# On Rationality of Kneading Determinants

Authors: Sheng Chen; Chao Xia
Journal: The Scientific World Journal
 (2014)

---

## Abstract

In this section,R denotes a ring with identity I. We study conditions under which I
−
A
λ and I
−
B
λ are left coprime or right coprime, where A
,
B
∈
R. As applications, we get sufficient conditions under which the Kneading determinant of a finite rank pair of operators on an infinite dimensional space is rational.


---

## Body

## 1. Introduction

IfA is a k
×
k matrix with rational coefficients, one has the well-known identity between formal power series:
(1)
exp
⁡
∑
n
≥
1
‍
-
tr
⁡
(
A
n
)
n
z
n
=
det
⁡
(
I
-
z
A
)
,
where tr
⁡
(
A
n
) denotes the trace of matrix A
n (matrix A raised to the nth power) and I denotes the k
×
k identity matrix. This identity plays a significant role in the discussion of an important problem in dynamical systems theory. For more details see [1, 2].We denote byℋ the infinite dimension vector space over ℚ; the space of linear forms on ℋ will be denoted, as usual, by ℋ
*, and the space of all linear endomorphisms on ℋ will be denoted by L
(
ℋ
). If ψ
∈
L
(
ℋ
) and n is a nonnegative integer, the nth iterate ψ
n is defined recursively by ψ
0
=
I
d
ℋ
∈
L
(
ℋ
), ψ
n
=
ψ
∘
ψ
n
-
1, for n
≥
1.The subspace ofL
(
ℋ
) whose elements are the linear endomorphism on ℋ with finite rank will be denoted by L
FR
(
ℋ
).Letq be a positive integer; we use the symbol h
¯ to denote an element of ℋ
q
=
ℋ
×
ℋ
×
ℋ
×
⋯
×
ℋ (q times) and the symbol α
¯ to denote an element of ℋ
*
q; that is
(2)
h
¯
=
(
h
1
,
…
,
h
q
)
,
α
¯
=
(
α
1
,
…
,
α
q
)
.
Given h
¯
∈
ℋ
q and α
¯
∈
ℋ
*
q, we define the finite rank endomorphism α
¯
⊗
h
¯
∈
L
FR
(
ℋ
) and the matrix M
(
α
¯
,
h
¯
)
∈
ℚ
q
×
q by setting
(3)
α
¯
⊗
h
¯
=
∑
p
=
1
q
‍
α
p
⊗
h
p
,
with the usual notation
(4)
α
∈
ℋ
*
,
h
∈
ℋ
:
(
α
⊗
h
)
(
x
)
=
α
(
x
)
u
,
x
∈
ℋ
,
M
(
α
¯
;
h
¯
)
=
(
α
1
(
h
1
)
⋯
α
1
(
h
q
)
⋮
⋱
⋮
α
q
(
u
1
)
⋯
α
q
(
h
q
)
)
.Definition 1 (see [3]).
A pair(
φ
,
ψ
)
∈
L
(
ℋ
)
×
L
(
ℋ
) is said to have finite rank if ψ
-
φ
∈
L
FR
(
ℋ
).Notice that if a pair(
φ
,
ψ
) has finite rank, then the pair (
φ
n
,
ψ
n
) also has finite rank for all n
≥
1, and therefore the trace of φ
n
-
ψ
n is defined.Definition 2 (see [3]).
For any pair(
φ
,
ψ
)
∈
L
(
ℋ
)
×
L
(
ℋ
) with finite rank, we define the Kneading determinant of (
φ
,
ψ
) as the following invertible element of ℚ
[
[
z
]
]:
(5)
Δ
(
φ
,
ψ
)
=
exp
⁡
∑
n
≥
1
‍
tr
⁡
(
φ
n
-
ψ
n
)
n
z
n
.Remark 3.
Kneading determinant was first studied by Milnor and Thurston in [4].Letℚ
[
[
z
]
]
q
×
q be the ring of the q
×
q matrices whose entries lie in ℚ
[
[
z
]
]. If φ
∈
L
(
ℋ
), u
¯
∈
ℋ
q, and α
¯
∈
ℋ
*
q, we define the matrix M
φ
(
α
¯
;
u
¯
)
∈
ℚ
[
[
z
]
]
q
×
q by
(6)
M
φ
(
α
¯
;
u
¯
)
=
(
∑
n
≥
0
‍
α
1
φ
n
(
u
1
)
z
n
⋯
∑
n
≥
0
‍
α
1
φ
n
(
u
q
)
z
n
⋮
⋱
⋮
∑
n
≥
0
‍
α
q
φ
n
(
u
1
)
z
n
⋯
∑
n
≥
0
‍
α
q
φ
n
(
u
q
)
z
n
)
.Lemma 4 (see [3]).
Let(
φ
,
ψ
)
∈
L
(
ℋ
)
×
L
(
ℋ
), u
¯
∈
ℋ
q, and α
¯
∈
ℋ
*
q such that ψ
-
φ
=
α
¯
⊗
u
¯. Denote by I the q
×
q identity matrix. Then,
(7)
Δ
(
φ
,
ψ
)
=
det
⁡
(
I
-
z
M
φ
(
α
¯
;
u
¯
)
)
holds in ℚ
[
[
z
]
].In general, any power series can be the Kneading determinant of some pair(
φ
,
ψ
) with finite rank (see [3]). So it is interesting to study conditions under which the Kneading determinant is a rational power series.The following is the main result of this paper.Theorem 5.
IfI
-
φ
λ and I
-
ψ
λ are left coprime or right coprime, and h
=
ψ
-
φ with finite rank, then Δ
(
φ
,
ψ
) is a rational function.


