PATTERNS GENERATION AND SPATIAL ENTROPY IN TWO-DIMENSIONAL LATTICE MODELS

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PATTERNS GENERATION AND SPATIAL ENTROPY IN TWO-DIMENSIONAL LATTICE MODELS^∗

JUNG-CHAO BAN^†, SONG-SUN LIN^‡, AND YIN-HENG LIN^§

Abstract. Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and Z2ℓ×2ℓ, ℓ ≥ 1, be a fixed finite square sublattice of Z². Function U: Z2ℓ×2ℓ→ S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A2 which is a q²×q² matrix, q = p^ℓ², can be defined. The recursive formulae of higher transition matrix An on Z2ℓ×nℓ have already been derived [4]. Now A^mn, m ≥ 1, contains all admissible patterns on Z(m+1)ℓ×nℓ which can be generated by B. In this paper, the connecting operator Cm, which comprises all admissible patterns on Z(m+1)ℓ×2ℓ, is carefully arranged. Cm

can be used to extend A^mn to A^mn+1recursively for n ≥ 2. Furthermore, the lower bound of spatial entropy h(A2) can be derived through the diagonal part of Cm. This yields a powerful method for verifying the positivity of spatial entropy which is important in examining the complexity of the set of admissible global patterns. The trace operator Tmof Cmcan also be introduced. In the case of symmetric A2, T2m gives a good estimate of the upper bound on spatial entropy. Combining Cm

with Tmhelps to understand the patterns generation problems more systematically.

Key words. Lattice dynamical systems, Spatial entropy, Patterns generation, Connecting operator, Trace operator

AMS subject classifications.Primary 37B50; Secondary 37B40

1. Introduction. Lattices are important in scientifically modelling underly- ing spatial structures. Investigations in this field have covered phase transition [11], [12], [34], [35], [36], [37], [38], [45], [46], [47], [48], chemical reaction [7], [8], [24], biology [9], [10], [21], [22], [23], [31], [32], [33] and image processing and pattern recognition [16], [17], [18], [19], [20], [25]. In the field of lattice dynamical systems (LDS) and cellular neural networks (CNN), the complexity of the set of all global patterns re- cently attracted substantial interest. In particular, its spatial entropy has received considerable attention [1],[2], [3], [4], [5], [13], [14], [15], [28], [29],[30], [39], [40], [41], [42], [43], [44].

The one dimensional spatial entropy h can be found from an associated transition matrix T. The spatial entropy h equals log ρ(T), where ρ(T) is the maximum eigenvalue of T.

In two-dimensional situations, higher transition matrices have been discovered in [30] and developed systematically [4] by studying the patterns generation problem.

This study extends our previous work [4]. For simplicity, two symbols on 2 × 2 lattice Z2×2 are considered. A transition matrix in the horizontal (or vertical)

∗Received October 18, 2005; accepted for publication July 25, 2006. This reserch is partially supported by the National Science Council, R.O.C.(Contract No. NSC 94-2115-M-009-002) and the National Center for Theoretical Sciences.

†The National Center for Theoretical Sciences, Hsin-Chu 300, Taiwan ([email protected].

edu.tw).

‡Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan ([email protected]).

§Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan ([email protected]).

497

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direction

(1.1) A2=







a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44





 ,

which is linked to a set of admissible local patterns on Z2×2 is considered, where aij ∈ {0, 1} for 1 ≤ i, j ≤ 4. The associated vertical (or horizontal) transition matrix B₂ is given by

(1.2) B2=







b11 b12 b13 b14

b21 b22 b23 b24

b31 b32 b33 b34

b41 b42 b43 b44







A₂ and B2 are connected to each other as follows.

(1.3) A₂=







b11 b12 b21 b22

b13 b14 b23 b24

b31 b32 b41 b42

b33 b34 b43 b44







=

A2;1 A2;2

A2;3 A2;4

,

and

(1.4) B₂=







a11 a12 a21 a22

a13 a14 a23 a24

a31 a32 a41 a42

a33 a34 a43 a44







=

B2;1 B2;2

B2;3 B2;4

.

Notably if A2 represents the horizontal (or vertical) transition matrix then B2

represents the vertical (or horizontal) transition matrix. Results that hold for A2 are also valid for B2. Therefore, for simplicity, only A2 is presented herein.

The recursive formulae for n-th order transition matrices An defined on Z2×n

were obtained [4] as follows

(1.5) A_n+1=







b11An;1 b12An;2 b21An;1 b22An;2

b13An;3 b14An;4 b23An;3 b24An;4

b31An;1 b32An;2 b41An;1 b42An;2

b33An;3 b34An;4 b43An;3 b44An;4







whenever

(1.6) A_n =

An;1 An;2

An;3 An;4

,

for n ≥ 2, or equivalently,

(1.7) An+1;α=

bα1An;1 bα2An;2

bα3An;3 bα4An;4

,

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for α ∈ {1, 2, 3, 4}. The number of all admissible patterns defined on Zm×nwhich can be generated from A2is now defined by

(1.8) Γm,n(A2) = |A^m−1_n |

= the summation of all entries in 2ⁿ× 2ⁿ matrix A^m−1_n . The spatial entropy h(A2) is defined as

(1.9) h(A2) = lim

m,n→∞

1

mnlog Γm,n(A2) = lim

m,n→∞

1

mnlog |A^m−1_n |.

