Examples

This subsection presents some examples to elucidate the methods described above.

Example 2.52 Consider

 are computed directly as follow:

Γ_B

where (n, l) is the greatest common divisor of n and l, are easily verified.

Consequently, for any n≥ 1, ζn(s) = exp

and the zeta function ζ(s) = Q^∞

Now, it is easy to check that lim

n→∞(bχn)ⁿ¹ = 2. Therefore, bS^∗ = ¹₂ as in (2.173) and (2.174) for bζ(s).

Theorem 2.45 implies that the zeta function ζ_B⁰(s) ofB given by (2.176) is

ζ_B⁰(s) = as described elsewhere [36].

However, (2.177) implies

Tb_n= I2ⁿ, (2.184) where I2ⁿ is the 2ⁿ× 2ⁿ identity matrix. Therefore,

ζbn(s) = (1− sⁿ)^−χn, (2.185) where χn is the cardinal number of In. Now, (2.181) and (2.185) imply bχn= χn, i.e.,

χ_n= 1 n

Xn l=1

2^(n,l). (2.186)

Note that (2.186) also follows from the identity (2.182).

Moreover, (2.184) implies

ζb_n(s) = exp

1

ntr(R_n) log(1− sⁿ)⁻¹

 . Therefore, (2.185) implies

1

ntr(Rn) = χn. (2.187)

Hence,

tr(Rn) = Xn

l=1

2^(n,l). (2.188)

The following example can also be solved explicitly and is helpful in elucidating the natural boundary and location of the poles of the zeta function.

Example 2.53 Consider

H₂ =









1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0









. (2.189)

Then,

V₂ =









1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0









= G⊗ G, (2.190)

where

G =



 1 1 1 0



 (2.191)

is the one-dimensional golden-mean matrix, which has eigenvalues

g = ¹⁺₂^√5 and g = ¹⁻₂^√5 =−g⁻¹. (2.192) Now,

He₂ = V2 and eV₂ = H2. (2.193) Then,

T₂ = V2◦ eH₂ = V2 = G⊗ G can be verified, and for any n≥ 2,

T_n= G⊗G ⊗ · · · ⊗ G⊗| {z }

n−1 times ⊗

G =ⁿ⁻¹⊗ G, (2.194)

which is the n− 1 times Kronecker product of G.

The spectrum of Tn is

Σ(Tn) ={g^n−jg^j|0 ≤ j ≤ n}, (2.195) which has n+1 members. The number of linearly independent symmetric eigenvectors of g^n−jg^j is

χn,j = χ(g^n−jg^j)

= P

i∈Iⁿ λn,i=g^n−jg^j

1.

(2.196)

Clearly, χn,0= χn,n= 1. Furthermore for any 1≤ j ≤ n − 1, by Burnside’s Lemma,

χ_n,j = 1 n

X

d|(j,n−j)

φ((j, n− j)/d)C_jd/(j,n−j)^nd/(j,n−j), (2.197)

where φ is the Euler totient function (2.36). The detailed proof of (2.197) is omitted

Now, consider bT_n and the associated zeta function bζ(s). Clearly,

b

T₂ = H2◦ eV2 = H2.

To study higher-order bT_n, n ≥ 3, the recursive formula of Hn must be obtained. Let

H_n=

The remaining matrix of bT_n can be verified to be a full matrix E_brn after the zero rows and columns have been deleted, where brⁿ is the sum of entries in the first row of bT_n. Hence, the maximum eigenvalue bλn of bT_n equals brⁿ, the other eigenvalues are zeros.

From (2.201), it is easy to verify

bλn+1 = bλn+ bλ_n−1 (2.202) with bλ2 = 3 and bλ3 = 4. Therefore,

ζbn(s) = (1− bλⁿsⁿ)⁻¹ (2.203) and

ζ(s) =b Y∞ n=1

(1− bλⁿsⁿ)⁻¹. (2.204)

Now, bλn and gⁿ must be compared. Let

gⁿ= αng + βn

with α2 = β2 = 1. Then, αn+1 = αn+ βn and βn+1 = αn. That

λb_n= α_n+ 2β_n can be verified and

bλn+1− gⁿ⁺¹ =− (√

5− 1)αn+1+ 2βn

(√

5− 1)αⁿ+ 2β_n−1

!

(bλn− gⁿ). (2.205) Equation (2.205) implies

bλ⁻_2n²ⁿ¹ < g⁻¹ < bλ⁻

1 2n+1

2n+1 . (2.206)

Equation (2.206) implies that the meromorphic extension bζ of ζ_B⁰ satisfies bS^∗ = g⁻¹ and has poles onn

bλ⁻_2n²ⁿ¹ e^πij/n: 0≤ j ≤ 2n − 1, n ≥ 1o

with the natural boundary|s| = g⁻¹. Furthermore, (2.199) and (2.204) lead an identity involving χn,j and g, the detail is omitted.

2.6 Equations on Z² with numbers in a finite field

This subsection briefly discusses the equations on Z² with numbers in a finite field, see [31; 36; 47]. The problems can be studied by applying the methods that were developed in the previous subsections. Lind [36] considered the following example.

Example 2.54 Consider F2 ={0, 1} and

X=n

x∈ F2^Z2 : xi,j + xi+1,j+ xi,j+1 = 0 for all i, j∈ Zo

. (2.207)

In this case, X is a compact group with coordinate-wise operations, and it is invariant under the natural Z²-shift action σ.

The equation

x_i,j + x_i+1,j + x_i,j+1 = 0 (2.208)

can be interpreted as a pattern generation problem on L-shape lattices: . Indeed, the solutions of (2.208) are given by

B(L) =

which consists of all even patterns on L-shape lattices. B(L) can be extended to Z2×2

as

can be easily verified.

Therefore,

He₂ = exactly a single 1 and each column has either two 1s or all 0s. Therefore, the eigenvalue λ of Tn is |λ| = 1 or λ = 0. With a rotationally symmetric eigenvector, Tⁿ generates the graph with equivalent classes Cn(i) as vertices and has m(n) disjoint cycles; each cycle has period pn,k ≥ 1, 1 ≤ k ≤ m(n). In computing, it is more efficient to compute λ ∈ Σ(τⁿ) with algebraic multiplicity χ(λ).

The following can be demonstrated

ζn(s) =

For n = 1 to 20, the numbers and periods of cycles are listed in Table 2.1.

n 1 2 3 4 5 6 7 8 p

q 1 1

1 1

1 2

1 1

1 3 1 1

1 2 2 1

1 7 3 1

1 1

9 10 11 12 13

1 7 2 4

1 3 6 1 1 4

1 31 1 3

1 2 4 2 1 5

1 63 1 5

14 15 16 17

1 2 7 14 3 4 1 20

1 3 15 4 4 72

1 1

1 5 15 1 3 256

18 19 20

1 2 7 14 2 1 4 259

1 511 1 27

1 3 6 12 1 1 4 272 p : the period of cycle.

q = q(p) : the number of cycles with period p.

Table 2.1.

From Table 2.1, ζn can be written for 1≤ n ≤ 20. For example,

ζ₁₄ = 1

(1− s¹⁴)³(1− s²⁸)⁴(1− s⁹⁸) (1− s¹⁹⁶)²⁰.

