
The first step is to construct the directed graph A[ as follows: The indices i and j will be called
the primary vertices. Suppose
Aij = a0 + a1 t + a2 t2 + . . . + an tn
Corresponding to the constant term a0 in Aij draw a0 arcs from i to j. These will be called constant
term routes. Corresponding to the power term ak tk in Aij where k > 0 draw ak simple paths of
length k from i to j, each having k edges and k − 1 secondary vertices. These paths from i to j will
be called power term routes. In particular, each route intersects the set of primary vertices only in its
starting vertex and its ending vertex. For example, the matrix
A=
gives rise to the graph A[
t + t2
t
1 + 2t3
0
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
@ : W
$
3
% % 1a
/2
where the constant term route is shown as a dotted arrow and the power term routes are shown as
solid arrows. The nonnegative polynomial matrix A and the graph A[ are essentially identical ways
of presenting the same data, and we will consider them as being the same.
We will explain two equivalent methods for constructing shift spaces from A[ . The first space
(P (A), σ(A)) will use the path space construction which generalizes the well known edge path construction explained in [LM]. This will be useful in showing how certain elementary row and column
operations on the matrix I − A give rise to topological conjugacies of shift spaces. The second
space (X(A), σ(A)) will come from a zero-one matrix A# of finite size. One of its uses is to show
(P (A), σ(A)) is actually a shift of finite type.
The Path Space Construction
Consider the set P of all sequences E = {En } = {(rn , tn )}, −∞ < n < ∞ , satisfying the
conditions
(1) Each rn is a route of A[ , and the end vertex of rn is the start vertex of rn+1 .
(2) Each tn is an integer and tn+1 = tn + |rn |, where |rn | is zero if rn is a constant
term route and is the number of arcs in rn if rn is a power term route.
The intuitive idea is that E is an infinite trip through A[ where at time tn the voyager is at the
starting vertex of the route rn and is about to traverse rn . Let Ω denote the (infinite) alphabet
consisting of all pairs (r, t) where r is a route of A[ and t is an integer. We give Ω the discrete
topology and endow P with the topology it inherits as a subset of the infinite product ΩZ . We say
E = {(rn , tn )} is equivalent to E 0 = {(rn0 , t0n )} and write E ∼ E 0 provided there is some integer
m so that rn+m = rn0 and tn+m = t0n for all n. Use the quotient topology to define
(2.2)
P (A) = P modulo ∼
We remark that P (A) is compact. To see this, let L be the length of the longest route in A[ . Let PL
denote the compact subspace of P consisting of those E satisfying 0 ≤ t0 ≤ L. Then the quotient
map from PL to P (A) is still onto, because the NZC implies that any E must have 0 ≤ tn ≤ L for
some n. Next define a shift map σ : P → P by the equation
(2.3)
σ(E)n = (rn , tn − 1)
4
M. BOYLE AND J.B.WAGONER
This respects the equivalence relation ∼ and induces a continuous shift map
(2.4)
σ(A) : P (A) → P (A)
The #-Construction
We will now construct a zero-one matrix A# with finite support from the graph A[ , and by
definition, we will let
(2.5)
(X(A), σ(A)) = {XA# , σA# }
where the bracket notation on the right denotes the standard “vertex shift” construction as explained
in [LM,W5] and in the Appendix.
Special Case A(0) = 0
This is just the edge path construction for the graph A[ . Namely, the set of states S # is the set of
arcs between primary and/or secondary vertices in A[ , and A# (α, β) = 1 iff the end vertex of α is
the start vertex of β.
General Case
The first step is to define the set S # of states of A# . The states will be quintuples
(2.6)
α = (i, α0 , j, α00 , k)
where
(1) Each of i, j, and k is a primary vertex,
(2) If i 6= j, then α0 is a connected path of constant term routes from i to j
and α00 is a subarc of a power term route from j to k.
(3) If i = j, then there are no paths of constant term routes from i to j because
of NZC. So we let α0 be the symbol φ, and we require α00 to be a subarc of
a power term route from j to k.
When i = j , the state α is really just the triple (j, α00 , k). So, for example, if Ajk = t, then α
is identified with the power term arc from j to k. But we also write α as a quintuple as in (2.6) to
preserve uniformity of notation. There are no states consisting only of a constant term path from one
vertex to another.
The next step is to define the transition function
(2.7)
A# : S # × S # → {0, 1}
Let α = (i, α0 , j, α00 , k) and β = (p, β 0 , q, β 00 , r). Then
A# (α, β) = 1
A# (α, β) = 0
when the conditions in Case 1
or Case 2 below hold
otherwise
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
5
Case 1. We have i = p, j = q, k = r, and α0 = β 0 . Both α00 and β 00 are subarcs on the same power
term route from j to k and the ending vertex of α00 is the starting vertex of β 00 .
Case 2. The subarc α00 is the last one along a power term route from j to k and therefore the ending
vertex of α00 is k. We require k = p, and we require that the subarc β 00 is the first one along a power
term route from q to r. In particular, the beginning vertex of β 00 is q.
The final step is to make A# into an n# × n# matrix where n# is the number of states. We do
this by
(2.8)
choosing an ordering of the set S #
#
Two different choices of orderings of S # will produce matrices A#
1 and A2 related by the equation
−1 #
A#
A1 Q
2 =Q
(2.9)
where Q is a permutation matrix. Q induces a conjugacy from (XA# , σA# ) to (XA# , σA# ) .
1
1
2
2
Comment NZC implies that the number of paths consisting only of constant term routes is finite.
Therefore S # is a finite set, and {XA# , σA# } will be a shift of finite type.
In the case where the constant term matrix A(0) = 0, there are no constant term routes and the
matrix A# is the same as the one constructed in [KRW]. If A is a matrix over Z + , then {tA}[ is
the graph associated to the matrix A and {tA}# is precisely the zero-one matrix describing the edge
0
path presentation matrix A of the shift of finite type associated to A. In particular, we have
(2.10)
(X(tA), σ(tA)) = {XA0 , σA0 }
See [LM] and the Appendix.
Example 2.11 Consider the matrix
A=
t
t
1
0
The graph A[ is
$
%
1e
2
There are three states:
The matrix A# is
α1
=
α2
=
α3
=
$
1
/2
1
2
/1
/1
6
M. BOYLE AND J.B.WAGONER

1
 1
1

0
0 
0
1
1
1
Hence
(X(A), σ(A)) = {XA# , σA# } = the full Bernoulli 2-shift
As discussed in the Appendix, we don’t require each row and each column of a matrix M to have a
nonzero entry in forming {XM , σM }.
Example 2.12 Consider the matrix
A=
t
1
t
0
The graph A[ is
$
%
1e
2
There are four states:
$
α1
=
α2
=
1
/2
α3
=
2
/1d
α4
=
1
/1
2
/2
The matrix A# is