The existence of the limit (1.9) has been shown in [4], [15], [30]. When h(A2) > 0, the number of admissible patterns grows exponentially with the lattice size m × n. In this situation, spatial chaos arises. When h(A2) = 0, pattern formation occurs.

To compute the double limit in (1.9), n ≥ 2 can be fixed initially and m allowed to tend to infinite [30] and [4]; then the Perron-Frobenius theorem is applied;

(1.10) lim

m→∞

1

mlog |A^m−1_n | = log ρ(An), which implies

(1.11) h(A2) = lim

n→∞

1

nlog ρ(An),

where ρ(M ) is the maximum eigenvalue of matrix M . An is a 2ⁿ× 2ⁿ matrix, so computing ρ(An) is usually quite difficult when n is larger. Moreover, (1.11) does not produce any error estimation in the estimated sequence 1

nlog ρ(An) and its limit h(A2). This causes a serious problem in computing the entropy. However, for a class of A2, the recursive formulae for ρ(An) can be discovered, along with a limiting equation to ρ^∗= exp(h(A2)), as in [4].

This study takes a different approach to resolve these difficulties. Previously, the double limit (1.9) was initially examined by taking the m-limit firstly as in (1.10).

Now, for each fixed m ≥ 2, the n-limit in (1.9) is studied. Therefore, the limit

(1.12) lim

n→∞

1

nlog |A^m−1_n | is considered. Write

(1.13) A^m_n =

Am,n;1 Am,n;2

Am,n;3 Am,n;4

.

The investigation of (1.12) would be simpler if a recursive formula such as (1.7) could be found for Am,n;α. The first task in this study is to solve this problem. For matrix multiplication, the indices of An;α, α ∈ {1, 2, 3, 4} are conveniently expressed as

(1.14) A_n =

An;11 An;12

An;21 An;22

. Then

(1.15) Am,n;α=

2^m−1

X

k=1

A^(k)_m,n;α,

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where

(1.16) A^(k)m,n;α= An;j1j2An;j2j3· · · An;jmjm+1,

(1.17) k = 1 +

m

X

i=2

2^m−i(ji− 1),

and

(1.18) α = 2(j1− 1) + jm+1.

A^(k)m,n;αin (1.16) is called an elementary pattern of order (m, n), and is a fundamental element in constructing Am,n;α in (1.15). Notably the elementary patterns are in lexicographic order, according to (1.17). As in [4], the following m-th order ordering matrix.

(1.19) X_m,n=

Xm,n;1 Xm,n;2

Xm,n;3 Xm,n;4

,

is represented to record systematically these elementary patterns, where (1.20) Xm,n;α= (A^(k)_m,n;α)^t_1≤k≤2m−1

is a 2^m−1 column vector.

The first main result of this study is to introduce the connecting operator Cm, and to use it to derive a recursive formula like (1.7) for A^(k)m,n;α. Indeed,

(1.21) C_m=







Cm;11 Cm;12 Cm;13 Cm;14

Cm;21 Cm;22 Cm;23 Cm;24

Cm;31 Cm;32 Cm;33 Cm;34

Cm;41 Cm;42 Cm;43 Cm;44







(1.22) =







Sm;11 Sm;12 Sm;21 Sm;22

Sm;13 Sm;14 Sm;23 Sm;24

Sm;31 Sm;32 Sm;41 Sm;42

Sm;33 Sm;34 Sm;43 Sm;44





 ,

where

(1.23)

Cm;ij =

ai1 ai2

ai3 ai4

◦ ⊗ˆ

B2;1 B2;2

B2;3 B2;4

m−2!

2×2

!

2^m−1×2^m−1

◦

E2^m−2×2^m−2⊗

a1j a2j

a3j a4j

2^m−1×2^m−1

is a 2^m−1× 2^m−1matrix where Ek×k is the k × k full matrix; ⊗ denotes the Kronecker product, ◦ denotes the Hadamard product and the product ˆ⊗ which involves both the Kronecker product and the Hadamard product, as stipulated by Definition 2.2.

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In Theorem 2.4, Cm;ijis shown to be ai1i2ai2i3· · · aimim+1, with i1= i and im+1= j. Therefore, all admissible paths of A2 from i to j with length m are arranged systematically in matrix Cm;ij. Now, the recursive formula is

(1.24) A^(k)_m,n+1;α=







2^m−1

X

l=1

(Sm;α1)klA^(l)_m,n;1

2^m−1

X

l=1

2^m−1

X

l=1

2^m−1

X

l=1





 ,

for m ≥ 2, n ≥ 2, 1 ≤ k ≤ 2^m−1 and 1 ≤ α ≤ 4. (1.24) is the generalization of (1.7).