Up to n = 20, the Taylor expansion of (2.216) at s = 0, which recovers Lind’s result [36] (p.438), is

ζ_B(s) = 1 + s + 2s²+ 4s³+ 6s⁴+ 9s⁵+ 16s⁶+ 24s⁷+ 35s⁸+ 54s⁹ (2.217) +78s¹⁰+ 110s¹¹+ 162s¹²+ 226s¹³+ 317s¹⁴+ 446s¹⁵+ 612s¹⁶

+834s¹⁷+ 1146s¹⁸+ 1543s¹⁹+ 2071s²⁰+· · · .

Further investigation is needed to understand τn and pn,k for large n. The results will appear elsewhere.

Lind [36] showed that the zeta function ζ⁰ defined by (2.207) is analytic in |s| < 1.

By (2.216), all poles of ζ appear on |s| = 1. Therefore, ζ is analytic in |s| < 1 with natural boundary |s| = 1.

In the following example, the harmonic patterns on square-cross lattice L: , which were studied by Ledrappier [31], are investigated.

Example 2.55 Let F2 ={0, 1} and

X=n

x∈ F2^Z2 : xi,j = x_i−1,j+ x_i,j−1+ xi+1,j + xi,j+1 for all i, j ∈ Zo

. (2.218) As in Example 2.54, the basic set on L is

B(L) =



 ^{x−1,0 x}_x^0,0_0,

−1

x1,0

x0,1

∈ F2^L : x0,0+ x_−1,0+ x_0,−1+ x1,0+ x0,1 = 0



, (2.219) which consists of all even patterns on a square-cross lattice. B(L) can be extended to Z_3×3 as

B =









x0,0

x₋1,0

x0,−1

x1,0

x0,1

x₋1,1 x1,1

x₋1,−1 x1,−1

∈ F2^Z3×3 : x0,0+ x_−1,0+ x_0,−1+ x1,0+ x0,1 = 0









. (2.220)

Then, that

Σ(B) = X (2.221)

can be easily verified.

Now, by (2.108), the associated trace operator T_n×3(B) can be constructed for n ≥ 1.

Furthermore, the rotational matrix R_n×2 is defined by (2.109). The number χ_n×2 of the equivalent classes of R_n×2 can be shown to be the number of n-bead necklaces with four colors. The formulae for χ_n×2, n ≥ 1, is given by (2.113) with m = 3.

As in Example 2.54, the reduced trace operator τ_n×3 of T_n×3 is more convenient for computing the n-th order zeta function ζn. The definition and results of the reduced trace operator for more symbols on larger lattices are similar to Definition 2.23 and Theorem 2.26.

By the same argument as in Example 2.54, let the graph generated by T_n×3 have m(n) disjoint cycles, each of period pn,k ≥ 1, for 1 ≤ k ≤ m(n). Then, the n-th order zeta function can be represented as

ζn(s) =

m(n)Y

k=1

1

1− s^npn,k. (2.222)

Hence,

ζ(s) = Y∞ n=1

m(n)Y

k=1

1

1− s^npn,k. (2.223)

Table 2.2 presents the numbers and periods of cycles of T_n×3. For brevity, only n = 1 to 9 are listed.

n 1 2 3 4 5

p q

1 3 1 1

1 3 1 3

1 2 3 6 2 2 2 2

1 3 6 1 7 8

1 3 5 15

7 7 9 9

6 7 8

1 2 3 4 6 12

2 6 6 8 10 48

1 3 9

1 1 260

1 3 6 12

1 7 88 640

9

1 2 3 6 7 14 21 42

2 2 2 2 260 390 260 390

p : the period of cycle.

q = q(p) : the number of cycles with period p.

Table 2.2.

Up to n = 16, the Taylor expansion of (2.223) at s = 0 is

ζ_B(s) = 1 + s + 2s²+ 5s³+ 7s⁴+ 17s⁵+ 32s⁶+ 46s⁷+ 84s⁸+ 140s⁹ (2.224) +229s¹⁰+ 384s¹¹+ 615s¹²+ 938s¹³+ 1483s¹⁴+ 2353s¹⁵+ 3563s¹⁶+· · · .

The analyticity and the natural boundary of the zeta function in (2.223) need further investigation. The results will appear elsewhere.

In the following example, we study the equation on the diagonal lattice L: and show that the rectangular zeta function ζ = bζ fails to describe poles and natural boundary of ζ⁰ but ζγ works well with γ =

It is easy to verify

T₁ = bT₁ =

Furthermore, for n≥ 3, we show that

T_n = bT_n= R^t_n. (2.231)

Indeed, by the recursive formula of Vn, it can be verified that Vn;i,j = 1 if and only if





i = 2j− 1 and 2j for 1≤ j ≤ 2ⁿ⁻¹,

i = 2(i− 2ⁿ⁻¹)− 1 and 2(i − 2ⁿ⁻¹) for 2ⁿ⁻¹+ 1 ≤ j ≤ 2ⁿ. (2.232) Therefore, by applying (2.26), Tn = [tn;i,j] with tn;i,j = 1 if and only if





i = 2j− 1 for 1≤ j ≤ 2ⁿ⁻¹,

i = 2(i− 2ⁿ⁻¹) for 2ⁿ⁻¹+ 1≤ j ≤ 2ⁿ. (2.233) Hence,

T_n= R^t_n. (2.234)

Therefore,

ζn(s) = 1

(1− sⁿ)^χn, (2.235)

where χn is the cardinal number of Iⁿ, and ζ(s) =

Y∞ n=1

1

(1− sⁿ)^χn. (2.236)

As in Example 2.52, lim

n→∞χnⁿ¹ = 2 and then S^∗ = ¹₂. On the other hand, consider

B^′ = (

0

0 0

0

,

0

0 1

1

,

1

1 0

0

,

1

1 1

1 )

. (2.237)

Then,

Σ(B^′) = Σ(B). (2.238)

In particular,

Γ_B^′







 n l 0 k





γ



 = 2^k. (2.239)

Therefore, as in Example 2.52,

ζγ;n= 1 1− 2sⁿ.

We can also use the construction of Tγ;n in Subsection 2.4 to study ζγ;n. Indeed, it is easy to see that

T_γ;1 =









1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1









. (2.240)

Therefore,

ζ_γ;1 = 1

1− 2s. (2.241)

Furthermore, for any n ≥ 2, after deleting the zero columns and rows of Tγ;n, Tγ;n is reduced to Tγ;1. Therefore,

ζγ;n= 1

1− 2sⁿ. (2.242)

Hence,

ζγ= Y∞ n=1

1

1− 2sⁿ. (2.243)

Then, ζγ has natural boundary with |s| = 1 and has poles n

2⁻ⁿ¹e^2πij/n : 0≤ j ≤ n − 1, n ≥ 1o .

Motivated by Examples 2.54∼2.56, given a finite field F and a set of finite lattice points L⊂ Z², consider the equation

P

(i,j)∈L

xi,j = 0 in F. (2.244)

Then, denote the solution set of (2.244) on Z² by

X(L) =



x∈ F^Z2 : X

(i,j)∈L

xi+k,j+l = 0, (k, l)∈ Z²



. (2.245)

Denoted by

B(L) =



x : L→ F : X

(i,j)∈L

xi,j = 0



, (2.246)

B(L) ⊂ F^L is the set of admissible local patterns.