1
 0

 1
0
1
0
1
0

0
1 

0 
1
0
1
0
1
which is just the matrix for the edge path presentation, as discussed in [LM], of the shift of finite
type associated to the graph
$
1e
%
2d
Hence, we again have
(X(A), σ(A)) = {XA# , σA# } = the full Bernoulli 2-shift
Equivalency of the Path Construction and the #-Constuction
Theorem 2.13. There is a topological conjugacy Φ : (P (A), σ(A)) → (X(A), σ(A)).
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
7
Here is how Φ and its inverse Ψ : (X(A), σ(A)) → (P (A), σ(A)) are constructed.
Definition of Φ: Let E = {(rh , th )} represent an infinite path in P (A) where −∞ < h < ∞.
We want to find an infinite allowable sequence Φ(E) = {Φ(E)n } of states in S # . The NZC
condition implies we can write the set of integers as the union of intervals [tp , tp+1 , . . . , tp+q+1 ]
where tp−1 < tp = . . . = tp+q < tp+q+1 and the intervals overlap only at their endpoints. Let
i be the start vertex of rp , j be the start vertex of rp+q , and k be the final vertex of rp+q . If
q = 0, let α0 = φ. If q > 0, let α0 be the path of constant term routes leading from i to j which
is the concatenation of rp , . . . , rp+q−1 . The power term route rp+q is the concatenation of subarcs
α100 , . . . , α`00 where tp+q+1 = tp+q +`. Then we let Φ(E)n = (i, α0 , j, αs00 , k) where s = n+1−tp+q
for tp+q ≤ n < tp+q+1 .
Definition of Ψ: Let x = {xp } be in X(A) where xp = (ip , αp0 , jp , αp00 , kp ). We want to get
Ψ(x) = {(rn , tn )} in P (A). The constant term paths αp0 and the power term routes αp00 fit together
compatibly to produce a sequence of primary vertices {vn } and a sequence of routes {rn } where rn
goes from vn to vn+1 . To establish the indexing of these routes which concatenate together, it suffices to specifiy that r0 is the power term route containing the arc α000 . To get the required sequence
of times {tn }, first let t0 = −(k − 1) = 1 − k where α000 is the k-th arc along r0 going from v0 to
v1 . Then we recursively define tn for n 6= 0 by the condition tn+1 = tn + |rn |.
Verification that Φ and Ψ commute with σ(A) and that they are continuous inverses of each other
is straightforward.
For future reference, we now generalize Nasu’s definition [N,W4] of simple automorphisms of
a shift of finite type. His work has played an important role. An elementary simple automorphism of (X(A), σ(A)) is one coming from an automorphism of the graph A[ which fixes the
primary vertices. A simple automorphism of (X(A), σ(A)) is one of the form αΘα−1 where
α : (X(A), σ(A)) → (X(B), σ(B)) is a topological conjugacy and Θ is an elementary simple
automorphism of (X(B), σ(B)). Composition is read from left to right.
Definition 2.14. Simp(σ(A)) is the subgroup of Aut(σ(A)) generated by simple automorphisms.
We briefly discuss in (4.20) below why this yields the same group as Nasu’s original definition.
3. P OSITIVE ROW AND C OLUMN O PERATIONS
In this section we discuss how positive row and column operations on matrices give rise to conjugacies between shifts of finite type defined by matrices satisfying NZC. In this section, as elsewhere
in this paper, composition is read from left to right. Also, we use the terms conjugacy and topological conjugacy interchangeabley.
Let A = {Aij } be a nonnegative polynomial matrix with finite support satisfying NZC (2.1). Fix
an entry Akl of A where k 6= l, and let b denote a polynomial in Z[t]. Let Tkl (b) denote the matrix
which has the entry b in the k th row and lth column and zeros elsewhere. Let I denote the infinite
identity matrix. Define
(3.1)
Ekl (b) = I + Tkl (b)
8
M. BOYLE AND J.B.WAGONER
Ekl (b) will be called a positive shear if b lies in Z + [t]. Define a new matrix of finite support B by
one of the equations
I −B
I −B
(3.2)
= Ekl (b)(I − A)
= (I − A)Ekl (b)
We will say that Eij (b) goes from I − A to I − B .
If a(t) =
P
r
ar tr and b(t) =
P
r br t
r
are polynomials, we define a ≤ b iff ar ≤ br for each r.
Positive Shear Lemma 3.3. Assume 0 ≤ b ≤ Akl . Then B is nonnegative and satisfies NZC.
Corresponding to the first and second equations in (3.2) respectively there are conjugacies
Lkl (b) : (X(A), σ(A)) → (X(B), σ(B))
Rkl (b) : (X(A), σ(A)) → (X(B), σ(B))
Lkl (b) and Rkl (b) are well defined up to multiplication on the left by elements in Simp(σ(A)) and
multiplication on the right by elements in Simp(σ(B)).
The relation of Lkl (b) and Rkl (b) to the well known operations of state splitting and merging in symbolic dynamics is that Lkl (b) comes from splitting off b incoming edges at the vertex l and Rkl (b)
comes from splitting off b outgoing edges at the vertex k. We will give a generalization of (3.3) in
(3.7) at the end of this section.
There is also a version of (3.3) overPZO[t], the set of zero-one polynomials. By definition, ZO[t]
consists of those polynomials a(t) = r ar tr where each coefficient ar is equal to zero or one. The
sum and product of two zero-one polynomials is not always a zero-one polynomial. Nevertheless,
there are still many cases when the equations (3.2) do hold over ZO[t]. This means that A has
zero-one polynomials as entries, b is a zero-one polynomial satisfying b ≤ Akl , and all the additions and multiplications occurring on the right hand side of (3.2) produce zero-one polynomials.
Finally, we require the matrix B defined in (3.2) to have zero-one polynomial entries. For example,
if (P, Q) : M → N is a strong shift equivalence in the category of zero-one matrices, then the
equations (4.12) through (4.15) below are examples where (3.2) holds over ZO[t].
Examination of the proof below of (3.3) shows