The recursive formula (1.24) immediately yields a lower bound on entropy.

Indeed, for any positive integer K and diagonal periodic cycle β1β2· · · βKβK+1, where βj∈ {1, 4} and βK+1= β1,

(1.25) h(A2) ≥ 1

mKlog ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβK+1).

Equation (1.25) implies h(A2) > 0, if a diagonal periodic cycle of β1β2· · · βKβ1 ap- plies, with a maximum eigenvalue of Sm;β1β2· · · Sm;βKβ1 that greater than one. This method powerfully yields the positivity of spatial entropy, which is hard in examining the complexity of patterns generation problems.

However, the subadditivity of Γm,n(A2) is known to imply

(1.26) h(A2) ≤ 1

mnlog Γm,n(A2)

as in [15]. Consequently, (1.8), (1.10) and (1.26) indicate an upper bound of entropy as

(1.27) h(A2) ≤ 1

nlog ρ(An), for any n ≥ 2.

However, the Perron-Frobenius theorem also implies

(1.28) lim sup

m→∞

1

mlog tr(A^m−1_n ) = log ρ(An),

where tr(M ) denotes the trace of matrix M [26], [27]. Therefore, (1.28) implies

(1.29) h(A2) = lim sup

m,n→∞

1

mnlog tr(A^m−1_n ).

In studying the double-limit of (1.29), for each fixed m ≥ 2, the n-limit in (1.29)

(1.30) lim sup

n→∞

1

nlog tr(A^m−1_n )

is first considered. (1.30) can be studied by introducing the following trace operator

(1.31) T_m=

Cm;11 Cm;22

Cm;33 Cm;44

.

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Then, a recursive formula for tr(A^m_n) can be verified

(1.32) tr(A^m_n) =

Tⁿ⁻²_m





trXm,2;1

trXm,2;4



 ,

for n ≥ 2, where tr(Xm,n;α) = (trA^(k)m,n;α)^t_1≤k≤2m−1 and |v| =

l

X

j=1

vj for vector v = (v1, · · · , vl)^t. Consequently, (1.29) and (1.32) yield

(1.33) h(A2) ≥ lim sup

m→∞

1

mlog ρ(Tm).

Notably, for a large class of A2, the limit sup in (1.28), (1.29), (1.30) and (1.33) can be replaced by limit. See section 3 for details.

Now, (1.33) can be applied to find the upper bounds of entropy. For example, when A2 is symmetric,

(1.34) h(A2) ≤ 1

2mlog ρ(T2m), for any m ≥ 1. Since

(1.35) T_n≤ Bn

can be shown for any n ≥ 2. Generally, (1.33) and (1.34) yield better approximation than (1.11) and (1.27).

In summary, this study yields lower-bound estimates of entropy like (1.25) by introducing connecting operators Cm, and upper-bound estimates of entropy like (1.34) by introducing trace operators Tm. This approach accurately and effectively yields the spatial entropy.

The rest of this paper is organized as follows. Section 2 derives the connecting operator Cmwhich can recursively reduce higher order elementary patterns to patterns of lower order. Then, the lower-bound of spatial entropy can be found by computing the maximum eigenvalues of the diagonal periodic cycles of sequence Sm;αβ. Section 3 addresses the trace operator Tmof Cm. The entropy can be calculated by computing the maximum eigenvalues of Tm. When A2 is symmetric, the upper-bounds of entropy are also found. Section 4 briefly discusses the theory for many symbols on larger lattices.

2. Connecting Operators.

2.1. Connecting operators and ordering matrices. This section derives connecting operators and investigates their properties. For clarity, two symbols on 2 × 2 lattice Z2×2 are examined first. Section 4 addresses more general situations.

Let A2 and B2 be defined as in (1.1)∼(1.4). The column matrices fA₂ and fB₂ of A₂ and B2 are defined by

(2.1) Af₂=







a11 a21 a12 a22

a31 a41 a32 a42

a13 a23 a14 a24

a33 a43 a34 a44







=

A˜2;1 A˜2;2

A˜2;3 A˜2;4

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and

(2.2) Bf₂=







b11 b21 b12 b22

b31 b41 b32 b42

b13 b23 b14 b24

b33 b43 b34 b44







=

B˜2;1 B˜2;2

B˜2;3 B˜2;4

,

respectively.

For matrices of higher order n ≥ 2, An, An+1 and An+1;α are defined as in (1.5)∼(1.7).

For matrix multiplication, the indices of An;α are conveniently expressed as

(2.3) A_n =

An;11 An;12

An;21 An;22

. Clearly, An;α= An;j1j2, where

(2.4) α = α(j1, j2) = 2(j1− 1) + j2.