Let Z_m×m be the smallest rectangular lattice that contains L. Let B be the set of all admissible patterns on Z_m×m that can be generated fromB(L). Then, the following can be easily verified;

X(L) = Σ(B). (2.247)

The results presented in previous subsections apply to Σ(B) and then to X(L). The above method can also be applied to any finite set of equations defined on L with numbers in F , since the solution set B(L) ⊂ F^L and can be extended to a unique admissible set B ⊆ F^Zm×m.

2.7 Square lattice Ising model with finite range interaction

This subsection extends the results presented in previous sections to the thermody-namic zeta function for a square lattice Ising model with finite range interaction, see Ruelle [45] and Lind [36]. For simplicity, the square lattice Ising model with nearest neighbor interaction is considered.

The square lattice Ising model with external field H, the coupling constant J in the horizontal direction, and the coupling constant J^′ in the vertical direction is now considered. Each site (i, j) of the square lattice Z² has a spin u_i,j with two possible values, +1 or −1. First, assume that the state space is {+1, −1}^Z2. Given a state U = {u^i,j}i,j∈Z in{+1, −1}^Z2, denoted by U_m×n = U

Zm×n ={u^i,j}0≤i≤m−1,0≤j≤n−1. Define the Hamiltonian (energy) E(Um×n) for U_m×n by

E(Um×n) =−J X

0≤i≤m−2 0≤j≤n−1

ui,jui+1,j − J^′ X

0≤i≤m−1 0≤j≤n−2

ui,jui,j+1− H X

0≤i≤m−1 0≤j≤n−1

ui,j. (2.248)

Therefore, the partition function Zm×n is defined by

Zm×n = X

Um×n

∈{+1,−1}^Zm×n

exp





K X

0≤i≤m−2 0≤j≤n−1

ui,jui+1,j + L X

0≤i≤m−1 0≤j≤n−2

ui,jui,j+1+ h X

0≤i≤m−1 0≤j≤n−1

ui,j





 , (2.249)

where K = J /kBT ,L = J^′/kBT , h = H/kBT , kB is Boltzmann’s constant and T is the temperature.

To the thermodynamic zeta function, given L =



-periodic states is defined by

ZL=Z Then, as in (1.31), the thermodynamic zeta function for the square lattice Ising model with nearest neighbor interaction can be defined by

ζ⁰(s)≡ ζIsing⁰ (s)≡ exp X

To simplify the notation, the subscript Ising is omitted in this subsection whenever such omission will not cause confusion.

As (1.8) and (1.9), for any n≥ 1, define the n-th order thermodynamic zeta function ζIsing;n(s) as

the thermodynamic zeta function ζIsing(s) is given by

ζ(s)≡ ζIsing(s)≡ Y∞ n=1

ζ_n(s). (2.253)

Since the discussion of ζn(s) is similar to that in Subsections 2.1 and 2.2, only the parts of the arguments that differ are emphasized. The results are outlined here and the details are left to the reader.

According to the spin ui,j ∈ {+1, −1} for i, j ∈ Z, replacing all the symbols ”0”

in (2.1) and (2.2) with the symbol ”−1” yields the ordering matrices XIsing;2×2 and Y_Ising;2×2.

The ordering matrix X_Ising;n×2, Y_Ising;n×2 and the cylindrical ordering matrix C_Ising;n×2 can be obtained in the same way. The recursive formulae for generating Y_Ising;n×2form Y_Ising;2×2 are as in (2.13).

Given L∈ L², (2.250) yields

Z^L = X

U ∈fixL({+1,−1}^Z2) Y

0≤i≤n−1 0≤j≤k−1

exp [ui,j(Kui+1,j+ Lui,j+1+ h)] . (2.254)

Based on (2.254), the associated horizontal transition matrix HIsing;2 = [aI;i,j]_4×4 and the vertical transition matrix VIsing;2 = [bI;i,j]_4×4 are defined as

H_Ising;2=









e^K+L−h e^−K−L−h e^K−L−h e^−K+L−h e^−K+L−h e^K−L−h e^−K−L−h e^K+L−h e^K+L+h e^−K−L+h e^K−L+h e^−K+L+h e^−K+L+h e^K−L+h e^−K−L+h e^K+L+h









= [a_I;i,j]_4×4, (2.255)

and

V_Ising;2 =









e^K+L−h e^−K−L−h e^−K+L−h e^K−L−h e^K−L−h e^−K+L−h e^−K−L−h e^K+L−h e^K+L+h e^−K−L+h e^−K+L+h e^K−L+h e^K−L+h e^−K+L+h e^−K−L+h e^K+L+h









= [bI;i,j]_4×4, (2.256)

respectively. Similar to (2.18) and (2.19), the associated column matrices eH_Ising;2 of H_Ising;2 and eV_Ising;2 of VIsing;2 are defined as

He_Ising;2=

Therefore, the trace operators TIsing;2 and bT_Ising;2 are defined as

T_Ising;2 = V_Ising;2◦ eH_Ising;2 and bT_Ising;2= H_Ising;2◦ eV_Ising;2. (2.259) The recursive formulas for TIsing;n and bT_Ising;n are similar to (2.26). Constructing T_Ising;2 and the rotational matrix R_n yield a similar result to that of Theorem 2.13 for Z

From Theorem 2.57, the n-th order thermodynamic zeta function ζIsing;n can now be obtained as follows.

Theorem 2.58 For any n≥ 1,

ζIsing;n = exp 1 n

X∞ k=1

tr T^k_Ising;nR_n
s^nk

!

. (2.261)

The Rn-symmetric property of TIsing;n is essential to the rationality of n-th order thermodynamic zeta function ζIsing;n.

Proposition 2.59 For any n≥ 1,

T_Ising;n;σ^l_(i),σ^l_(j)= TIsing;n;i,j (2.262) for all 1≤ i, j ≤ 2ⁿ and 0≤ l ≤ n − 1.

Similarly, the associated reduced trace operator τIsing;n can be defined as in (2.100).

Finally, by the arguments presented in Subsection 2.2, the rationality of the n-th order thermodynamic zeta function ζIsng;n is established as follows.

Theorem 2.60 For n≥ 1,

ζIsing;n(s) = Y

λ∈Σ(TIsing;n)

(1− λsⁿ)^−χ(λ) (2.263)

= (det (I− sⁿτIsing;n))⁻¹, (2.264) where χ(λ) is the number of linear independent symmetric eigenvectors and generalized eigenvectors of TIsing;n with eigenvalue λ. Furthermore,

ζIsing(s) = Y∞ n=1

(det (I− sⁿτIsing;n))⁻¹. (2.265)

The state space {+1, −1}^Z2 is extended to the shift of finite type given by B ⊆ {+1, −1}^Z2×2. Given B ⊆ {+1, −1}^Z2×2 and L =



 n l 0 k



 Z² ∈ L2, the partition

function for B with



 n l 0 k



-periodic patterns is defined as

ZL(B) = ZB







 n l 0 k







 =

X thermodynamic zeta function is defined by

ζ_Ising;B⁰ (s)≡ exp X

Similar to (2.252) and (2.253), for any n ≥ 1, the n-th order thermodynamic zeta function ζ_Ising;B;n(s) is defined as

ζ_Ising;B;n(s)≡ exp

and the thermodynamic zeta function ζ_Ising;B(s) is given by

ζ_Ising;B(s)≡ Y∞ n=1

ζ_Ising;B;n(s). (2.269)

Equations (2.15), (2.255) and (2.256) are combined to define the associated hori-zontal transition matrix and vertical transition matrix as follows.

