PII. S0161171204309026 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON THE CLASS OF SQUARE PETRIE MATRICES
INDUCED BY CYCLIC PERMUTATIONS
BAU-SEN DU
Received 2 September 2003
Letn≥2 be an integer and letP= {1,2,...,n,n+1}. LetZpdenote the finite field{0,1,2,..., p−1}, wherep≥2 is a prime. Then every mapσonPdetermines a realn×nPetrie matrix
Aσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσ is acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2
are all equal tonk=0xk. As a consequence, we obtain that ifσis acyclicpermutation onP,
then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.
2000 Mathematics Subject Classification: 15A33, 15A36.
1. Introduction. Throughout this paper, letn≥2 be a fixed integer and let P = {1,2,...,n,n+1}. For every integer 1≤i≤n, letJi=[i,i+1]. Letσ be a map from
P into itself. The linearization ofσ on P is defined as the continuous mapfσ from
[1,n+1]into itself such thatfσ(k)=σ (k)for every integer 1≤k≤n+1 andfσ is linear onJifor every integer 1≤i≤n. LetAσ=(aij)be thereal n×nmatrix defined byaij=1 iffσ(Ji)⊃Jjandaij=0 otherwise. The definition ofAσmay seem opaque. But if we takeJi’s as the vertices of a directed graph and draw an arrow from the vertex
Jito the vertexJjiffσ(Ji)⊃Jj, thenAσwill be the adjacency matrix [4, page 17] of the resulting directed graph. For example, the adjacency matrix of the cyclic permutation
σ: 1→2→5→4→3→1 is given as
Aσ=
0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0
. (1.1)
In the theory of discrete dynamical systems on the interval, this adjacency matrixAσ turns out to contain much information on the dynamical properties of the mapfσ. For example, for some special types (including cyclic permutations) ofσ, ifxn+n−1
Due to the continuity offσ, it is clear that such matricesAσhave entries either zeros or ones such that the ones in eachrowoccur consecutively. Actually, we haveaij=1 for allai≤j≤bi−1, whereai=min{fσ(i),fσ(i+1)}andbi=max{fσ(i),fσ(i+1)}, and
aij=0 elsewhere. For our purpose, we define a Petrie matrix [5] to be a matrix whose entries are either zeros or ones such that the ones in eachrowoccur consecutively. So, the matrixAσ induced by a mapσ onPis a square Petrie matrix whose determinant is easily seen (by induction) [7] to be either 0 or±1. For any prime numberp≥2, let
Zp= {0,1,2,...,p−1}denote the usual finite field and letWZp= { n
i=1riJi|ri∈Zp,1≤
i≤n}be then-dimensional vector space overZpwith{Ji|1≤i≤n}as a set of basis. Then the matrixAσ(mod 2)defines a linear transformationψσ onWZ2 such that, for every integer 1≤i≤n,ψσ(Ji)=nj=1aijJj.
If bothσ andρare just permutations onP, then it is easy to see thatAσmay not be similar toAρoverZ2. But if bothσandρarecyclicpermutations onP, then we show, in
this paper, thatAσis similar toAρoverZ2(butAσmay not be similar toAρoverZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to
n k=0xk.
As a consequence, we obtain that ifσ is acyclicpermutation, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero (not true in general ifσ is not cyclic) with constant term±1.
2. On the Petrie matrix Aσ overZ2 with any map σ on P. In the following, we
let[x:y]denote the closed interval on the real line withxandy as endpoints. For integers 1≤k < j≤n+1, we let[k,j]denote the elementji=−k1JiofWZ2and callkand
jthe endpoints (this terminology will be used in the proof ofTheorem 3.2inSection 3) of the elementji=−1kJi. Part (2) of the following lemma is proved in [4, pages 22-23], which will be needed inSection 3. Here, we present a different proof (see also [3]).
Lemma2.1. Letn,P,Ji’s,σ,fσ,WZ
2,ψσ,Aσ be defined as inSection 1. Letρbe a
map fromPinto itself and letψρandAρbe defined similarly. Then the following hold. (1)Let1≤k < j≤n+1be any integers. Then for any element[k,j]=ij=−k1JiinWZ2,
ψσ([k,j])=[fσ(k):fσ(j)].
