On Rationality of Kneading Determinants

In this section, R denotes a ring with identity I. We study conditions under which I − Aλ and I − Bλ 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.


Introduction
If is a × matrix with rational coefficients, one has the well-known identity between formal power series: where tr( ) denotes the trace of matrix (matrix raised to the th power) and denotes the × 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 by H 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 (H). If ∈ (H) and is a nonnegative integer, the th iterate is defined recursively by 0 = Id H ∈ (H), = ∘ −1 , for ≥ 1.
The subspace of (H) whose elements are the linear endomorphism on H with finite rank will be denoted by Let be a positive integer; we use the symbol ℎ to denote an element of H = H × H × H × ⋅ ⋅ ⋅ × H ( times) and the symbol to denote an element of H * ; that is Given ℎ ∈ H and ∈ H * , we define the finite rank endomorphism ⊗ ℎ ∈ FR (H) and the matrix M( , ℎ) ∈ Q × by setting with the usual notation ∈ H * , ℎ ∈ H : ( ⊗ ℎ) ( ) = ( ) , ∈ H, ) .
Notice that if a pair ( , ) has finite rank, then the pair ( , ) also has finite rank for all ≥ 1, and therefore the trace of − is defined.

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Lemma 4 (see [3]). Let ( , ) ∈ (H) × (H), ∈ H , and ∈ H * such that − = ⊗ . Denote by I the × identity matrix. Then, In general, any power series can be the Kneading determinant of some pair ( , ) 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. If − and − are left coprime or right coprime, and ℎ = − with finite rank, then Δ ( , ) is a rational function.

Coprimeness of − and −
In this section, denotes a ring with identity . We discuss conditions under which − and − are left coprime or right coprime, where , ∈ .
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Proof of the Main Theorem
In this section, , ∈ (H), where H is an infinite dimensional vector space over Q, and we denote by (H) the ring of linear transforms on H.
We use induction to prove the conclusion. If The proof is complete. Now we will give the proof of Theorem 5.
Proof. If − and − are coprime, by Lemma 8, there is a finite dimensional spacẽsuch that For more details see [5]. So by Lemma 4, det(I− M ( ; )) is rational; we get the conclusion.