For m ≥ 2, the elementary pattern in the entries of A^m_n is represented by An;j1j2An;j2j3· · · An;jmjm+1,

where js∈ {1, 2}. A lexicographic order for multiple indices Jm+1= (j1j2· · · jmjm+1) is introduced, using

(2.5) χ(Jm+1) = 1 +

m

X

s=2

2^m−s(js− 1).

Now,

(2.6) A^(k)_m,n;α= An;j1j2An;j2j3· · · An;jmjm+1, where

(2.7) α = α(j1, jm+1) = 2(j1− 1) + jm+1

and

(2.8) k = χ(Jm+1)

is given in (2.5). Notably, (2.5) and (2.8) do not involve jm+1 but (2.7)does.

Therefore, A^m_n can be expressed as

(2.9) A^m_n =

Am,n;1 Am,n;2

Am,n;3 Am,n;4

, where

(2.10) Am,n;α=

2^m−1

X

k=1

A^(k)_m,n;α.

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Furthermore,

(2.11) Xm,n;α= (A^(k)_m,n;α)^t_1≤k≤2m−1.

1 ≤ k ≤ 2^m−1, Xm,n;αis a 2^m−1column-vector that consists of all elementary patterns in Am,n;α. The ordering matrix Xm,nof A^m_n is now defined by

(2.12) X_m,n=

Xm,n;1 Xm,n;2

Xm,n;3 Xm,n;4

.

The ordering matrix Xm,n allows the elementary patterns to be tracked during the reduction from A^m_n+1 to A^m_n. This careful book-keeping provides a systematic way to generate the admissible patterns and later, lower-bound estimates of spatial entropy.

The following simplest example is studied first to illustrate the above concept.

Example 2.1. For m = 2, the following can easily be verified;

(2.13) A²_n=

A²_n;11+ An;12An;21 An;11An;12+ An;12An;22

An;21An;11+ An;22An;21 An;21An;12+ A²_n;22

,

and

(2.14)

A⁽¹⁾_2,n;1 = A²_n;11, A⁽²⁾_2,n;1= An;12An;21, A⁽¹⁾_2,n;2 = An;11An;12, A⁽²⁾_2,n;2 = An;12An;22, A⁽¹⁾_2,n;3 = An;21An;11, A⁽²⁾_2,n;3 = An;22An;21, A⁽¹⁾_2,n;4 = An;21An;12, A⁽²⁾_2,n;4 = A²n;22.









 .

Therefore,

(2.15)

X2,n;1=

A²_n;11 An;12An;21

, X2,n;2 =

An;11An,12

An;12An;22

,

X2,n;3=

An;21An;11

An;22An;21

, X2,n;4 =

An;21An,12

A²_n;22

.









 .

Applying (1.7), and by a straightforward computation,

(2.16) X2,n+1;1=

A²_n+1;11 An+1;12An+1;21

=







b²₁₁A²_n;1+ b12b13An;2An;3 b11b12An;1An;2+ b12b14An;2An;4

b13b11An;3An;1+ b14b13An;4An;3 b13b12An;3An;2+ b²₁₄A²_n;4

b21b31A²_n;1+ b22b33An;2An;3 b21b32An;1An;2+ b22b34An;2An;4

b23b31An;3An;1+ b24b33An;4An;3 b23b32An;3An;2+ b24b34A²_n;4







Clearly, the j1j2entries of A²_n+1;11and An+1;12An+1;21in (2.16) consist of entries of X2,n;α in (2.14) with α = α(j1, j2) in (2.4). Moreover, the terms in (2.16) can be

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rearranged in terms of X2,n;α by exchanging the second row in the first matrix with the first row in the second matrix in (2.16) as follows.

(2.17)







b²₁₁ b12b13

b21b31 b22b33

A²_n;1 An;2An;3

b11b12 b12b14

b21b32 b22b34

An;1An;2

An;2An;4

b13b11 b14b13

b23b31 b24b33

An;3An;1

An;4An;3

b13b12 b²₁₄ b23b32 b24b34

An;3An;2

A²_n;4





 Applying (1.1), (1.2) and (2.1), (2.17) can be rewritten as







a²₁₁ a12a21

a13a31 a14a41

A²_n;11 An;12An;21

a11a12 a12a22

a13a32 a14a42

An;11An;12

An;12An;22

a21a11 a22a21

a23a31 a24a41

An;21An;11

An;22An;21

a21a12 a²₂₂ a23a32 a24a42

An;21An;12

A²_n;22







(2.18) =

(B2;11◦ ˜A2;11)X2,n;1 (B2;11◦ ˜A2;12)X2,n;2

(B2;12◦ ˜A2;11)X2,n;3 (B2;12◦ ˜A2;12)X2,n;4

.