Remark 3.4. In the category of matrices over ZO[t], the conjugacies Lkl (b) and Rkl (b) are well
defined without any simple automorphism ambiguity.
Indication of Proof
We will discuss L = Lkl (b) in the context of (P (A), σ(A)). For simplicity, let b = tp be a single
route from the primary vertex k to the primary vertex l of length p ≥ 0. The matrix B (i.e., the graph
B [ ) is obtained from the matrix A by first deleting the route b between k and l and then inserting
a route r0 of length |r0 | = p + |r| from k to q for each route r starting at l and ending at q. Every
infinite trip through A corresponds uniquely to a concatenation of routes in B and this determines
L. More precisely, let E = {(rn , tn )} be in P (A). Consider a part of E like
. . . , (rn , tn ), (rn+1 , tn+1 ), (rn+2 , tn+2 ), . . .
where rn = b and rn+1 = r as above. Delete (rn+1 , tn+1 ) and replace (rn , tn ) with (r0 , tn ). Note
that tn+2 = tn + |r0 |. The result is just a subsequence E 0 of E where some of the items have been
changed. Now renumber this subsequence in an increasing fashion so that it becomes a sequence
E 0 = {En0 } where n runs through all the integers. The equivalence class modulo ∼ of E 0 does not
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
9
depend on this renumbering, and we define L(E) = E 0 .
We now discuss a generalization of the Positive Shear Lemma. If P = {Pij } and Q = {Qij } are
nonnegative polynomial matrices, we say P ≤ Q provided Pij ≤ Qij for each pair of indices (i, j).
Assume X is the conjugate of an upper triangular matrix by a permutation. Define an invertible
matrix
(3.5)
E(X) = I + X
Let A be an nonnegative polynomial matrix satisfying NZC as in (2.1). Define a new matrix B by
one of the equations
(3.6)
I −B
I −B
= E(X)(I − A)
= (I − A)E(X)
Generalized Positive Shear Lemma 3.7. Assume 0 ≤ X ≤ A and that
X 2 = 0. Then B is nonnegative and satisfies NZC . Corresponding to the first and second equations
in (3.6) respectively there are conjugacies
L(X) : (X(A), σ(A)) → (X(B), σ(B))
R(X) : (X(A), σ(A)) → (X(B), σ(B))
In the category of zero-one matrices, L(X) and R(X) are well defined. In the category of matrices over Z + [t], L(X) and R(X) are well defined up to multiplication on the left by elements in
Simp(σ(A)) and multiplication on the right by elements in Simp(σ(B)).
The proof is a straightforward generalization of the proof for the Positive Shear Lemma. The idea is
that the condition X 2 = 0 allows the construction in (3.3) to be done at various places in A simultanously in view of the following lemma.
Lemma 3.8. Let X be a nonnegative polynomial matrix satisfying the condition X 2 = 0. Then
there are disjoint sets of indices I and J such that Xij = 0 unless i is in I and j is in J.
If i < j whenever i is in I and j is in J, then X is an upper triangular matrix. More generally, X is
the conjugate of an upper triangular matrix by a permutation.
Proof Let I be the set of those indices i such that Xij 6= 0 for some j. Let J be the set of those
indices j such that Xij 6= 0 for some i. The condition X 2 = 0 implies I ∩ J = 0.
4. C ONJUGACIES
The purpose of this section is to show that conjugacies between shifts of finite type are generated
by the positive row and column type conjugacies Lkl (b) and Rkl (b) of (3.3). Polynomial matrices
A, B, etc. will have finite support.
Theorem 4.1. Let A and B be nonnegative polynomial matrices satisfying NZC. Any conjugacy
∆ : (X(A), σ(A)) → (X(B), σ(B)) can be written as a composition
∆=
n
Y
i=1
Cki li (bi )i
10
M. BOYLE AND J.B.WAGONER
where A0 = A, An = B, and Cki li (bi ) is a positive row or column conjugacy Lkl (b) or Rkl (b) as
in (3.3) which goes from I − Ai−1 to I − Ai if i = 1 and from I − Ai to I − Ai−1 if i = −1.
This will be a consequence of the next two results. The first is a special case of (4.1).
Proposition 4.2. Let A and B be zero-one matrices. Any conjugacy
∆ : (X(tA), σ(tA)) → (X(tB), σ(tB)) can be written as a composition
∆=
n
Y
Cki li (t)i
i=1
where A0 = tA, An = tB, and Cki li (t) is a positive row or column conjugacy as in (3.3) which
goes from I − Ai−1 to I − Ai if i = 1 and from I − Ai to I − Ai−1 if i = −1. Moreover, each
Ai = tDi where Di is a zero-one matrix.
Proposition 4.3. Let A be a nonnegative polynomial matrix satisfying NZC. There is a path of
positive row and column operations over Z + [t] connecting I − A and I − tA# .
We first show how (4.1) follows from (4.2) and (4.3). Then we prove (4.2) and (4.3).
Proof of (4.1)
Let A be any nonnegative polynomial matrix satisfying NZC. Since A(0) has
no cycles, there is a permutation matrix P such that P A(0)P −1 is upper triangular. Consequently,
there is a product E = EA of elementary matrices Eij (b) where b is a positive integer such that if
we define A0 by the equation I − A0 = E(I − A), then
A0 (0) = 0 and the positivity conditions in (3.3) are satisfied as each term
Eij (b) in E is applied successively to I − A reading from right to left.
(4.4)
From (3.3) we know that corresponding to E there is a conjugacy from (X(A), σ(A)) to (X(A0 ), σ(A0 )).
More specifically, this means there is a conjugacy
θA : (XA# , σA# ) → (XA# , σA# )
(4.5)
0
0
coming from the positive row operations in E. Next we apply (4.3) to obtain a conjugacy
#
ρA : (X(A0 ), σ(A0 )) → (X(tA#
0 ), σ(tA0 ))
(4.6)
coming from positive row and column operations connecting I − A0 and I − tA#
0 . Similarly, we get
conjugacies θB and ρB . Now write the conjugacy ∆ : (X(A), σ(A)) → (X(B), σ(B)) in the form
−1
−1 −1
∆ = θA ρA (ρ−1
A θA ∆θB ρB )ρB θB
−1