H_Ising;2(B) = HIsing;2◦ H2(B) (2.270) and

V_Ising;2(B) = VIsing;2◦ V2(B). (2.271) Therefore, the trace operator TIsing;n(B) and the associated reduced trace operator τIsing;n(B) can be defined for all n ≥ 1 as above. Since all arguments for ζIsing;B;n are similar to those above; the final result is as follows.

Theorem 2.61 For n≥ 1,

ζ_Ising;B;n(s) = Y

λ∈Σ(TIsing;n(B))

(1− λsⁿ)^−χ(λ) (2.272)

= [det (I− sⁿτIsing;n(B))]⁻¹, (2.273) where χ(λ) is the number of linear independent symmetric eigenvectors and generalized eigenvectors of TIsing;n(B) with eigenvalue λ. Moreover,

ζ_Ising;B(s) = Y∞ n=1

[det (I− sⁿτ_Ising;n(B))]⁻¹. (2.274) Remark 2.62 The results in this subsection hold for models with finite range interac-tion.

3 Zeta functions for higher-dimensional shifts of finite type

This section studies the zeta functions for d-dimensional shifts of finite type, d≥ 3.

3.1 Three-dimensional shifts of finite type

In this subsection, the zeta functions for three-dimensional shifts of finite type are investigated.

3.1.1 Periodic patterns, trace operator and rotational matrices

This subsection studies the properties of the periodic patterns and derives trace

op-erator and rotational matrices. Furthermore, Γ_B













a1 b12 b13

0 a2 b23

0 0 a₃











can be expressed

in terms of the trace of the products of the trace operator and rotational matrices . For clarity, two symbols on 2× 2 × 2 lattice Z2×2×2 are examined first. For given positive integers N1, N2 and N3, the rectangular lattice ZN1×N2×N3 is defined by

Z_N

1×N2×N3 ={(n1, n2, n3) : 0≤ ni ≤ Ni− 1, 1 ≤ i ≤ 3} . In particular,

Z2×2×2={(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

Define the set of all global patterns on Z³ with two symbols {0, 1} by

Σ³₂ ={0, 1}^Z3 =

U | U : Z³ → {0, 1}

.

Here, Z³ ={(n1, n2, n3) : n1, n2, n3 ∈ Z}, the set of all three-dimensional lattice points (vertices). The set of all local patterns on ZN1×N2×N3 is defined by

ΣN1×N2×N3 ={U|Z_N1×N2×N3 : U ∈ Σ³2},

and a local pattern of a global pattern U on ZN1×N2×N3 is denoted by

UN1×N2×N3 ≡ U|Z_N1×N2×N3 = (uα1,α2,α3)_0≤α

i≤Ni−1,1≤i≤3,

where u_α1,α2,α3 ∈ {0, 1}. To simplify the notation, the subscripts of UN1×N²×N³ and (uα1,α2,α3)_{0≤αi≤Ni−1,1≤i≤3} are omitted whenever such omission will not cause confusion.

Now, for any givenB ⊂ Σ2×2×2, B is called a basic set of admissible local patterns.

In short,B is a basic set. A local pattern UN1×N2×N3 = (uα1,α2,α3) is calledB-admissible if for any vertex (lattice point) (n1, n2, n3) with 0 ≤ ni ≤ Ni− 2, 1 ≤ i ≤ 3, there exist a 2× 2 × 2 admissible local pattern (βk1,k2,k3)_{0≤k1,k2,k3≤1}∈ B such that

un1+k1,n2+k2,n3+k3 = βk1,k2,k3

for 0≤ k1, k2, k3 ≤ 1.

Given a lattice L∈ L3 with Hermite normal form,

L =







a1 b12 b13

0 a2 b23

0 0 a3





Z³, (3.1)

where a_i ≥ 1 for 1 ≤ i ≤ 3 and 0 ≤ bij ≤ ai− 1 for i + 1 ≤ j ≤ 3. A global pattern

U = (uα1,α2,α3)_{α1,α2,α3∈Z} is called L-periodic or







a₁ b₁₂ b₁₃ 0 a2 b23

0 0 a3





-periodic if for every

α1, α2, α3 ∈ Z

u_{α1+a1p+b12q+b13r,α2+a2q+b23r,α3+a3r}= u_α1,α2,α3 (3.2)

for all p, q, r ∈ Z.

 are closely related as

fol-lows.

where (m, n) is the greatest common divisor of m and n and [p, q] is the least common

multiple of p and q. Then,



-periodic patterns are



Proof. By (3.2), the



-periodic pattern is easily identified as



Given a basic set B ⊂ Σ2×2×2, defined on cubic lattice Z_2×2×2, the L-periodic patterns that are B-admissible must be verified on Z2×2×2. For n1, n2, n3 ∈ Z, let Z2×2×2((n₁, n₂, n₃)) be the cubic lattice with the smallest vertex (n₁, n₂, n₃):

Z_2×2×2((n1, n2, n3)) = {(n¹+ k1, n2+ k2, n3+ k3) : 0≤ k¹, k2, k3 ≤ 1} .

Now, the admissibility of L-periodic patterns is demonstrated to be verified on finite cubic lattices.

Proposition 3.2 An L-periodic pattern U is B-admissible if and only if

U |Z2×2×2((α1,α2,α3))∈ B for 0≤ αi ≤ ai− 1, 1 ≤ i ≤ 3.

Proof. Sine B ⊂ Σ2×2×2, it is sufficient to prove

U |Z2×2×2((α1,α2,α3)): α1, α2, α3 ∈ Z

= 

U |Z2×2×2((α1,α2,α3)): 0≤ αi ≤ ai− 1, 1 ≤ i ≤ 3 . The proof follows easily from (3.2). The details are left to the reader.

According to Proposition 3.2, the admissibility of an L-periodic pattern U is deter-mined by U |Z(a1+1)×(a2+1)×(a3+1)= (u_α1,α2,α3) and U |Z(a1+1)×(a2+1)×(a3+1) has the periodic property that is given by (3.2), which can be divided into two parts:





u_a1,α2,α3 = u_0,α2,α3 uα1,a2,α3 = u[α1−b12]_a1,0,α3

(3.3) for 0≤ αⁱ ≤ aⁱ, 1≤ i ≤ 3, where [m]ⁿ≡ m (mod n);

uα1,α2,a3 =











u_{[α1−b12−b13]a1,0,0} if α₂− b23 = a₂

u[α1−b13]_a1,α2−b23,0 if 0≤ α2− b23≤ a2− 1 u[α1+b12−b13]_a1,α2−b23+a2,0 if −a²+ 1≤ α² − b²³ ≤ −1

(3.4)

for 0≤ α1 ≤ a1, 0 ≤ α2 ≤ a2.