Therefore, after the entries of X2,n+1;1as in (2.17) or (2.18) have been permuted, X2,n+1;1 can be represented by a 2 × 2 matrix

(2.19) Xˆ2,n+1;1≡ P(X2,n+1;1) ≡

X2,n+1;1;1 X2,n+1;1;2

X2,n+1;1;3 X2,n+1;1;4

, where

(2.20)

X2,n+1;1;1= S2;11X2,n;1, X2,n+1;1;2= S2;12X2,n;2, X2,n+1;1;3= S2;13X2,n;3, X2,n+1;1;4= S2;14X2,n;4









 and

(2.21)

S2;11= B2;11◦ Ã2;11≡ C2;11, S2;12= B2;11◦ Ã2;12≡ C2;12, S2;13= B2;12◦ Ã2;11≡ C2;21, S2;14= B2;12◦ Ã2;12≡ C2;22,









 .

The above derivation indicates that X2,n+1;α can be reduced to X2,n;β via multiplication with connecting matrices C2;αβ. This procedure can be extended to introduce the connecting operator Cm= [ Cm;αβ ], for all m ≥ 2.

Before Cmis introduced, three products of matrices are defined as follows.

Definition 2.2. For any two matrices M = (Mij) and N = (Nkl), the Kronecker product (tensor product) M ⊗ N of M and N is defined by

(2.22) M⊗ N = (MijN).

For any n ≥ 1,

⊗Nⁿ= N ⊗ N ⊗ · · · ⊗ N,

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n-times in N.

Next, for any two m × m matrices

P= (Pij) and Q = (Qij)

where Pij and Qij are numbers or matrices, the Hadamard product P ◦ Q is defined by

(2.23) P◦ Q = (Pij· Qij),

where the product Pij· Qij of Pij and Qij may be a multiplication between numbers, between numbers and matrices or between matrices whenever it is well-defined.

Finally, product ˆ⊗ is defined as follows. For any 4 × 4 matrix

(2.24) M₂=







m11 m12 m21 m22

m13 m14 m23 m24

m31 m32 m41 m42

m33 m34 m43 m44







=

M2;1 M2;2

M2;3 M2;4

and any 2 × 2 matrix

(2.25) N=

N1 N2

N3 N4

,

where mij are numbers and Nk are numbers or matrices, for 1 ≤ i, j, k ≤ 4, define

(2.26) M₂⊗N =ˆ







m11N1 m12N2 m21N1 m22N2

m13N3 m14N4 m23N3 m24N4

m31N1 m32N2 m41N1 m42N2

m33N3 m34N4 m43N3 m44N4





 .

Furthermore, for n ≥ 1, the n + 1 th order of transition matrix of M2 is defined by M_n+1≡ ˆ⊗Mⁿ₂ = M2⊗Mˆ 2⊗ · · · ˆˆ ⊗M2,

n-times in M2. More precisely, M_n+1 = M2⊗( ˆˆ ⊗Mⁿ⁻¹₂ ) =

M2;1◦ ( ˆ⊗Mⁿ⁻¹₂ ) M2;2◦ ( ˆ⊗Mⁿ⁻¹₂ ) M2;3◦ ( ˆ⊗Mⁿ⁻¹₂ ) M2;4◦ ( ˆ⊗Mⁿ⁻¹₂ )

(2.27) =







m11Mn;1 m12Mn;2 m21Mn;1 m22Mn;2

m13Mn;3 m14Mn;4 m23Mn;3 m24Mn;4

m31Mn;1 m32Mn;2 m41Mn;1 m42Mn;2

m33Mn;3 m34Mn;4 m43Mn;3 m44Mn;4







=

Mn+1;1 Mn+1;2

Mn+1;3 Mn+1;4

,

where

M_n= ˆ⊗Mⁿ⁻¹₂ =

Mn;1 Mn;2

Mn;3 Mn;4

. Here, the following convention is adopted,

⊗Mˆ ⁰₂= E2×2.

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Definition 2.3. For m ≥ 2, define (2.28)

C_m=







Cm;11 Cm;12 Cm;13 Cm;14

Cm;21 Cm;22 Cm;23 Cm;24

Cm;31 Cm;32 Cm;33 Cm;34

Cm;41 Cm;42 Cm;43 Cm;44







=







Sm;11 Sm;12 Sm;21 Sm;22

Sm;13 Sm;14 Sm;23 Sm;24

Sm;31 Sm;32 Sm;41 Sm;42

Sm;33 Sm;34 Sm;43 Sm;44





 ,

where

(2.29)

Cm;αβ =

aα1 aα2

aα3 aα4

◦ ⊗ˆ

B2;1 B2;2

B2;3 B2;4

m−2!

2×2

!