where composition is read from left to right. Apply (4.2) to express ρ−1
A θA ∆θB ρB as a product of
conjugacies coming from positive row and column operations.
Proof of (4.2)
By definition a conjugacy ∆ from (X(tA), σ(tA)) to (X(tB), σ(tB)) is a
conjugacy from (X{tA}# , σ{tA}# ) to (X{tB}# , σ{tB}# ). For any zero-one matrix M we have
0
{tM }# = M where M 0 is the edge path presentation or subdivision matrix associated to M as
discussed in [LM] and the Appendix. We also have XM 0 = X(tM ) and a canonical topological
conjugacy X(tM ) ' P (tM ). From (7.14) we know that ∆ is the subdivision of a conjugacy from
(XA , σA ) to (XB , σB ). This means that
Y
0
0
(4.7)
∆=
c(Ri , Si )i
i
0
0
where (Ri , Si ) is the subdivision of a strong shift equivalence (Ri , Si ) in the category of zero-one
0
0
matrices. It therefore suffices to prove (4.2) for c(R , S ) where (R, S) : A → B is a strong shift
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
11
equivalence in the category of zero-one matrices. Corresponding to the strong shift equivalence
equations
A = RS and B = SR
(4.8)
we will construct below a conjugacy
e(R, S) : {X{tA}# , σ{tA}# } → {X{tB}# , σ{tB}# }
defined by the equation
e(R, S) = f (R, S)f (I, B)−1
(4.9)
where, for any strong shift equivalence (U, V ) : M → N , we let
(4.10)
f (U, V ) = R21 (tV )−1 L12 (U )−1 R12 (U )L12 (tV ) = L12 (U )−1 R12 (U )
Composition is read from left to right.
0
0
Proposition 4.11. c(R , S ) = e(R, S)
Construction of e(R, S)
We will construct e(R, S) for strong shift equivalences over Z + , where in general it will be well
defined modulo simple isomorphisms. The proof shows there is no such ambiguity in the special
case when (R, S) : A → B is a strong shift equivalence in the category of zero-one matrices.
Let M = {Mij } be a nonnegative integer matrix of infinite size, but with only finitely many
nonzero entries. As in the Appendix, we let the support of M be the largest integer s = supp(M )
such that either Msk 6= 0 or Mks 6= 0 for some integer k. If A = 0, let supp(A) = 1. Given
(R, S) : A → B over Z + , let n be greater than or equal to supp(A), supp(B), supp(R), and
supp(S). Let I denote the n × n identity matrix. To obtain f (R, S) we now define the terms occurring in (4.10). The key set of equations involves 2n × 2n matrices.
Polynomial Strong Shift Equivalence Equations
(4.12)
(4.13)
(4.14)
(4.15)
0
I
I
−tS
I − tRS
−tS
I
0
R
I
I
−tS
I
tS
0
I
−R
I
I
−tS
I
0
I
tS
−R
I
R
I
0
I
=
=
0
I − tSR
I − tRS
0
I − tRS
−tS
=
I
−tS
=
0
I
0
I
0
I − tSR
I
0
0
I − tSR
We then apply the Generalized Positive Shear Lemma (3.7) to each of the equations (4.12), (4.13),
(4.14), and (4.15) respectively to obtain the conjugacies R21 (tS), L12 (R), R12 (R), and L12 (tS).
12
M. BOYLE AND J.B.WAGONER
From the path space construction, we have equalities of shifts of finite type
tRS 0
tRS 0
0
0
0
0
P
=P
and
P
=P
0
0
tS 0
tS tSR
0 tSR
Moreover, the construction in the Positive Shear Lemma shows that
R21 (tS) = Identity and L12 (tS) = Identity
This completes the construction of f (P, Q).
Proof of (4.11)
We compute the effect of L12 (R) and R12 (R) on an element z = {zi } of
0 R
P
tS 0
Consider the diagram
(4.16)
ai+1 t
/ ki+2
/ ki+1
{=
}>
{
}
}
{{
}}
{{
{
si t }}}
ri+1
ri
{{
}
{{ si+1 t
}}
{
}
{
}
{{
}}
{{
}}
/ li+1
li
c t
ki
ai t
i
Write z in the form
z = {. . . , ri , tsi , ri+1 , tsi+1 , . . .}
where ri is one of the Rki li constant term arcs from ki to li and si t is one of the Sli ki+1 power term
arcs of length one from li to ki+1 . We have 1 ≤ ki ≤ n and n + 1 ≤ li ≤ 2n. Let x = {xi } in
tA 0
tA 0
XA0 ' P (tA) = P
=P
0 0
tS 0
denote the image of z under L12 (R). Then xi = (ki , ai t, ki+1 ) where ai t is one of the Aki ki+1
power term arcs corresponding to the pair (ri , si t). Let
0 0
C=
0 B
Let w = {wi } in
0 0
0
0
XC 0 ' P (tC) = P
=P
0 tB
tS tB
denote the image of z under R12 (R). Then wi = (li , ci t, li+1 ) where ci t is one of the Cli li+1 power
term paths corresponding to the pair (si t, ri+1 ). We have
w = f (R, S)(x)
0
0
Let y = {yi } be the image of x under c(R , S ) in
XB 0 ' P (tB) = P
tB
0
0
0
0
0
Write yi = (pi , bi t, pi+1 ). Examination of the proof of the construction of c(R , S ) in [W4,Section
2] shows that w is equal to y except for a permutation of indices, i.e., li = pi + n and ci = bi for all
−∞ < i < ∞. So now we let P = I and Q = B, and we examine the construction of f (I, B) as
above to conclude that
w = f (I, B)(y)
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
13
Therefore, y is the image of x under e(R, S).
Proof of (4.3) We may assume A(0) = 0 in view of (4.4). Then the proof that there is a path of
positive row and column operations from I − A to I − tA# follows from repeated applications of
(4.19) below.
Suppose there is a path of length n in A from one primary vertex p to another primary vertex
q going thru intermediate primary vertices as is illustrated in the following diagram where we take
n = 3. This is essentially the general case.
(4.17)
p
t
/r
t
/s
t
/q
There may be other ingoing and outgoing edges at p and q, and we may also have p = q. But assume
there are no ingoing or outgoing edges at the vertices r and s other than those shown. Let B be
obtained by replacing this path by one of length n connecting p to q thru secondary vertices as in the
diagram
(4.18)
p
/
/
/q
t3
Claim 4.19. There is a path of positive row and column operations over Z + [t] from I − A to I − B.
Proof
There are two cases depending on whether p 6= q or p = q.
Case 1: p 6= q . Write the vertices of A in the
matrix representation of I − A looks like