Notably, (uα1,α2,α3)_{0≤α1≤a1,0≤α2≤a2},α3 has the same structure (3.3) for all 0≤ α³ ≤ a³, which fact is useful in constructing the cylindrical ordering matrix. Then, the set of all local patterns in Σa1+1,a2+1,a3+1 that satisfy the periodic property (3.3) is denoted by Pa1,a2;b12;a3+1. However, (3.4) is important in allowing patterns in Pa1,a2;b12;a3+1 to become L-periodic and it will be used to define the rotational matrices later.

Now, the counting function for Un1×n2×n3 = (uα1,α2,α3) in Σn1×n2×n3, n1, n2, n3 ≥ 1, is defined by

ψ (Un1×n2×n3) = 1 +

nX1−1 α1=0

nX2−1 α2=0

nX3−1 α3=0

uα1,α2,α32^{n1n2(n3−1−α3)+n1(n2−1−α2)+n1−1−α1}. (3.5)

Similar to (3.5), the counting function ψ for patterns U in Pn1,n2;l;1, 0≤ l ≤ n1− 1, is defined by

ψ U

≡ ψ U |Z_n1×n2×1

. (3.6)

Notably, ψ is bijective from Pn1,n2;l;1 to {i | 1 ≤ i ≤ 2ⁿ¹ⁿ²}.

Given n1, n2 ≥ 1, 0 ≤ l ≤ n¹ − 1, h ≥ 1, a local pattern U in Pⁿ1,n2;l;h can be represented as

U = U₀⊕zU₁⊕z· · · ⊕zU_h−1, (3.7) where Ui ∈ Pn1,n2;l;1, 0 ≤ i ≤ h − 1, and U^′ ⊕z U^′′ means that U^′′ is put on the top (in the z-direction) of U^′. Therefore, the cylindrical ordering matrix Cn1,n2;l;h = [C_{n1,n2;l;h;i,j}]_2n1n2×2n1n2 of patterns in P_n1,n2;l;h is defined by

Cn1,n2;l;h;i,j =

U0⊕^z · · · ⊕^z U_h−1 | ψ(U⁰) = i and ψ(U_h−1) = j

. (3.8) In particular, for h = 2, C_n1,n2;l;2 can be applied to construct the associated trace operator. Notably the set Cn1,n2;l;2;i,j contains exactly one pattern.

Now, given B ⊂ Σ2×2×2, the associated trace operator Tn1,n2;l(B) = [tⁿ1,n2;l;i,j], with tn1,n2;l;i,j ∈ {0, 1}, can be defined by

tn1,n2;l;i,j = 1 if and only if the pattern in Cn1,n2;l;2;i,j is B-admissible. (3.9)

Remark 3.3 Given L^′ =







a₁ b₁₂ 0 0 a2 0 0 0 a3





Z³, (3.3) and (3.4) easily verify that

U|Z_a1+1,a2+1,a3+1 : U is L^′-periodic

= 

U = U0⊕z· · · ⊕zUa3 ∈ Pa1,a2;b12;a3+1 : U0 = Ua3

.

(3.10)

Furthermore, given B ⊂ Σ2×2×2, from Proposition 3.2 and the construction of the transition matrix T_a1,a2;b12(B),

Γ_B













a₁ b₁₂ 0 0 a2 0 0 0 a3











= tr T^a_a³_1,a2;b12(B)

. (3.11)

The shift maps and the related rotational matrices are considered below for general

L =







a1 b12 b13

0 a₂ b₂₃ 0 0 a3





Z³.

Let n1, n2 ≥ 1, 0 ≤ l ≤ n1−1; the shift (to the left) in the x-direction of any pattern U = (uα1,α2,0) in Pn1,n2;l;1, uα1,α2,0 ∈ {0, 1}, is defined by

σx;n1,n2;l((uα1,α2,0)) =

u⁽¹⁾_α1,α2,0

0≤α1≤n1,0≤α2≤n2

where

u⁽¹⁾_α1,α2,0 =





u[α1+1−l]n1,0,0 if α2 = n2,

u[α1+1]_n1,α2,0 if 0≤ α² ≤ n² − 1. (3.12) Similarly, the shift (to the below) in the y-direction is defined by

σy;n1,n2;l((uα1,α2,0)) =

u⁽²⁾_α1,α2,0

0≤α1≤n1,0≤α2≤n2

where

u⁽²⁾_α1,α2,0 =





u[α1−l]n1,α2+1−n2,0 if α2+ 1 ≥ n2,

u_{[α1]n1,α2+1,0} if 0≤ α²+ 1≤ n²− 1. (3.13) Notably, σ_x;n1,n2;l and σ_y;n1,n2;l are automorphisms on P_n1,n2;l;1.

The following example illustrates σx;n1,n2;l and σy;n1,n2;l. Example 3.4 Let

U = (uα1,α2,0)≡ ∈ P^3,2;1;1

u0,0,0

u0,0,0

u0,0,0

u1,0,0

u1,0,0

u2,0,0

u2,0,0

u2,0,0

u0,1,0

u0,1,0 u1,1,0 u2,1,0

be a local pattern that lies on the plane {(z¹, z2, 0) : z1, z2 ∈ Z}. Now, consider σ^x;3,2;1 and σ_y;3,2;1 which are acting on U . Then it is easy to see

σx;3,2;1 U

=

u1,0,0

u1,0,0

u1,0,0

u2,0,0

u2,0,0

u0,0,0

u0,0,0

u0,0,0

u1,1,0

u1,1,0 u2 ,1

,0 u0,1,0

and

σ_y;3,2;1 U

= .

u0,1,0

u0,1,0

u0,1,0

u1,1,0

u1,1,0

u2,1,0

u2,1,0

u2,1,0

u2,0,0

u2,0,0 u0 ,0

,0 u1,0,0

Moreover, both σ_x;3,2;1 U

and σ_y;3,2;1 U

are also belong to P_3,2;1;1.

From (3.12) and (3.13), for 0 ≤ rⁱ ≤ nⁱ− 1, i = 1, 2, the following can be straight-forwardly verified;

σ_x;n^r1 _1,n2;l σ^r_y;n² _1,n2;l((uα1,α2,0))

=

u⁽³⁾_α1,α2,0

0≤α1≤n1,0≤α2≤n2

where

u⁽³⁾_α1,α2,0 =





u_{[α1+r1−l]n1,α2+r2−n2,0} if n₂ ≤ α2+ r₂ ≤ 2n2− 1,

u[α1+r1]_n1,α2+r2,0 if 0≤ α2+ r2 ≤ n2− 1. (3.14) Furthermore,

σy;n1,n2;l◦ σ^x;n1,n2;l = σx;n1,n2;l◦ σ^y;n1,n2;l (3.15) and

σ_x;nⁿ¹_1,n2;l = σ^l_x;n1,n2;l σ_y;nⁿ²_1,n2;l

= identity map. (3.16)

Hence,

σ_x;n⁻¹_1,n2;l ≡ σx;nⁿ¹⁻¹1,n2;l and σ_y;n⁻¹_1,n2;l ≡ σx;n^l 1,n2;l σⁿ_y;n²⁻¹_1,n2;l

. (3.17)

Therefore, for 0≤ rⁱ ≤ nⁱ− 1, i = 1, 2,

σ_x;n^−r1_1,n2;l σ^−r_y;n²_1,n2;l((uα1,α2,0))

=

u⁽⁴⁾_α1,α2,0

0≤α1≤n1,0≤α2≤n2

where

u⁽⁴⁾_α1,α2,0 =











u[α1−r1−l]n1,0,0 if α2− r2 = n2,

u_{[α1−r1]n1,α2−r2,0} if 0≤ α²− r² ≤ n²− 1, u_{[α1−r1+l]n1,α2−r2+n2,0} if −n2 + 1≤ α2− r2 ≤ −1.