## 2. Coprimeness ofI
-
A
λ and I
-
B
λ

In this section,R denotes a ring with identity I. We discuss conditions under which I
-
A
λ and I
-
B
λ are left coprime or right coprime, where A
,
B
∈
R.We sayI
-
A
λ and I
-
B
λ are left coprime if there exist polynomials X
(
λ
)
,
Y
(
λ
)
∈
R
[
λ
] such that
(8)
X
(
λ
)
(
I
-
A
λ
)
+
Y
(
λ
)
(
I
-
B
λ
)
=
I
.Proposition 6.
I
-
A
λ and I
-
B
λ are left coprime if and only if there exist X
0
,
X
1
,
…
,
X
m
∈
R such that
(9)
X
m
H
+
X
m
-
1
H
B
+
X
m
-
2
H
B
2
+
⋯
+
X
0
H
B
m
+
B
m
+
1
=
0
,
for H
=
A
-
B.Proof.
IfI
-
A
λ and I
-
B
λ are left coprime, there exist X
(
λ
)
=
X
0
+
X
1
λ
+
⋯
+
X
m
λ
m and Y
(
λ
)
=
Y
0
+
Y
1
λ
+
⋯
+
Y
s
λ
s such that
(10)
X
(
λ
)
(
I
-
A
λ
)
+
Y
(
λ
)
(
I
-
B
λ
)
=
I
.
We can assume thats equals m. And we have
(11)
X
0
+
Y
0
=
I
,
X
i
A
+
Y
i
B
-
X
i
+
1
-
Y
i
+
1
=
0
,
i
=
1,2
,
…
,
m
-
1
,
X
m
A
+
Y
m
B
=
0
.
So,(12)
Y
0
=
I
-
X
0
.
If we writeH
=
A
-
B, then,
(13)
Y
1
=
-
X
1
+
X
0
H
+
B
,
Y
2
=
-
X
2
+
X
1
H
+
X
0
H
B
+
B
2
,
⋮
Y
m
=
-
X
m
+
X
m
-
1
H
+
⋯
+
X
0
H
B
m
-
1
+
B
m
.
So,(14)
X
m
H
+
X
m
-
1
H
B
+
X
m
-
2
H
B
2
+
⋯
+
X
0
H
B
m
+
B
m
+
1
=
0
.
Conversely, if there existX
0
,
X
1
,
…
,
X
m
∈
R such that
(15)
X
m
H
+
X
m
-
1
H
B
+
X
m
-
2
H
B
2
+
⋯
+
X
0
H
B
m
+
B
m
+
1
=
0
,
let
(16)
X
(
λ
)
=
X
0
+
X
1
λ
+
⋯
+
X
m
λ
m
,
Y
0
=
I
-
X
0
,
Y
1
=
-
X
1
+
X
0
H
+
B
,
Y
2
=
-
X
2
+
X
1
H
+
X
0
H
B
+
B
2
,
⋮
Y
m
=
-
X
m
+
X
m
-
1
H
+
⋯
+
X
0
H
B
m
-
1
+
B
m
,
and let
(17)
Y
(
λ
)
=
Y
0
+
Y
1
λ
+
⋯
+
Y
m
λ
m
.
Then,(18)
X
(
λ
)
(
I
-
A
λ
)
+
Y
(
λ
)
(
I
-
B
λ
)
=
I
.Now we give the definition of right coprime. We sayI
-
A
λ and I
-
B
λ are right coprime if there exist polynomials X
(
λ
)
,
Y
(
λ
)
∈
R
[
λ
] such that
(19)
(
I
-
A
λ
)
X
(
λ
)
+
(
I
-
B
λ
)
Y
(
λ
)
=
I
.Proposition 7.
I
-
A
λ and I
-
B
λ are right coprime if and only if there exist X
0
,
X
1
,
…
,
X
s
∈
R such that
(20)
H
X
s
+
B
H
X
s
-
1
+
B
2
H
X
s
-
2
+
⋯
+
B
s
H
X
0
+
B
s
+
1
=
0
,
for H
=
A
-
B.Proof.
It is similar to the proof of Proposition6.