2^m−1×2^m−1

◦

E2^m−2×2^m−2⊗

a1β a2β

a3β a4β

2^m−1×2^m−1

. Similarly, for B2, define

(2.30)

U_m=







Um;11 Um;12 Um;13 Um;14

Um;21 Um;22 Um;23 Um;24

Um;31 Um;32 Um;33 Um;34

Um;41 Um;42 Um;43 Um;44







=







Wm;11 Wm;12 Wm;21 Wm;22

Wm;13 Wm;14 Wm;23 Wm;24

Wm;31 Wm;32 Wm;41 Wm;42

Wm;33 Wm;34 Wm;43 Wm;44





 ,

where

(2.31)

Um;αβ =

bα1 bα2

bα3 bα4

◦ ⊗ˆ

A2;1 A2;2

A2;3 A2;4

m−2!

2×2

!

2^m−1×2^m−1

◦

E₂m−2×2^m−2⊗

b1β b2β

b3β b4β

2^m−1×2^m−1

. S_m= [Sm;αβ] and Wm= [Wm;αβ].

Now Cm+1 can be found from Cmby a recursive formula, as in (1.7).

Theorem 2.4. For any m ≥ 2 and 1 ≤ α, β ≤ 4,

(2.32) Cm+1;αβ =

aα1Cm;1β aα2Cm;2β

, and

(2.33) Um+1;αβ=

bα1Um;1β bα2Um;2β

bα3Um;3β bα4Um;4β

.

Proof. By (2.27),

⊗Bˆ ^m−1₂ = B2⊗( ˆˆ ⊗B^m−2₂ ) =

B2;1◦ ( ˆ⊗B^m−2₂ ) B2;2◦ ( ˆ⊗B^m−2₂ ) B2;3◦ ( ˆ⊗B^m−2₂ ) B2;4◦ ( ˆ⊗B^m−2₂ )

. Therefore,

Cm+1;αβ= (B2;α◦ ( ˆ⊗B^m−1₂ )) ◦ (E2^m−1×2^m−1⊗ ˜A2;β)

=

aα1(B2;1◦ ˆ⊗B^m−2₂ ) aα2(B2;2◦ ˆ⊗B^m−2₂ ) aα3(B2;3◦ ˆ⊗B^m−2₂ ) aα4(B2;4◦ ˆ⊗B^m−2₂ )

◦ (E2^m−1×2^m−1⊗ ˜A2;β)

=

.

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A similar result also holds for Um;αβ; the details are omitted here. The proof is complete.

Notably, (2.32) implies Cm;ij is ai1i2ai2i3· · · aimim+1 with i1 = i and im+1 = j.

C_m;ij consist of all words(or paths) of length m starting from i and ending at j.

Indeed, the entries of Cm and Bm+1 are the same. However, the arrangements are different. Cmcan also be used to study the primitivity of An, n ≥ 2, as in [6].

That the recursive formula (1.24) holds remains to be shown. Indeed, in (2.6) substituting n for n + 1 and using (1.7),

(2.34)

A^(k)_m,n+1;α

= An+1;j1j2An+1;j2j3· · · An+1,jmjm+1

=

m

Y

i=1

bαi1An;11 bαi2An;12

bαi3An;21 bαi4An;22

where αi = α(ji, ji+1), for 1 ≤ i ≤ m. After m matrix multiplications are executed in (2.34),

(2.35) A^(k)_m,n+1;α=

"

A^(k)_m,n+1;α;1 A^(k)_m,n+1;α;2 A^(k)_m,n+1;α;3 A^(k)_m,n+1;α;4

#

where

(2.36) A^(k)_m,n+1;α;β =

2^m−1

X

l=1

K(m; α, β; k, l)A^(l)_m,n;β

is a linear combination of A^(l)_m,n;β with the coefficients K(m; α, β; k, l) which are products of bαlj, 1 ≤ l ≤ m. K(m; α, β; k, l) must be studied in more details.

Note that

(2.37) A^m_n+1=

Am,n+1;1 Am,n+1;2

Am,n+1;3 Am,n+1;4

=







2^m−1

X

k=1

A^(k)_m,n+1;1

2^m−1

X

k=1

A^(k)_m,n+1;2

2^m−1

X

k=1

A^(k)_m,n+1;3

2^m−1

X

k=1

A^(k)_m,n+1;4







=







P2^m−1

k=1 A^(k)_m,n+1;1;1 P2^m−1

k=1 A^(k)_m,n+1;1;2 P2^m−1

k=1 A^(k)_m,n+1;2;1 P2^m−1

k=1 A^(k)_m,n+1;2;2 P2^m−1

k=1 A^(k)_m,n+1;1;3 P2^m−1

k=1 A^(k)_m,n+1;1;4 P2^m−1

k=1 A^(k)_m,n+1;2;3 P2^m−1

k=1 A^(k)_m,n+1;2;4 P2^m−1

k=1 A^(k)_m,n+1;3;1 P2^m−1

k=1 A^(k)_m,n+1;3;2 P2^m−1

k=1 A^(k)_m,n+1;4;1 P2^m−1

k=1 A^(k)_m,n+1;4;2 P2^m−1

k=1 A^(k)_m,n+1;3;3 P2^m−1

k=1 A^(k)_m,n+1;3;4 P2^m−1

k=1 A^(k)_m,n+1;4;3 P2^m−1

k=1 A^(k)_m,n+1;4;4







Now, Xm,n+1;α;β is defined as

(2.38) Xm,n+1;α;β = (A^(k)_m,n+1;α;β)^t.