(3.18)

Now, the two rotational matrices R_x;n1,n2;l and R_y;n1,n2;l are defined as follows.

Definition 3.5 The 2ⁿ¹ⁿ² × 2ⁿ¹ⁿ² x-rotational matrix Rx;n1,n2;l = [Rx;n1,n2;l;i,j], Rx;n1,n2;l;i,j ∈ {0, 1}, is defined by

Rx;n1,n2;l;i,j = 1 if and only if i = ψ(U ) and j = ψ(σx;n1,n2;l(U)), (3.19) where U ∈ Pⁿ1,n2;l;1. From (3.19), for convenience, denote by

j = σx(i). (3.20)

Similarly, the 2ⁿ¹ⁿ² × 2ⁿ¹ⁿ² y-rotational matrix Ry;n1,n2;l = [Ry;n1,n2;l;i,j], Ry;n1,n2;l;i,j ∈ {0, 1}, is defined by

Ry;n1,n2;l;i,j = 1 if and only if i = ψ(U ) and j = ψ(σy;n1,n2;l(U )), (3.21) where U ∈ Pn1,n2;l;1. From (3.21), for convenience, denote by

j = σ_y(i). (3.22)

Obviously, Rx;n1,n2;l and Ry;n1,n2;l are permutation matrices. By (3.16), Rⁿ_x;n¹ _1,n2;l = R^l_x;n1,n2;lRⁿ_y;n²_1,n2;l = I2ⁿ¹ⁿ², where In is the n× n identity matrix.

Example 3.6 Let n1 = 2, n2 = 1 and l = 1,

Rx;2,1;1 = Ry;2,1;1 =









1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1







 .

Then,

R_x;2,1;1² = R_x;2,1;1R_y;2,1;1 = I₄ but R_y;2,1;1 6= I4.

The following proposition shows the permutation characters of Rx;n1,n2;land Ry;n1,n2;l. Proposition 3.7 Let M = [Mi,j]2ⁿ¹ⁿ²×2ⁿ¹ⁿ² be a matrix where Mi,j denotes a number or a pattern or a set of patterns. Then

(MRx;n1,n2;l)_i,j = M_i,σ⁻¹_{x (j)} and (MRy;n1,n2;l)_i,j = M_i,σy⁻¹_(j). (3.23) Furthermore, for any r ≥ 1

MR^r_x;n1,n2;l

i,j = M_i,σ^−r_{x (j)} and MR^r_y;n1,n2;l

i,j = M_i,σ^−r_{y (j)}. (3.24) Proof. For any 1≤ i, j ≤ 2ⁿ¹ⁿ², by (3.20),

(MR_x;n1,n2;l)_i,j =P

q

M_i,qR_{x;n1,n2;l;q,j}

= M_i,σ⁻_x¹_(j)R_{x;n1,n2;l;σx}⁻¹_(j),j

= M_i,σ⁻¹_{x (j)}. Similarly,

(MRy;n1,n2;l)_i,j =P

q

Mi,qRy;n1,n2;l;q,j

= M_i,σ⁻_y¹_(j)R_y;n1,n2;l;σ⁻_y¹_(j),j

= M_i,σ⁻¹_{y (j)}.

Applying (3.23) r times yields (3.24). The proof is complete.

Now, the following lemma can be obtained.

Lemma 3.8 Given L =







a1 b12 b13

0 a₂ b₂₃ 0 0 a3





Z³,

nU |Z(a1+1)×(a2+1)×(a3+1): U is L-periodico

= 

U = U₀⊕z · · · ⊕zU_a3 ∈ Pa1,a2;b12;a3+1 : U_a3 = σ_x;a^−b13_1,a2;b12 σ_y;a^−b23_1,a2;b12(U₀)
(3.25) Proof. From (3.4) and (3.18),

U = U0⊕^z · · · ⊕^zUa3 ∈ P^a1,a2;b12;a3+1 : Ua3 = σ_x;a^−b13_1,a2;b12 σ_y;a^−b23_1,a2;b12(U0)

= 

U ∈ Pa1,a2;b12;a3+1 : U satisfies (3.4) .

Then, by the construction of Pa1,a2;b12;a3+1, the last set is equal to

{U ∈ Σ^a1+1,a2+1,a3+1 : U satisfies (3.3) and (3.4)}

= {U ∈ Σa1+1,a2+1,a3+1 : U satisfies (3.2)} . Therefore, (3.25) follows. The proof is complete.

Proposition 3.2, 3.7 and Lemma 3.8 yield the following main results for

Γ_B where ♯S is the cardinal number of set S.

Then, Proposition 3.7 and the construction of Ta1,a2;b12(B), Rx;a1,a2;b12 and Ry;a1,a2;b12

easily yield (3.26). Equation (3.27) holds from (3.26) and (3.28). The proof is complete.

The (a1, a2; b12)-th zeta function ζa1,a2;b12(s) can now be obtained as follows.

Theorem 3.10 Given a basic set B ⊂ Σ2×2×2. For ai ≥ 1, 1 ≤ i ≤ 3, 0 ≤ b^ij ≤ aⁱ− 1, i + 1≤ j ≤ 3,

ζa1,a2;b12(s) = exp 1 a1a2

X∞ a3=1

1 a3

tr T^a_a³_1,a2;b12(B)Ra1,a2;b12

s^a1a2a3

!

. (3.29)

Proof. The results follow from Theorem 3.9.

3.1.2 Rationality of ζa1,a2;b12

This subsection proves that ζa1,a2;b12 is a rational function. First, the rotational symmetry of Ta1,a2;b12 is introduced.

Theorem 3.11 Given B ⊂ Σ2×2×2. Denote by T_a1,a2;b12(B) = [ta1,a2;b12;i,j]. For a1, a2 ≥ 1, 0 ≤ b12 ≤ a1− 1,

t_{a1,a2;b12;σx}⁻¹_(i),σx⁻¹_(j)= ta1,a2;b12;i,j (3.30) and

t_{a1,a2;b12;σy}⁻¹_(i),σy⁻¹_(j)= ta1,a2;b12;i,j (3.31) for all 1≤ i, j ≤ 2^a1a2. Furthermore,

t_a1,a2;b12;σ−r1

x (^{σ−r2y (i)})^,σ−r1x (^{σ−r2y (j)}) = t^{a1,a2;b12;i,j} (3.32) for all 1≤ i, j ≤ 2^a1a2, −a¹ + 1≤ r¹ ≤ a¹− 1 and −a²+ 1 ≤ r² ≤ a²− 1.

Proof. The proof of (3.31) is similar to that of (3.30) and omitted. We now prove (3.30).