## 3. Proof of the Main Theorem

In this section,φ
,
ψ
∈
L
(
ℋ
), where ℋ is an infinite dimensional vector space over ℚ, and we denote by L
(
ℋ
) the ring of linear transforms on ℋ.Lemma 8.
Suppose thatI
-
φ
λ and I
-
ψ
λ are left coprime or right coprime, h
=
ψ
-
φ is of finite rank, and W
=
Im
(
h
). Then there exists a finite dimensional space W
~ containing W such that (
φ
k
-
ψ
k
)
(
ℋ
)
⊂
W
~ with k
=
1,2
,
….Proof.
First we assume that ifI
-
φ
λ and I
-
ψ
λ are right coprime, then by Proposition 7 we have X
0
,
X
1
,
…
,
X
s
∈
L
(
ℋ
) such that
(21)
h
X
m
+
φ
h
X
m
-
1
+
φ
2
h
X
m
-
2
+
⋯
+
φ
m
h
X
0
+
φ
m
+
1
=
0
.
WriteW
~
=
W
+
φ
W
+
φ
2
W
+
⋯
+
φ
m
W; W
~ is an invariant subspace of ℋ under h. It is not very hard to check that φ
(
W
~
)
=
φ
m
+
1
W
+
φ
W
+
φ
2
W
+
⋯
+
φ
m
W
⊂
W
~. So W
~ is invariant under the operator φ.
We use induction to prove the conclusion.
Ifk
=
1, then (
φ
k
-
ψ
k
)
(
ℋ
)
=
W.
Suppose that fork
≤
l we have (
φ
k
-
ψ
k
)
(
ℋ
)
⊂
W
+
φ
W
+
⋯
+
φ
m
W. Then for k
=
l
+
1, we have (
φ
k
-
ψ
k
)
(
ℋ
)
=
φ
(
φ
l
-
ψ
l
)
(
ℋ
)
-
h
ψ
l
(
ℋ
)
⊂
φ
(
W
~
)
+
W
⊂
W
~. We get the conclusion in this case.
Now we assume thatI
-
φ
λ and I
-
ψ
λ are left coprime; then there exist s
∈
ℤ and x
0
,
x
1
,
…
,
x
s
∈
L
(
ℋ
) such that
(22)
x
s
h
+
x
s
-
1
h
φ
+
x
s
-
2
h
φ
2
+
⋯
+
x
0
h
φ
s
+
φ
s
+
1
=
0
,
holds. Take
(23)
W
^
=
W
+
x
0
W
+
⋯
+
x
s
W
,
W
~
=
W
^
+
φ
W
^
+
φ
2
W
^
+
⋯
+
φ
s
W
^
.
Notice when l
≥
s
+
1, φ
l
(
ℋ
)
⊂
x
0
W
+
x
1
W
+
⋯
+
x
s
W.
We use induction to prove the conclusion.
Ifk
=
1, then (
φ
k
-
ψ
k
)
(
ℋ
)
=
W.
Suppose that fork
≤
l we have (
φ
k
-
ψ
k
)
(
ℋ
)
⊂
W
+
φ
W
+
⋯
+
φ
m
W. Then for k
=
l
+
1, we have (
φ
k
-
ψ
k
)
(
ℋ
)
=
φ
l
h
(
ℋ
)
-
(
φ
l
-
ψ
l
)
ψ
(
ℋ
)
⊂
φ
l
(
W
)
+
W
~
⊂
W
~.
The proof is complete.Now we will give the proof of Theorem5.Proof.
IfI
-
φ
λ and I
-
ψ
λ are coprime, by Lemma 8, there is a finite dimensional space W
~ such that
(24)
(
ψ
k
-
φ
k
)
(
ℋ
)
⊂
W
~
,
and we have φ
(
W
~
)
⊂
W
~. Define
(25)
φ
~
=
Pr
(
φ
)
|
W
~
:
W
~
⟶
W
~
.
Supposeq
=
rank
⁡
(
φ
-
ψ
); we denote by I
q the identity operator. Then p
(
z
)
=
det
⁡
(
I
q
-
z
φ
~
)
∈
ℚ
[
z
]. For any i
,
j
∈
{
1,2
,
…
,
q
}, we have ∑
n
≥
0
‍
α
i
φ
n
(
u
j
)
z
n
=
f
i
j
(
z
)
/
p
(
z
) for some f
i
j
(
z
)
∈
ℚ
[
z
]. For more details see [5]. So by Lemma 4, det
⁡
(
I
-
z
M
φ
(
α
¯
;
u
¯
)
) is rational; we get the conclusion.Example 9.
Supposeℋ is the ring of countable infinite matrix with finite nonzero entries in each column.
Let(26)
φ
=
(
α
1
α
2
α
3
⋯
A
0
0
⋯
0
A
0
⋯
⋮
⋮
⋱
⋱
)
,
h
=
(
α
1
α
2
α
3
⋯
0
0
0
⋯
0
0
0
⋯
⋮
⋮
⋱
⋱
)
,
ψ
=
φ
+
h
,
where A
=
(
0
1
0
0
0
1
0
0
0
) and α
1
,
α
2
,
α
3
,
… are three-dimensional row vectors. We see that (
φ
-
h
)
3
=
0. So I
-
ψ
λ and I
-
φ
λ are left coprime. It is easy to check that
(27)
Δ
(
φ
,
ψ
)
=
1
-
2
z
1
-
z
.



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*Source: 504160-2014-02-18.xml*