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As in (2.17), the entries of Xm,n+1;α are rearranged into a new matrix

(2.39) Xˆm,n+1;α≡ P(Xm,n+1;α) ≡

Xm,n+1;α;1 Xm,n+1;α;2

Xm,n+1;α;3 Xm,n+1;α;4

.

From (2.36) and (2.38),

(2.40) Xm,n+1;α;β = K(m; α, β)Xm,n;β

where

K(m; α, β) = (K(m; α, β; k, l)), 1 ≤ k, l ≤ 2^m−1,

is a 2^m−1× 2^m−1 matrix. Now, K(m; α, β) = Sm;αβ must be shown as follows.

Theorem 2.5. For any m ≥ 2 and n ≥ 2, let Sm;αβ be given as in (2.28) and (2.29). Then,

K(m; α, β) = Sm;αβ, i.e.,

(2.41) Xm,n+1;α;β= Sm;αβXm,n;β, or equivalently, the recursive formula (1.24) holds. That is,

(2.42) A^(k)_m,n+1;α=







2^m−1

X

l=1

2^m−1

X

l=1

2^m−1

X

l=1

2^m−1

X

l=1





 .

Moreover, for n = 1,

(2.43) A^(k)_m,2;α=







2^m−1

X

l=1

(Sm;α1)kl 2^m−1

X

l=1

(Sm;α2)kl 2^m−1

X

l=1

(Sm;α3)kl 2^m−1

X

l=1

(Sm;α4)kl







for any 1 ≤ k ≤ 2^m−1 and α ∈ {1, 2, 3, 4}.

Proof. The result is proven by the induction on m.

When m = 2, and α = 1, (2.41) was proven as in Example 2.1. The case with α = 2, 3 and 4 can also be proved analogously; the details are omitted.

Now, (2.41) ia assumed to hold for m; the goal is to show that it also holds for m + 1. Since

A^m+1_n+1 = An+1· A^m_n+1=

An+1;1 An+1;2

An+1;3 An+1;4

Am,n+1,1 Am,n+1;2

Am,n+1,3 Am,n+1;4

,

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(2.11) implies Xm+1,n+1;1=

An+1;1Xm,n+1;1

An+1;2Xm,n+1;3

, Xm+1,n+1;2=

An+1;1Xm,n+1;2

An+1;2Xm,n+1;4

,

Xm+1,n+1;3=

An+1;3Xm,n+1;1

An+1;4Xm,n+1;3

, and Xm+1,n+1;4=

An+1;3Xm,n+1;2

An+1;4Xm,n+1;4

. For α = 1, by induction on m,

(An+1;1P(Xm,n+1;1), An+1;2P(Xm,n+1;3))^t

=







b11An;1 b12An;2

b13An;3 b14An;4

Sm;11Xm,n;1 Sm;12Xm,n;2

b21An;1 b22An;2

b23An;3 b24An;4







=







b11Sm;11An;1Xm,n;1 b11Sm;12An;1Xm,n;2







+













Hence Xm+1,n+1;1 can be represented by a matrix Xˆm+1,n+1;1≡ P(Xm+1,n+1;1) ≡

Xm+1,n+1;1,1 Xm+1,n+1;1,2

Xm+1,n+1;1,3 Xm+1,n+1;1,4

=

2

6

4

b11Sm;11 b12Sm;13

b21Sm;31 b22Sm;33

An;1Xm,n;1

An;2Xm,n;3

b11Sm;12 b12Sm;14

b21Sm;32 b22Sm;34

An;1Xm,n;2

An;2Xm,n;4

b13Sm;11 b14Sm;13

b23Sm;31 b24Sm;33

An;3Xm,n;1

An;4Xm,n;3

b13Sm;12 b14Sm;14

b23Sm;32 b24Sm;34

An;3Xm,n;2

An;4Xm,n;4

3

7

5

Once again, (1.1), (1.2) and (2.1) can be used to recast the matrix ˆXm+1,n+1;1as







a11Cm;11 a12Cm;21

a13Cm;31 a14Cm;41

Xm+1,n;1

a11Cm;12 a12Cm;22

a13Cm;32 a14Cm;42

Xm+1,n;2

a21Cm;11 a22Cm;21

a23Cm;31 a24Cm;41

Xm+1,n;3

a21Cm;12 a22Cm;22

a23Cm;32 a24Cm;42

Xm+1,n;4







According to Theorem 2.4, the above matrix becomes

=

Cm+1;11Xm+1,n;1 Cm+1;12Xm+1,n;2

Cm+1;21Xm+1,n;3 Cm+1;22Xm+1,n;4

=

Sm+1;11Xm+1,n;1 Sm+1;12Xm+1,n;2

Sm+1;13Xm+1,n;3 Sm+1;14Xm+1,n;4

.