1 − a(t) −t

0
1


0
0

 −c(t)
0
∗
0
order p, r, s, q followed by the other vertices. The
0
−t
1
0
0

−b(t)
∗
0
0 

−t
0 

1 − d(t) ∗ 
∗
∗
where the last row and column indicate the remaining portion of I
of I − B looks like

1 − a(t) 0 0 −b(t) − t3 ∗

0
1 0
0
0


0
0
1
0
0

 −c(t)
0 0 1 − d(t) ∗
∗
0 0
∗
∗
− A. The matrix representation






A sequence of positive row and column operations from I − A to I − B is (from left to right)
Rsq (t) , Rrs (t) , Rrq (t2 ) , Lpr (t) , Lps (t2 )
14
M. BOYLE AND J.B.WAGONER
Case 2: p = q . Write the vertices of A in the order p, r, s followed by the other vertices. The matrix
representation of I − A looks like


1 − a(t) −t 0 ∗

0
1 −t 0 



−t
0
1 0 
∗
0
0 ∗
The matrix representation of I − B looks like

1 − a(t) − t3

0


0
∗

0 0 ∗
1 0 0 

0 1 0 
0 0 ∗
A sequence of positive row and column operations from I − A to I − B is (from left to right)
Rrs (t) , Lpr (t) , Lps (t2 ) , Rsp (t)
Discussion of simple automorphisms 4.20.
Here is an example illustrating why the definition of simple automorphisms in (2.14) yields the
same group as Nasu’s original definition in [N]. Consider a portion of A[ which has m power term
routes of length two going from the primary vertex p to the primary vertex q as in the following
diagram.
/ /q
p
(4.21)
mt2
Any permutation α of these m routes determines a permutation β of the m routes of length one from
the primary vertex r to the primary vertex q in the diagram below.
(4.22)
p
t
/r
mt
/q
Observe that β is a simple automorphism in the sense of [N]. Let B [ be the obtained from A[ by
replacing (4.21) with (4.22). The conjugacy between (X(A), σ(A)) and (X(B), σ(B)) intertwining
α and β is Lpr (t) followed by Rrq (mt). This procedure is called zipping.
5. Z ETA F UNCTIONS AND D IMENSION M ODULES
This section generalizes the material in [KRW] on zeta functions and dimension modules to shifts
of finite type built from nonnegative polynomial matrices A with finite support satisfying NZC.
The Zeta Function
We define the zeta function of (X(A), σ(A)) to be
(5.1)
ζA (t) = ζA# (t)
By Proposition 4.3 and the formulas in (3.2), we know that
(5.2)
det(I − tA# ) = det(I − A)
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
15
Consequently
det(tI − A# ) = tv det(I − A(t−1 ))
(5.3)
where v denotes the number of vertices in A# and where A(t−1 ) is obtained from A by substituting
t−1 for t. The Bowen-Lanford formula [Sm] for the zeta function yields
ζA (t) =
(5.4)
1
det(I − A)
and we also have
Entropy of (X(A), σ(A)) = log(λA )
(5.5)
where λA is the largest root of tv det(I − A(t−1 )).
The Dimension Module
We will first present the dimension module (G(A), G+ (A), s(A)) of (X(A), σ(A)) from an algebraic viewpoint, and then we will show in (5.16) that this agrees with the usual of the dimension
group of {XA# , σA# }. Let n be a positive integer. Let F n be the standard free, left Z[t]-module of
rank n. Let F ∞ denote the free, left Z[t]-module which is direct sum of countably many copies of
Z[t]. Elements of F n and F ∞ will be written as row vectors. Define
G(A) = Coker(I − A) = F ∞ /Image(I − A)
(5.6)
Let n ≥ supp(A). The inclusion F n ⊂ F ∞ induces an isomorphism
(5.7)
F n /Image(I(n, n) − A(n, n)) = F ∞ /Image(I − A)
where, as in the Appendix, I(n, n) and A(n, n) are the finite n×n-matrices obtained by considering
just the first n rows and columns of I and A. Therefore, G(A) is a finitely generated Z[t]-module.
We let
(5.8)
s(A) =
the endomorphism of the Z[t]-module G(A)
coming from left multiplicition by t−1
Let F+∞ denote the subset of F ∞ consisting of elements that have all coordinates nonnegative.
Define
(5.9)
G+ (A) = Image of F+∞ in G(A)
Let X be a polynomial matrix of finite support and assume there is a permutation matrix P such
that P XP −1 is upper triangular with diagonal entries equal to zero. The matrix I + X is invertible
over Z[t]. Let A be a polynomial matrix of finite support. As in (3.6), define the matrix B by one of
the equations
(5.10)
I −B
I −B
=
=
(I + X)(I − A)
(I − A)(I + X)
Corresponding to the first and second equations respectively, we have two commutative diagrams of
exact sequences of left Z[t]-modules which induce isomorphisms of dimension modules. The first
16
M. BOYLE AND J.B.WAGONER
diagram is
(5.11)
/ F∞
O
0
I−A
I+X
/ F∞
0
/ F∞
O
πA
/ G(A)
O
I
I−B
/0
I
/ F∞
πB
/ G(B)
/0
In this case, G(A) = G(B) and the induced isomorphism of dimension modules is the identity. The
second diagram is
(5.12)
/ F∞
0
I−A
πA
/ F∞
I−B
/ G(A)
/0
g(X)
I+X
I
0
/ F∞
/ F∞
πB
/ G(B)
/0
which yields the isomorphism of dimension modules
g(X) : G(A) → G(B)
(5.13)
Proposition 5.14. Assume A is a nonnegative polynomial matrix satisfying NZC. If 0 ≤ X ≤ A
and X 2 = 0, then
g(X) : (G(A), G+ (A), s(A)) → (G(B), G+ (B), s(B))
is an isomorphism of dimension module triples.
Proof Since X ≥ 0 we know that I + X takes F+∞ into F+∞ . Therefore, g(X) takes G+ (A)
into G+ (B). It is injective, because g(X) is an isomorphism. It remains to verify that g(X) takes
G+ (A) onto G+ (B). The condition X 2 = 0 yields the matrix equation
I = (A − X)(I + X) + (I − A)(I + X) = (A − X)(I + X) + (I − B)
So modulo Image(I − B) we see that any element w of G+ (B) is of the form
w = v(I + X)
where v = w(A − X) lies in
F+∞
because A − X ≥ 0.
Corollary 5.15. The action of t on G(A) is an isomorphism which takes G+ (A) bijectively to itself.
In particular, G(A) is a module over the ring Z[t, t−1 ] of Laurent polynomials and s(A) is an
isomorphism of the pair (G(A), G+ (A)). Consequently, under the hypotheses of (5.14), g(X) is an
isomorphism of Z[t, t−1 ]-modules.
Proof First assume A(0) = 0. Then A = tB = Bt for some nonnegative polynomial matrix
B. Modulo Image(I − A), we know that I = tB = Bt. So the action of t is invertible on
G(A). The action of t takes G+ (A) to itself. So does the action of t−1 , because t−1 = B modulo
Image(I − A). Therefore, the action of t is a bijection of G+ (A) to itself. The general case follows
from this special case using (5.14), because we know from (4.3) that there is a sequence of positive
row and column operations connecting I − A and I − tA# .
If M is a matrix over Z + , let (GM , G+
M , sM ) denote the dimension group triple defined by direct
limits as in [LM].
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