(2)ψρ◦ψσ=ψρ◦σand(Aσ)(Aρ)≡Aρ◦σ(mod 2). Consequently, ifσis a permutation
onP, thenψσ is invertible with inverseψσ−1 andAσ is nonsingular with determinant ±1.
Proof. It follows from the definition of ψσ in Section 1 that ψσ(Ji)= [fσ(i) :
fσ(i+1)]for every integer 1≤i≤n. Thus, we obtain thatψσ([k,j])=ψσ(ji=−1kJi)= j−1
i=kψσ(Ji)=ji=−1k[fσ(i):fσ(i+1)]=[fσ(k):fσ(j)]since 1+1=0 inZ2. This proves
part (1).
By part (1),ψσ([k,j])=[fσ(k):fσ(j)]. Similarly,ψρ([k,j])=[fρ(k):fρ(j)]. So,
(ψρ◦ψσ)(Ji)=ψρ([fσ(i): fσ(i+1)])=[fρ(fσ(i)):fρ(fσ(i+1))]=[(ρ◦σ )(i) :
(ρ◦σ )(i+1)]=ψρ◦σ(Ji)since, on the finite setP,fσ=σ andfρ=ρ. This shows that
ON THE CLASS OF SQUARE PETRIE MATRICES... 1619
3. On the Petrie matrixAσ with any cyclic permutationσ onP. We will need the following elementary result. We include its proof for completeness.
Lemma3.1. Let1≤j≤nbe any fixed integer and letbdenote the greatest common
divisor ofjandn+1. Lets=(n+1)/b. For every integer1≤k≤s−1, let1≤mk≤n
be the unique integer such thatkj≡mk(modn+1). Then themk’s are all distinct and {mk|1≤k≤s−1} = {kb|1≤k≤s−1}.
Proof. LetB= {mk|1≤k≤s−1}andC= {kb|1≤k≤s−1}. For every integer 1≤k≤s−1, sincej/b and (n+1)/b are relatively prime, the congruence equation
(j/b)x≡k(mod(n+1)/b)has a solution in 1≤x≤s−1=(n+1)/b−1. Consequently, for every integer 1≤k≤s−1, the congruence equation jx≡kb(modn+1) has a solution in 1≤x ≤s−1. Since 1≤kb≤n for every 1≤k≤s−1, we obtain that
C ⊂B. Since bothB and C contain exactly s−1 elements, we have B =C. That is, {mk|1≤k≤s−1} = {kb|1≤k≤s−1}. This completes the proof.
Theorem3.2. Letn,P,Ji’s,σ,fσ,WZ
2,ψσ,Aσ be defined as inSection 1. Assume
thatσ is also a cyclic permutation onP. Then the following hold.
(1) For every integer1≤i≤n,nk=0ψk
σ(Ji)=0. Consequently,kn=0ψkσ(w)=0for
allw∈WZ2.
(2) Let1≤i≤n−1and1≤j≤nbe two fixed integers such that1≤i < fσj(i)≤n
and letJ=[i,fσj(i)]=f j σ(i)−1
k=i Jk. Assume thatj andn+1are relatively prime.
Then the set{ψk
σ(J)|0≤k≤n−1}is a basis forWZ2.
(3) For any cyclic permutationsσ andρonP,ψσ andψρare similar onWZ2.
Con-sequently, the Petrie matrices overZ2of all cyclic permutations onPare similar to one another and have the same characteristic polynomialnk=0xk.
(4) The coefficients of the characteristic polynomial ofAσ are all odd integers (and
hence nonzero) with constant term±1.
Remark3.3. Part (3) of the above theorem does not hold if the Petrie matrices of
cyclic permutations are over the finite fieldZp for any prime p≥3. For example, if
P = {1,2,3,4,5}, σ denotes the cyclic permutation 1→2→5→4→3→1, and ρ
denotes the cyclic permutation 1→2→3→4→5→1, thenAσ andAρare not similar overZpfor any primep≥3 because the characteristic polynomials ofAσ andAρare
x4−x3−3x2−3x−1 andx4−x3−x2−x−1, respectively, which are distinct overZ p for any primep≥3.