Given 1≤ i, j ≤ 2^a1a2, Ca1,a2;b12;2;i,j and C_{a1,a2;b12;2;σx}⁻¹_(i),σx⁻¹_(j) contain only one pat-tern respectively. Let

U = U0 ⊕^z U1 = (uα1,α2,α3)∈ C^a1,a2;b12;2;i,j

with ψ(U₀) = i and ψ(U₁) = j, and

U^′ = U^′₀⊕zU^′₁ = (u^′_α1,α2,α3)∈ Ca1,a2;b12;2;σ⁻¹x (i),σ⁻¹x (j)

with ψ(U^′₀) = σ⁻¹_x (i) and ψ(U^′₁) = σ_x⁻¹(j). SinceB ⊂ Σ2×2×2 and (3.9), to prove (3.30) is equal to prove

{(uⁿ1+k1,n2+k2,k3)_0≤k1,k2,k3≤1 : 0≤ n¹ ≤ a¹− 1, 0 ≤ n² ≤ a²− 1}

= {(u^′n1+k1,n2+k2,k3)_0≤k1,k2,k3≤1 : 0≤ n¹ ≤ a¹− 1, 0 ≤ n² ≤ a²− 1}.

(3.33)

Since ψ(U0) = i and ψ(U^′₀) = σ_x⁻¹(i), by (3.18),

u^′_α1,α2,0 =





u_{[α1−1−b12]a1,0,0} if α₂ = a₂,

u[α1−1]a1,α2,0 if 0≤ α2 ≤ a2− 1.

Similarly, from ψ(U1) = j and ψ(U^′₁) = σ_x⁻¹(j),

u^′_α1,α2,1 =





u[α1−1−b12]_a1,0,1 if α2 = a2,

u_{[α1−1]a1,α2,1} if 0≤ α² ≤ a²− 1.

Then, (3.33) is directly obtained.

Therefore, (3.30) and (3.31) hold. For 0 ≤ r¹ ≤ a¹− 1 and 0 ≤ r² ≤ a² − 1, by applying (3.31) r₂ times and (3.30) r₁ times, (3.32) holds. From (3.15), (3.16) and (3.17), (3.32) follows. The proof is complete.

To study the rationality of ζa1,a2;b12, we need more definitions and properties about the two shifts in (3.20) and (3.22) as follows.

Given a1, a2 ≥ 1, 0 ≤ b¹² ≤ a¹− 1, for 1 ≤ i ≤ 2^a1a2, the equivalent classC^a1,a2;b12(i) of i is defined by

C^a1,a2;b12(i)≡

σ^−r_x ¹ σ_y^−r2(i)

: 0≤ r¹ ≤ a¹− 1, 0 ≤ r² ≤ a²− 1

. (3.34)

Clearly,

either C^a1,a2;b12(i) =C^a1,a2;b12(j) or C^a1,a2;b12(i)∩ C^a1,a2;b12(j) = ∅. (3.35) The cardinal number of Ca1,a2;b12(i) is denoted by ω_a1,a2;b12;i. Let i be the smallest element in its equivalent class, and the index set Ia1,a2;b12 is defined by

Ia1,a2;b12 ={i : 1 ≤ i ≤ 2^a1a2, i≤ j for all j ∈ Ca1,a2;b12(i)} . (3.36) Therefore,

{j : 1 ≤ j ≤ 2^a1a2} = ∪

i∈I_a1,a2;b12Ca1,a2;b12(i). (3.37) The cardinal number of I^a1,a2;b12 is denoted by χa1,a2;b12.

The following example illustrates C2,2;j(i).

Example 3.12

The equivalent classes are invariant under the two shift maps. Therefore, the fol-lowing proposition is directly obtained and the proof is omitted.

Proposition 3.13 Given a1, a2 ≥ 1 and 0 ≤ b¹² ≤ a¹ − 1. Let N ≡ 2^a1a2 and

Definition 3.14 For a1, a2 ≥ 1, 0 ≤ b12 ≤ a1−1, the reduced trace operator τa1,a2;b12 =

The following theorem expresses the average of Γ_B in terms of the trace of the reduced trace operator τ and plays a crucial role in proving the rationality of ζa1,a2;b12. The proof here is simpler and more straightforward than the proofs in Subsection 2.2 for d = 2.

Now, by Eq. (3.37), the last sum becomes

. Then, by Theorem 3.11,

aP1−1 Therefore, Eq. (3.41) is equal to

1

According to Proposition 3.13, Eq. (3.43) is equal to

X

For any qa3−1 ∈ Ca1,a2;b12(ka3−1), there exist 0 ≤ r1 ≤ a1 − 1 and 0 ≤ r2 ≤ a2 − 1 such that

qa3−1 = σ_x^−r1 σ^−r_y ²(ka3−1)
. Then, by Theorem 3.11,

P

q∈C_a1,a2;b12(i)

tq_a3−1,q = P

q∈C_a1,a2;b12(i)

t_σ−r1

x (^{σ−r2y (k}a3−1))^,q

= P

q∈C_a1,a2;b12(i)

t_k

a3−1,σ^r1x (^σyr2(q))

= P

q∈C_a1,a2;b12(i)

tk_a3−1,q. Therefore,

aP3−1 j=1

P

qj∈C_a1,a2;b12(kj)

ti,q1· · · t^qa3−2,q_a3−1 P

q∈C_a1,a2;b12(i)

tq_a3−1,q

!

=

aP3−2 j=1

P

qj∈Ca1,a2;b12(kj)

ti,q1· · · P

q_a3−1∈Ca1,a2;b12(k_a3−1)

tq_a3−2,q_a3−1

! P

q∈Ca1,a2;b12(i)

tk_a3−1,q

!

= ^aP³⁻²

j=1

P

qj∈Ca1,a2;b12(kj)

t_i,q1· · · P

q_a3−1∈Ca1,a2;b12(k_a3−1)

t_{ka3−2,qa3−1}

! P

q∈Ca1,a2;b12(i)

t_ka3−1,q

!

...

= P

q1∈C_a1,a2;b12(i)

ti,q1

! "

a3Q−1 j=2

P

qj∈C_a1,a2;b12(kj)

tkj−1,qj

!# P

q∈C_a1,a2;b12(i)

tk_a3−1,q

!

= τa1,a2;b12;i,k1τa1,a2;b12;k1,k2· · · τ^a1,a2;b12;k_a3−1,i

Finally, (3.44) is equal to

P

i∈I_a1,a2;b12 aP3−1

j=1

P

kj∈I_a1,a2;b12

τa1,a2;b12;i,k1τa1,a2;b12;k1,k2· · · τa1,a2;b12;k_a3−1,i

= tr τ_a^a₁³_,a2;b12

= P

λ∈Σ(τ_a1,a2;b12)

χ_a1,a2;b12(λ)λ^a3. The proof is complete.

Therefore, the rationality of ζa1,a2;b12 and ζ can be obtained as follows.

Theorem 3.16 For a1, a2 ≥ 1, 0 ≤ b¹²≤ a¹ − 1,

ζa1,a2;b12(s) = (det (I − s^a1a2τa1,a2;b12))⁻¹

= Q

λ∈Σ(τ_a1,a2;b12)

(1− λs^a1a2)^{−χa1,a2;b12(λ)},

(3.45)

and

ζ(s) = Q^∞

a1=1

Q∞ a2=1

a1Q−1 b12=0

(det (I− s^a1a2τa1,a2;b12))⁻¹

= Q^∞

a1=1

Q∞ a2=1

a1Q−1 b12=0

Q

λ∈Σ(τ_a1,a2;b12)

(1− λs^a1a2)^{−χa1,a2;b12(λ)}.