17
Proposition 5.16. Assume A is a nonnegative polynomial matrix satisfying NZC. There is an isomorphism of triples
(G(A), G+ (A), s(A)) = (GA# , G+
, s #)
A# A
Proof
As in [KRW] or [LM] we know that there is an isomorphism
(G(tA# ), G+ (tA# ), s(tA# )) = (GA# , G+
, s #)
A# A
The result now follows from (5.14) and (4.3).
An irreducible component of a directed graph G is a subgraph H such that any two vertices i and
j in H may be joined by a directed path from i to j using edges in H and, moreover, H is contained
in no larger subgraph with this property.
Corollary 5.17. Let A and B be nonnegative polynomial matrices satisfying NZC. Then the following conditions are equivalent:
(G) There is an isomorphism between the Z[t, t−1 ]-modules G(A)
and G(B)
(E) There is a sequence of row and column operations over Z[t].
connecting I − A and I − B
If, in addition, the graphs A[ and B [ each have one irreducible component which is primitive, then
both (G) and (E) are equivalent to
(G+ ) There is an isomorphism between the dimension module triples
(G(A), G+ (A), s(A)) and (G(B), G+ (B), s(B))
Proof That condition (E) implies (G) comes from (5.14). We now show that (G) implies (E). From
(5.16) we know that (G) implies there is an isomorphism (GA# , sA# ) ' (GB # , sB # ). The proof of
Krieger’s result [LM,7.5.8] shows that A# and B # are shift equivalent over Z. Effros and Williams
showed this implies A# and B # are strong shift equivalent over Z. See [W3]. The PSSE show that
I − tA# and I − tB # are connected by row and column operations over Z[t]. Therefore, so are
I − A and I − B by (4.3).
Clearly (G+ ) implies (G). We want to show (G) implies (G+ ) under the assumption that A[ and
B each have one irreducible component which is primitive. Let M be a nonnegative polynomial
matrix satisfying NZC, and let M0 be obtained as in (4.4). From the state splitting and merging
construction in the proof of (3.3), the graph M [ has the same number of primitive components as
does M0[ . So from (4.4) and (5.14) we may assume A(0) = B(0) = 0. From (5.16) we know that
there are isomorphisms of dimension module triples
[
(G(M ), G+ (M ), s(M )) ' (GM # , G+
, s # ) ' (GM # , G+ # , sM # )
M# M
nd
Mnd
nd
#
for M = A and M = B where, as in the Appendix, Mnd
is a nondegenerate zero-one matrix con#
nected to M by a path of strong shift equivalences in RS(ZO). Since A(0) = B(0) = 0, the
graphs A# and B # come from A[ and B [ by considering all vertices to be primary. Therefore the
#
graphs A#
nd and Bnd are primitive. For primitive matrices P and Q, it is well known [LM] that
+
isomorphism of (GP , sP ) and (GQ , sQ ) implies isomorphism of (GP , G+
P , sP ) and (GQ , GQ , sQ ).
18
M. BOYLE AND J.B.WAGONER
6. C LASSIFICATION
This section summarizes the discussion of conjugacy and eventual conjugacy [LM] in terms of
row and column operations on polynomial matrices of finite support.
Theorem 6.1. Let A and B be nonnegative polynomial matrices satisfying NZC. There is a conjugacy between (X(A), σ(A)) and (X(B), σ(B)) iff there is a path of positive row and column
operations over Z + [t] connecting I − A and I − B.
Theorem 6.2. Let A and B be nonnegative polynomial matrices satisfying NZC. Assume the graphs
A[ and B [ each have one irreducible component which is primitive. There is an eventual conjugacy
between (X(A), σ(A)) and (X(B), σ(B)) iff there is a path of row and column operations over
Z[t] connecting I − A and I − B.
Here is an example of (6.1) for the matrices
A=
t
t
1
0
and B =
t
1
t
0
Both give the shifts conjugate to (X2 , σ2 ) which comes from the matrix
2t 0
C=
0 0
I − C is obtained from I − A by first multiplying on the left by E12 (1) and then on the right by
E21 (t). I − C is obtained from I − B by first multiplying on the left by E12 (t) and then on the right
by E21 (1).
Let A and B be nonnegative integral matrices of finite support. Recall from [LM] the following
well known facts: (XA , σA ) and (XB , σB ) are conjugate iff A and B are strong shift equivalent over
Z + ; (XA , σA ) and (XB , σB ) are eventually conjugate iff the matrices A and B are shift equivalent
over Z + ; primitive matrices A and B are shift equivalent over Z + iff they are strong shift equivalent
over Z. Parallel to (6.1) and (6.2) we have the following results.
Theorem 6.3. Let A and B be nonnegative integral matrices. Then A and B are strong shift
equivalent over Z + iff there is a path of positive row and column operations over Z + [t] connecting
I − tA and I − tB.
Theorem 6.4. Assume A and B are integral matrices. Then A and B are strong shift equivalent
over Z iff there is a path of row and column operations over Z[t] connecting I − tA and I − tB.
Proof of (6.1)
This is really just a combination of the results (3.3) and (4.1).
Proof of (6.2) If there is a path of row and column operations over Z[t] connecting I − A and
I − B, then the dimension modules G(A) and G(B) are isomorphic. Acccording to (5.16) and
(5.17) this implies there is an isomorphism between the dimension module triples (GA# , G+
, s #)
A# A
+
and (GB # , GB # , sB # ). Therefore, we have an eventual conjugacy between (X(A), σ(A)) and
(X(B), σ(B)). Conversely, if (X(A), σ(A)) and (X(B), σ(B)) are eventually conjugate, then
there is an isomorphism of dimension module triples (GA# , G+
, s # ) and (GB # , G+
, s # ).
A# A
B# B
Using (5.16) and (5.17) , we conclude there is a sequence of row and column operations over Z[t]
connecting I − A and I − B.
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
19
0
Proof of (6.3) This is a special case of (6.1), because {tA}# = A and because there is the subdi0
vision strong shift equivalence (RA , SA ) : A → A as in (2.1) of [W4]. See the Appendix.
Proof of (6.4) This just like the proof in (5.17). If I − tA and I − tB are connected by row and
column operations, then (G(tA), s(tA)) = (GA , sA ) and (G(tB), s(tB)) = (GB , sB ) are isomorphic. Conversely if the dimension groups are isomorphic, A and B are strong shift equivalent over
Z. The PSSE show that I − tA and I − tB are connected by row and column operations over Z[t].
7. A PPENDIX
The purpose of this section is to clarify a certain nondegeneracy condition which has appeared
in earlier discussions of shifts of finite type and their automorphisms in the context of strong shift