Proof. For any fixed integer 1≤i≤n, let 1≤j≤nbe the unique integer such that
fσj(i)=i+1, and soJi=[i,i+1]=[i,fσj(i)]. Letbdenote the greatest common divisor ofj andn+1 and lets=(n+1)/b. For every integer 1≤k≤s−1, let 1≤mk≤nbe the unique integer such thatkj≡mk(modn+1). Then, byLemma 3.1, we obtain that {mk|1≤k≤s−1} = {kb|1≤k≤s−1}. Letm0=0. Then{mk|0≤k≤s−1} = {kb| 0≤k≤s−1}. Hence, the set{0,1,2,3,...,n−1,n}is the disjoint union of the sets{mk+
m|0≤k≤s−1}, 0≤m≤b−1. Therefore,sk−1=0ψmk
σ (Ji)=sk−1=0ψσkj(Ji) (sincekj≡
mk(modn+1))=[i:fσj(i)]+[fσj(i):fσ2j(i)]+[fσ2j(i):fσ3j(i)]+ ··· +[fσ(s−2)j(i) :
fσ(s−1)j(i)]+[fσ(s−1)j(i):i]=0. So,n=0ψσ(Ji)=mb−=10ψmσ(sk−=10ψmσk(Ji))=0. This
For the proof of part (2), we first show that ifEis a nonempty subset of{1,2,3,..., n−1,n}such thatJ+k∈Eψkσ(J)=0, thenE= {1,2,3,...,n−1,n}. Indeed, for every integer 1≤k≤n, let 1≤mk≤nbe the unique integer such thatkj≡mk(modn+1). Assume thatm1=j∉E. Then, for anym∈E,m =0,j. Sinceψmσ(J)=ψmσ([i,fσj(i)])=
[fm
σ(i):fσm+j(i)], the endpoints ofψmσ(J)do not contain the pointfσj(i). Thus, in the expression ofψm
σ(J)as a sum of the basis elementsJk’s, it contains either both the basis elementsJfj
σ(i)−1andJfσj(i)or none of them. But, sinceJ=[i,f j
σ(i)]=Ji+Ji+1+ ···+Jfj
σ(i)−1contains the elementJfσj(i)−1, but not the elementJfσj(i), in its expression as a sum of the basis elementsJk’s, we obtain that in the expression ofJ+m∈Eψσm(J) as a sum of the basis elementsJk’s, the coefficient ofJfj
σ(i)−1is different from that of
Jfj
σ(i)by 1. This implies thatJ+
m∈Eψmσ(J) =0, which is a contradiction. Therefore,
m1=j∈E.
Thus,
0=J+ m∈E
ψm σ(J)
=J+ψjσ(J)+ m∈E\{m1}
ψm σ(J)
= i,fσj(i)+ fσj(i):fσ2j(i)+
m∈E\{m1}
ψm σ(J)
= i:fσ2j(i)+
m∈E\{m1}
ψm σ(J).
(3.1)
Proceeding in this manner finitely many times, we obtain that{m1,m2,...,mn−1} ⊂E
and
0=J+ m∈E
ψm σ(J)
= i:f2j σ (i)+
m∈E\{m1}
ψm σ(J)
= i:f3j σ (i)+
m∈E\{m1,m2}
ψm σ(J)
= ··· = i:fσnj(i)
+
m∈E\{m1,m2,...,mn−1}
ψm σ(J).