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The cases with α = 2, 3 and 4 can also be considered analogously (2.41) follows.

Next, (2.42) follows easily from (2.35), (2.36) and (2.41).

Equation (2.43) remains to be shown. If the 2 × 2 matrix

(2.44) A₁≡

A1;11 A1;12

A1;21 A1;22

≡

A1;1 A1;2

A1;3 A1;4

≡

1 1 1 1

is introduced, then the previous argument also hold for n = 1. Hence, (2.43) holds.

The proof is complete.

For any positive integer p ≥ 2, applying Theorem 2.5 p times permits the elementary patterns of A^m_n+p to be expressed as the product of a sequence of Sm;βiβi+1 and the elementary patterns in A^m_n. The elementary pattern in A^m_n+p is first studied.

For any p ≥ 2 and 1 ≤ q ≤ p − 1, define

(2.45) A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_q =

"

A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_q_;1 A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_q_;2 A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_q_;3 A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_q_;4

# .

Then

(2.46) A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_p=

2^m−1

X

l1=1

· · ·

2^m−1

X

lp=1

(

p

Y

i=1

K(m; βi−1, βi; li−1, li))A^(l_m,n;β^p⁾ _p,

where β0 = α and l0= k can be easily verified. Therefore, for any p ≥ 1, a generalization for (2.37) can be found for A^m_n+p as a 2^p+1× 2^p+1 matrix

(2.47) A^m_n+p=Am,n+p;α;β1;β2··· ;β_p where

(2.48) Am,n+p;α;β1;β2··· ;βp=

2^m−1

X

k=1

A^(k)_m,n;α;β₁_;β₂_{··· ;β}_p.

In particular, if α; β1, β2· · · , βp∈ {1, 4}, then Am,n+p;α;β1;β2··· ;βp lies on the diagonal of A^m_n+p in (2.47).

Now, define

(2.49) Xm,n+p;α;β1;β2;··· ;βp= (A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_p)^t. Therefore, Theorem 2.5 can be generalized to

Theorem 2.6. For any m ≥ 2, n ≥ 2 and p ≥ 1,

(2.50) Xm,n+p;α;β1;β2··· ;βp= Sm;αβ1Sm;β1β2· · · Sm;β_p−1βpXm,n;βp

where α, βi∈ {1, 2, 3, 4} and 1 ≤ i ≤ p.

(16)

Proof. From (2.46), (2.40) and (2.42),

A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_p =

2^m−1

X

l1=1

· · ·

2^m−1

X

lp=1

(

p

Y

i=1

K(m; βi−1, βi; li−1, li))A^(l_m,n;β^p⁾ _p

=

2^m−1

X

l1=1

· · ·

2^m−1

X

lp=1

(

p

Y

i=1

(Sm;β_i−1βi)l_i−1li)A^(l_m,n;β^p⁾ _p

=

2^m−1

X

l1=1

· · ·

2^m−1

X

lp=1

(Sm;β0β1)l0l1(Sm;β1β2)l1l2· · · (Sm;β_p−1βp)l_p−1lpA^(l_m,n;β^p⁾ _p

=

2^m−1

X

lp=1

(Sm;β0β1Sm;β1β2· · · Sm;β_p−1βp)l0lpA^(l_m,n;β^p⁾ _p

=

2^m−1

X

lp=1

(Sm;αβ1Sm;β1β2· · · Sm;β_p−1βp)klpA^(l_m,n;β^p⁾ _p

is derived. By (2.49), then

Xm,n+p;α;β1;β2;··· ;β_p= (A^(k)_m,n+p;α;β₁_;β₂_{;··· ;β}_p)^t

= (

2^m−1

X

lp=1

(Sm;αβ1Sm;β1β2· · · Sm;β_p−1βp)klpA^(l_m,n;β^p⁾ _p)^t

= Sm;αβ1Sm;β1β2· · · Sm;β_p−1βpXm,n;βp. The proof is complete.

2.2. Lower bound of entropy. In this subsection, the connecting operator Cm

is employed to estimate the lower bound of entropy, and in particular, to verify the positivity of entropy.

First, recall some properties of Γm,nand spatial entropy.

Γm,n satisfies the subadditivity in m and n:

(2.51) Γm1+m2,n≤ Γm1,nΓm2,n, and

(2.52) Γm,n1+n2≤ Γm,n1Γm,n2, or equivalently,

(2.53) |A^m_n¹^+m²| ≤ |A^m_n¹||A^m_n²| and

(2.54) |A^m_n₁_+n₂| ≤ |A^m_n₁||A^m_n₂|, for positive integers m, n, m1, n1, m2and n2. Here

(2.55) A₁=

1 1 1 1

is applied.