equivalence theory. The main reason for this is that relaxing the nondegeneracy condition gives algebraic flexibility which is necessary for relating strong shift equivalence theory to the positive row
and column operation approach to shifts of finite type and their automorphisms which is developed
in this paper.
Let P = (Pij ) be a zero-one matrix of finite support where the indices i and j run through the
positive integers. This means that Pij 6= 0 for at most finitely many pairs of indices (i, j). The
support of P is the least integer n = supp(P ) such that Pij = 0 if either i > n or j > n. If P = 0,
we let supp(P ) = 1. Any finite n × n matrix will be considered as an infinite matrix of support less
than or equal to n by appending identically zero rows and columns to it. As in [LM], construct the
shift of finite type {XP , σP } from P as follows:
bi-infinite sequences x = {xk } where
(7.1)
XP
=
P (xk , xk+1 ) = 1 for −∞ < k < ∞.
If i is a state ( i.e., a positive integer) such that the ith row or the ith column is zero, then i does not
appear as a symbol xk in any x in XP . The shift map σP : XP → XP is defined by σP (x) = y
where yk = xk+1 . XP has the product topology which makes it a Cantor set and σP is the shift
homeomorphism. {XP , σP } is the vertex shift associated to P . More generally, if P is a matrix over
Z + , then there is the edge path shift of finite type (XP , σP ) associated to P by the formula
(XP , σP ) = {XP 0 , σP 0 }
(7.2)
0
where P is the zero-one matrix of the edge path presentation of the directed graph arising from P as
in [LM]. Namely, the states are the arcs in the directed graph associated to P , and P 0 (α, β) = 1 iff
the end vertex of α is the start vertex of β. Strictly speaking, an order of the arcs is chosen as well to
get the matrix P 0 . If another order is chosen, corresponding new P 0 will be conjugate to the first one
by a permutation, and the resulting vertex shifts will be topologically conjugate. If P is a zero-one
matrix, there is a topological conjugacy between {XP , σP } and {XP 0 , σP 0 }. See the discussion of
subdivision below.
Let Q be a zero-one matrix of finite support. A strong shift equivalence (R, S) : P → Q in the
category ZO of zero-one matrices with finite support is a zero-one matrix R of finite support and
zero-one matrix S of finite support satisfying the strong shift equivalence equations
(7.3)
P = RS and Q = SR
This data produces a conjugacy
(7.4)
c(R, S) : {XP , σP } → {XQ , σQ }
20
M. BOYLE AND J.B.WAGONER
in the following way . Let x = {xk } be in XP , and let y = {yk } in XQ be the image of x under
c(R, S). We know that
X
1 = A(xk , xk+1 ) =
R(xk , p)S(p, xk+1 )
p
Since the matrices A, R, and S are zero-one, there is exactly one p for which R(xk , p)S(p, xk+1 ) =
1. We set
(7.5)
yk = p
A strong shift equivalence (R, S) : P → Q of matrices with finite support over Z + is defined
similarly and induces a conjugacy
(7.6)
c(R, S) : (XP , σP ) → (XQ , σQ )
which is well defined up to simple automorphisms. See [W4, Section 2].
Now as in [W2,W4] construct the spaces RS(ZO) and RS(Z + ) of strong shift equivalences for
zero-one or Z + matrices with finite support. We then have the following versions of R.F. Williams’
results in strong shift equivalence theory.
Theorem 7.7. Up to topological conjugacy shifts of finite type are in one-to-one correspondence
with the path components of π0 (RS(ZO)) of RS(ZO). The inclusion RS(ZO) ⊂ RS(Z + ) induces
a bijection of path components π0 (RS(ZO)) = π0 (RS(Z + )).
Theorem 7.8. Let A and B be zero-one matrices. Any topological conjugacy α : {XA , σA } →
{XB , σB } can be written as a composition
Y
α=
c(Ri , Si )i
i
corresponding to a path of strong shift equivalences connecting A and B in RS(ZO).
The point of these theorems is to eliminate the following nondegeneracy condition which has
often been tacitly and/or explicitly assumed in the literature. Let X = {Xij } be a matrix with finite
support. Let X(m, n) be the finite m × n matrix obtained by considering the first m rows and the
first n columns of X. We say X is an m × n matrix provided
(7.9)
Xij = 0 if i > m or j > n
and we say an m × n matrix X is nondegenerate provided
(7.10)
no row or column of X(m, n) is entirely zero
Let RSnd (ZO) and RSnd (Z + ) denote the subspaces of RS(ZO) and RS(Z + ) formed by taking
vertices to be nondegenerate m × m matrices P and edges (R, S) : P → Q between a nondegenerate m × m matrix P and a nondegenerate n × n matrix Q to be a nondegenerate m × n matrix
R and a nondegenerate n × m matrix S satisfying the strong shift equivalence equations (7.3).
The nondegeneracy condition we are referring to is that Theorem 7.7 and Theorem 7.8 are usually
stated with RSnd (ZO) and RSnd (Z + ) instead of RS(ZO) and RS(Z + ). The technical reason for
this is the requirement that all atoms of a Markov partition are assumed to be nonempty sets. See
[W1,W2,W4,BW].
Proposition 7.11. Let A 6= 0 be a zero-one or Z + matrix of finite support. Then there is a path in
RS(ZO), respectively RS(Z + ), from A to some nondegenerate matrix And .
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
21
Suppose P is an m × m matrix such that P (m, m) is invertible. We let P −1 denote the m × m
matrix such that P −1 (m, m) = P (m, m)−1 . Let X be an m × m matrix. We say X is trimmable
provided there is a row or a column of X(m, m) which is entirely zero. Proposition 7.11 results
from repeated application of
Lemma 7.12. Assume X is a trimmable m × m zero-one or Z + matrix which is not identically zero.
There is an m × m permutation matrix P and a two step path of strong shift equivalences
(P, P −1 X) : X → P −1 XP
(R, S) : P −1 XP → Y
in RS(ZO) or RS(Z + ) respectively where Y is an n × n matrix with n < m.
Proof We give the proof for the case when X(m, m) has a zero column. The argument when it
has a zero row is similar. Let P be an m × m matrix such that P (m, m) is a permutation matrix and
the last column of P −1 XP is xero. Let n = m − 1 and write
Y 0
P −1 XP =
U 0
where Y is a n × n matrix and U is a 1 × n matrix. Define the m × n matrix R and the n × m matrix
S to be
Y
R=
S= I 0
U
where I is the n × n identity matrix. Then we have the strong shift equivalence equations