(3.2)
In particular, 0= [i: fσnj(i)]+m∈E\{m1,m2,...,mn−1}ψmσ(J). If m∈ E and m = mn, then, as above, sincem =0 andm =mn≡nj(modn+1), the endpoints ofψσm(J) do not contain the pointfσnj(i). Hence, in the expression ofψmσ(J)as a sum of the basis elementsJk’s, it contains either both the basis elementsJfnj
σ (i)−1 andJfσnj(i) or none of them. But, since[i,fσnj(i)]=Ji+Ji+1+ ··· +Jfnj
σ (i)−1 contains the element
Jfnj
σ (i)−1, not the element Jfσnj(i), in its expression as a sum of the basis elements
Jk’s, we obtain that in the expression of[i:fσnj(i)]+m∈E\{m1,m2,...,mn−1}ψmσ(J) as a sum of the basis elements Jk’s, the coefficient of Jfnj
ON THE CLASS OF SQUARE PETRIE MATRICES... 1621
ofJfnj
σ (i) by 1. This implies that [i:f nj
σ (i)]+m∈E\{m1,m2,...,mn−1}ψmσ(J) =0, which is a contradiction. Thus, mn =nj ∈E. Since, by assumption,j and n+1 are rela-tively prime, we see that, byLemma 3.1, {m1,m2,...,mn} = {1,2,...,n−1,n}. Since {m1,m2,...,mn} ⊂E⊂ {1,2,...,n−1,n}, we obtain thatE= {1,2,...,n−1,n}. This
proves our assertion.
Now, assume thatnk=−01α(k)ψkσ(J)=0, whereα(k)=0 or 1 inZ2, 0≤k≤n−1. If
α(0)=0 andα() =0 for some integer 1≤ < n−1, letbe the smallest such integer; then, sinceψσ is invertible byLemma 2.1(2), we obtain thatJ+kn=1−−1α(k)ψkσ(J)= 0. So, without loss of generality, we may assume thatα(0) =0. That is, we assume that J+nk=1−1α(k)ψσk(J)=0. Let E = {k|1≤k≤n−1,α(k) =0}. Then, we have
J+k∈Eψkσ(J)=0. But then it follows from what we have just proved above that
E= {1,2,...,n−1,n}. This contradicts the assumption thatE⊂ {1,2,...,n−1}. So, the set{ψk
σ(J)|0≤k≤n−1}is linearly independent and hence, by [8], is a basis forWZ2. This proves part (2).
Letθdenote the cyclic permutation 1→2→3→ ··· →i→i+1→ ··· →n→n+1→1 onPand letσbe any cyclic permutation onP. Choose any fixed integer 1≤j≤nsuch thatj and n+1 are relatively prime and letJ=[1,fσj(1)]. Then, by part (2), the set {ψk
σ(J)|0≤k≤n−1}is a basis forWZ2. Letφbe the linear transformation onWZ2 defined byφ(Jk)=ψkσ−1(J), 1≤k≤n. Then φis an isomorphism onWZ2. Further-more,(φ◦ψθ)(Jn)=φ(nk=1Jk)=nk=1φ(Jk)=nk=1ψσk−1(J)=ψnσ(J)(by part (1))=
ψσ(ψnσ−1(J))=ψσ(φ(Jn))=(ψσ◦φ)(Jn)and, for every integer 1≤k≤n−1,(φ◦
ψθ)(Jk)=φ(ψθ(Jk))=φ(Jk+1)=ψkσ(J)=ψσ(ψσk−1(J))=ψσ(φ(Jk))=(ψσ◦φ)(Jk). Thus,ψσis similar toψθthroughφ. Since the property of similarity is obviously tran-sitive, we obtain that ifρis any cyclic permutation onP, thenψσ andψρ are similar onWZ2. Consequently, by [8], the Petrie matrices (overZ2) of all cyclic permutations on
P are similar to one another and so have the same characteristic polynomialnk=0xk sincenk=0xkis easily verified to be the characteristic polynomial of the Petrie matrix
AθoverZ2. This proves part (3).
Finally, letσ be a cyclic permutation on P. Since Aσ is a realn×n matrix with entries either zeros or ones, the coefficients of the characteristic polynomial ofAσare all integers. By taking every entry inAσmodulo 2 and applying part (3) and the fact that the determinants of Petrie matrices are either 0 or±1, we obtain that the characteristic polynomial of Aσ(mod 2)is equal to nk=0xk. Consequently, the coefficients of the
characteristic polynomial ofAσare all odd integers with constant term±1. This proves part (4) and completes the proof ofTheorem 3.2.
Acknowledgment. The author would like to thank Professor Peter Jau-Shyong
Shiue for his interest and many helpful suggestions which led to the improvement of this paper.
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