(3.46)

Proof. By using the power series

− log(1 − t) = X∞ n=1

tⁿ

n, (3.47)

equation (3.45) follows from (1.34) and Theorem 3.15. Equation (3.46) follows form (1.35) and (3.45).

The following example is used to demonstrate the application of the above result.

Example 3.17 Consider

B = {U2×2×2= (uα1,α2,α3)∈ Σ2×2×2: u0,0,j = u1,0,j = u0,1,j = u1,1,j for j = 0, 1} . Clearly, the set P(B) of all B-admissible and periodic patterns is

U = (uα1,α2,α3)∈ Σ³2 : ui,j,k = u0,0,k for all i, j, k ∈ Z . Then, it is easy to verify that

Γ_B However, (3.48) and (3.49) can be obtained from (3.45) and (3.46). The trace operator

χ_a1,a2;b12×χ_a1,a2;b12

.

Therefore,

ζa1,a2;b12(s) = (det (I − s^a1a2τa1,a2;b12))⁻¹

= (1− 2s^a1a2)⁻¹

and Equations (3.48) and (3.49) are recovered.

3.1.3 Zeta functions in inclined coordinates

This subsection presents the zeta function with respect to inclined coordinates, de-termined by applying the unimodular transformations in GL3(Z). Z³ is known to be invariant under the unimodular transformation in GL3(Z). Indeed, Lind [36] proved that the zeta function ζ_B⁰ is independent of a choice of basis for Z³. Recall that

GLd(Z) =n

γ = [γij]_1≤i,j≤d : γij ∈ Z for 1 ≤ i, j ≤ d and |det(γ)| = 1o .

This subsection presents the construction of the trace operator Tγ;a1,a2;b12(B) and the reduced trace operator τγ;a1,a2;b12(B), and then determines ζγ;a1,a2;b12 and ζ_B;γ. Finally, For simplicity, onlyB ⊂ Σ2×2×2with two symbols are considered. The general cases can be treated analogously.

For a given γ =

∈ GL³(Z), the lattice points in γ-coordinates are

(1, 0, 0)γ = (γ11, γ12, γ13), (0, 1, 0)γ = (γ21, γ22, γ23), (0, 0, 1)γ = (γ31, γ32, γ33), and the unit vectors are



Notably, when γ =

, standard rectangular coordinates are used and the

subscript γ is omitted.

The matrix Mγ is defined by

Mγ =

and the zeta function ζ_B;γ with respect to γ is defined by

ζ_B;γ(s)≡

The following introduces the cylindrical ordering matrix, the trace operator and the rotational matrices. The proofs of the results as in previous subsections are omitted.

Fix a γ ∈ GL3(Z). Let Zγ;n1×n2×n3 be the n1× n2 × n3 lattice with the basis

Since the basic setB ⊂ Σ2×2×2, the L_γ-periodic patterns that areB-admissible must be verified on Z_2×2×2. Let (n1, n2, n3)γ = (m1, m2, m3),

Z2×2×2((n₁, n₂, n₃)_γ) ={(m1 + k₁, m₂+ k₂, m₃+ k₃) : 0≤ k1, k₂, k₃ ≤ 1} . Now, the admissibility is demonstrated to have to be verified on finite lattice as follows.

Proposition 3.18 Given γ =



is B-admissible if and only if

U |Z2×2×2 ((α1, α2, α3)γ)∈ B

According to Proposition 3.18, the admissibility of an Lγ-periodic pattern U is determined by U |Zγ;ba1×ba2×ba3= u_{(α1,α2,α3)γ}

0≤αⁱ≤baⁱ−1,1≤i≤3 and U |Zγ;ba1×ba2×ba3 has the the periodic condition that is given by (3.52), which can be divided into two parts: (i) for 0 ≤ αⁱ ≤ baⁱ − 1, 1 ≤ i ≤ 3 and p, q ∈ Z, if 0 ≤ α¹ + a1p + b12q ≤ ba¹ − 1 and 0≤ α2+ a₂q≤ ba2− 1,

u(α1+a1p+b12q,α2+a2q,α3)γ = u(α1,α2,α3)γ; (3.55)

(ii) for 0≤ αi ≤ bai−1, 1 ≤ i ≤ 3, p, q ∈ Z and r ∈ Z\{0}, if 0 ≤ α1+a1p+b12q+b13r≤ ba1 − 1, 0 ≤ α²+ a2q + b23r≤ ba²− 1 and 0 ≤ α³+ a3r≤ ba³− 1,

u(α1+a1p+b12q+b13r,α2+a2q+b23r,α3+a3r)γ = u(α1,α2,α3)γ. (3.56) Then, for h ≥ 1, the set of all local patterns on Zγ;ba1×ba2×h that satisfy (3.55) with 0≤ α3 ≤ h − 1 is denoted by Pγ;a1,a2;b12;h.

Similar to (3.6), the counting function ψ_γ for patterns Uγ in Pγ;a1,a2;b12;h is defined by

ψ_γ Uγ

= 1 +

aX1−1 α1=0

aX2−1 α2=0

Xh−1 α3=0

u(α1,α2,α3)γ2^{a1a2(h−1−α3)+a1(a2−1−α2)+a1−1−α1}. A local pattern Uγ in Pγ;a1,a2;b12;h can be represented as

U_γ = U_γ;0⊕γ3 U_γ;1⊕γ3 · · · ⊕γ3 U_γ;h−1,

where Uγ;i ∈ P^γ;a1,a2;b12;1, 0 ≤ i ≤ h − 1, and U^′γ ⊕^z U^′′_γ means that U^′′_γ is put on the top (in the γ3-direction) of U^′_γ. For 0 ≤ i ≤ j ≤ h − 1, let Uγ;i:j = Uγ;i ⊕γ3

· · · ⊕^γ3 Uγ;j. Therefore, for h ≥ ba³, the cylindrical ordering matrix Cγ;a1,a2;b12;h = [C_{γ;a1,a2;b12;h;i,j}]_{2a1a2(h−1)×2a1a2(h−1)} of patterns in P_{γ;a1,a2;b12;h} is defined by

Cγ;a1,a2;b12;h;i,j =

Uγ ∈ Pγ;a1,a2;b12;h: ψ_γ(U_γ;0:ba3−2) = i and ψ_γ(U_{γ;h−ba3+1:h−1}) = j . In particular, for h = ba³, Cγ;a1,a2;b12;ba3 can be used to construct the associated trace operator. Notably the set C_{γ;a1,a2;b12;ba3;i,j} either contains exactly one pattern or is an

Uγ ∈ Pγ;a1,a2;b12;h: ψ_γ(U_γ;0:ba3−2) = i and ψ_γ(U_{γ;h−ba3+1:h−1}) = j . In particular, for h = ba³, Cγ;a1,a2;b12;ba3 can be used to construct the associated trace operator. Notably the set C_{γ;a1,a2;b12;ba3;i,j} either contains exactly one pattern or is an

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