P −1 XP = RS and Y = SR
Proof of (7.7) One result in R.F.Williams’ paper [Wi] as formulated in [W2] is that the isomorphism classes of shifts of finite type are in one-to-one correspondence with the path components of
π0 (RSnd (ZO)). So the first step in proving (7.7) is to show that
π0 (RSnd (ZO)) → π0 (RS(ZO))
is a bijection. Proposition 7.11 shows this map is onto. To show this is one-to-one, suppose
there is a path Γ of strong shift equivalences in RS(ZO) connecting the nondegenerate m × m
matrix A to the nondegenerate n × n matrix B. The construction (7.4) above yields a topological conjugacy α : (XA , σA ) → (XB , σB ). Then Williams’ work as formulated in [W2] says
that α arises from a path Γnd in RSnd (ZO) connecting A and B. To verify the last statement
of (7.7), observe that the proof in [W4] showing π0 (RSnd (ZO)) = π0 (RSnd (Z + )) also shows
π0 (RS(ZO)) = π0 (RS(Z + )).
Proof of (7.8) Let α : {XA , σA } → {XB , σB } be an isomorphism where A and B are zero-one
matrices with finite support. Use Proposition 7.11 to find paths ΓA and ΓB in RS(ZO) from A
and B to nondegenerate matrices And and Bnd respectively. Let θA : {XA , σA } → {XAnd , σAnd }
denote the conjugacy which is the product of the conjugacies c(R, S)±1 corresponding to the edges
in ΓA . Similarly, let θB : {XB , σB } → {XBnd , σBnd } denote the conjugacy which is the product of
the conjugacies c(R, S)±1 corresponding to the edges in ΓB . Then, reading composition from left
to right,
−1
−1
α = θA (θA
αθB )θB
By the version of (7.8) using RSnd (ZO), we know that the isomorphism
−1
θA
αθB : {XAnd , σAnd } → {XBnd , σBnd }
22
M. BOYLE AND J.B.WAGONER
is a product of conjugacies c(R, S)±1 corresponding to the edges in a path from And to Bnd in
RSnd (ZO).
While the argument for (7.7) does produce some path Γnd , connecting A and B in RSnd (ZO), we
have not shown that Γ is homotopic keeping endpoints fixed to a path in RSnd (ZO). As explained
in [W5] we know that
Aut(σA ) = π1 (RSnd (ZO), A)
Aut(σA )/Simp(σA ) = π1 (RSnd (Z + ), A)
Question 1 Are the natural homomorphisms
π1 (RSnd (ZO), A) → π1 (RS(ZO), A)
π1 (RSnd (Z + ), A) → π1 (RS(Z + ), A)
isomorphisms ?
Question 2 Are the inclusions RSnd (ZO) ⊂ RS(ZO) and RSnd (Z + ) ⊂ RS(Z + ) homotopy
equivalences ?
We end this Appendix with a discussion of subdivision which is needed in this paper. Let A
be a zero-one matrix. There are two ways to associate a shift of finite type to A. The first is the
vertex shift {XA , σA } as given above. The second is the edge path construction {XA0 , σA0 } as in
0
[LM,W4]. In (2.1) of [W4] there is the subdivision strong shift equivalence (RA , SA ) : A → A
which induces the subdivision isomorphism sdA : {XA , σA } → {XA0 , σA0 }. If (R, S) : A → B is
a strong shift equivalence of zero-one matrices, the diagram (2.1) of [W4] produces a commutative
diagram of conjugacies
{XA , σA }
(7.13)
sdA
{XA0 , σA0 }
0
/ {XB , σB }
c(R,S)
sdB
0
0
c(R ,S )
/ {X 0 , σ 0 }
B
B
0
The conjugacy c(R , S ) is called the subdivision of c(R, S). In the category of zero-one matrices
0
0
there is no ambiguity in c(R , S ) because there is ony one choice to be made in (2.5) of [W4]. An
immediate consequence of (7.13) is
Proposition 7.14. Any conjugacy β : {XA0 , σA0 } → {XB 0 , σB 0 } is the subdivision of the conjugacy α : {XA , σA } → {XB , σB } where α = sdA βsd−1
(reading composition from left to right).
B Q
Q
0
0
This means that if we write α = i c(Ri , Si )i , then β = i c(Ri , Si )i
R EFERENCES
[B1]
[B2]
[BL]
[BW]
M.Boyle, Symbolic dynamics and matrices in combinatorial and graph-theoretic problems in linear algebra, IMA
Volumes in Mathematics and Its Applications,Vol.50 (1993), editors R.A. Brualdi, S. Friendlander, and V. Klee,
1-38
, Positive K-theory in symbolic dynamics, Dynamics and Randomness, editors A. Maass, S. Martinez,
and J. San Martin, Kluwer, 2002, 31-52
M.Boyle and D.Lind, Small polynomial matrix presentations of nonnegative matrices , Linear Algebra Applications,
355, 2002, 49-70
L. Badoian and J.B.Wagoner, Simple connectivity of the Markov partition space, Pac. Jour. Math. , 193, 2000, No.1,
1-3
POSITIVE ALGEBRAIC K-THEORY AND SHIFTS OF FINITE TYPE
[KOR]
23
K.H.Kim, N. Ormes, and F.W.Roush, The spectra of nonnegative integer matrices via formal power series, JAMS,
13, 2000, No.4, 773-806
[KR]
K.H.Kim and F.W.Roush, Williams’ conjecture is false for irreducible subshifts, Annals of Math(2),149,1999, No.2,
545-558
[KRW] K.H.Kim, F.W.Roush, and J.B.Wagoner,Characterization of inert actions on periodic points, Part I and Part II,
Forum Mathematicum 12 ,2000, 565-602 and 671-712
[LM]
D.Lind and B.Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995
[M]
B. Marcus, R. Roth, and P. Siegel, Constrained systems and coding for recording channels, Chapter 20, Volume II
of Handbook of Coding Theory (ed., V.S. Pless and W. C. Huffman), Elsevier Press, 1998
[N]
M. Nasu,Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts, Springer Lecture Notes in Math 1342, 1988
[Sh]
C.Shannon, A mathematical theory of communication, Bell Sys.Tech Jour., 27,1948, 379-423, reprinted by University of Illinois Press,1963
[Sm]
S.Smale, Differential Dynamical Systems, BAMS, Vol.73, No.6, 1967, 747-847
[W1]
J.B.Wagoner, Markov partitions and K2 , Publ. Math. IHES, No. 65, 1987
[W2]
, Triangle inequalities and symmetries of a subshift of finite type, Pacific J. Math. 144(1990), 331350.
[W3]
, Higher dimensional shift equivalence is the same as strong shift equivalence over the integers, Proc.
A.M.S. 109 (1990), 527-536
, Eventual finite order generation for the kernel of the dimension group representation , Trans. A.M.S.
[W4]
(1990), 331-350
[W5]
, Strong shift equivalence theory and the shift equivalence problem, BAMS, 36, 1999, No.3, 271-296
, Strong Shift Equivalence Theory preprint from UC Berkeley, to appear in Symbolic Dynamics,
[W6]
AMS Proceedings of Symposia in Applied Mathematics
[Wi]
R.F.Williams, Classification of subshifts of finite type, Annals of Math.(2) 98 (1973), 120-153; Errata ibid. 99
(1974), 380-381
D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF M ARYLAND , C OLLEGE PARK , MD 20742
E-mail address: [email protected]
D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF C ALIFORNIA , B ERKELEY, CA 94720
E-mail address: [email protected]