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Pattern Generation and Spatial Entropy in
Two-dimensional Lattice Models
Jung-Chao Ban1, Song-Sun Lin2, Yin-Heng Lin3 ?
1 The National Center for Theoretical Sciences, Hsin-Chu, Taiwan.
2 Department of Applied Mathematics, National Chiao Tung University, Hsin Chu Taiwan. 3 Department of Applied Mathematics, National Chiao Tung University, Hsin Chu Taiwan.
Received: date / Accepted: date
Abstract:
1. Introduction
Lattices are important in scientifically modelling the underlying spatial struc-tures. Investigations on this subject have been published in biology [9], [10], [20], [21], [22], [30], [31], [32]; chemical reaction and phase transition [7], [8], [23], and image processing and pattern recognition [15], [16], [17], [18], [19], [24]. In lattice dynamical systems (LDS) and cellular neural networks (CNN), the complexity of the set of all global patterns has received considerable attention recently, in particular, to study its spatial entropy [1],[2], [3], [4], [5], [12], [13], [14], [27], [28],[29], [38], [39], [41].
As is well-known the one dimensional spatial entropy h can be found with 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 discov-ered in [29] and developed completely in [4] by studying the pattern generation problem.
This study extends our previous work [4]. For simplicity, two symbols on 2 × 2 lattice Z2×2 are addressed. A transition matrix in horizontal (or vertical)
direction A2= a11a12a13a14 a21a22a23a24 a31a32a33a34 a41a42a43a44 , (1.1)
? The authors would like to thank the National Science Council, R.O.C. and the National
which is linked to a set of admissible local patterns on Z2×2is considered, where
aij ∈ {0, 1} for 1 ≤ i, j ≤ 4. The associated vertical (or horizontal) transition
matrix B2is denoted by B2= b11b12b13b14 b21b22b23b24 b31b32b33b34 b41b42b43b44 (1.2)
A2and B2 are connected to each other as follows.
A2= b11b12b21b22 b13b14b23b24 b31b32b41b42 b33b34b43b44 = A2;1 A2;2 A2;3 A2;4 , (1.3) and B2= a11a12a21a22 a13a14a23a24 a31a32a41a42 a33a34a43a44 = B2;1 B2;2 B2;3 B2;4 . (1.4)
Notably if A2 represents the horizontal (or vertical) transition matrix then
B2 represents the vertical (or horizontal) transition matrix. Results which hold
for A2are also valid for B2. Therefore, for simplicity, only A2 is stated herein.
The recursive formulae for n-th order transition matrices Andefined on Z2×n
were obtained [4] using the following procedure:
An+1= b11An;1b12An;2b21An;1b22An;2 b13An;3b14An;4b23An;3b24An;4 b31An;1b32An;2b41An;1b42An;2 b33An;3b34An;4b43An;3b44An;4 (1.5) whenever An = An;1 An;2 An;3An;4 , (1.6) for n ≥ 2, or equivalently, An+1;α= bα1An;1bα2An;2 bα3An;3bα4An;4 , (1.7)
for α ∈ {1, 2, 3, 4}. The number of all admissible patterns defined on Zm×nwhich
can be generated from A2 is now defined by
Γm,n(A2) = |Am−1n |
= the summation of all entries in 2n× 2n
matrix Am−1
n .
(1.8) The spatial entropy h(A2) is defined as
h(A2) = lim m,n→∞ log Γm,n(A2) mn =m,n→∞lim log |Am−1 n | mn . (1.9)
The existence of the limit (1.9) has been shown in [4], [14], [29]. When h(A2) > 0,
the number of admissible patterns grows exponentially with the lattice sizes m × n. In this situation, spatial chaos arises. When h(A2) = 0, pattern formation
occurs.
To compute the double limit in (1.9), [29] and [4] first fixed n ≥ 2 and let m tend to infinite; then use Perron-Frobenius theorem,
lim m→∞ log |Am−1 n | m = log ρ(An), (1.10) which implies h(A2) = lim n→∞ log ρ(An) n , (1.11)
where ρ(M ) is the maximum eigenvalue of matrix M . Since Anis 2n×2nmatrix,
computing ρ(An) is usually quite difficult when n is larger. Moreover, (1.11) does
not produce any error estimate for the sequence log ρ(An)
n and its limit h(A2). This leads a great deficiency 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(A
2)), see as in [4].
This study takes a different approach to resolve these difficulties. Previously, the double limit (1.9) was 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, i.e., the limit
lim n→∞ log |Am−1 n | n (1.12) is considered. Write Amn = Am,n;1Am,n;2 Am,n;3Am,n;4 . (1.13)
The investigation of (1.12) could be simpler if a recursive formula like (1.7) could be found for Am,n;α. The first task of this study is to solve this problem.
For matrix multiplication, the indices of An;α, α ∈ {1, 2, 3, 4} are conveniently
expressed as An = An;11 An;12 An;21An;22 . (1.14) Then Am,n;α= 2m−1 X k=1 A(k)m,n;α, (1.15) where A(k)m,n;α= An;j1j2An;j2j3· · · An;jmjm+1, (1.16) k = 1 + m X i=2 2m−i(ji− 1), (1.17) and α = 2(j1− 1) + jm+1. (1.18)
A(k)m,n;α in (1.16) is called an elementary pattern of order (m, n), a fundamental
element in constructing Am,n;α in (1.15). Notably the elementary patterns are
in lexicographic order according to (1.17). The following m-th order ordering matrix, see as in [4]. Xm,n= Xm,n;1Xm,n;2 Xm,n;3Xm,n;4 , (1.19)
is represented to record these elementary patterns systematically, where Xm,n;α= (A(k)m,n;α)
t
1≤k≤2m−1 (1.20)
is a 2m−1 column vector.
The first main result of this study is to introduce the reduction operator Cm,
and a recursive formula like (1.7) can be found for A(k)m,n;α. Indeed,
Cm= Cm;11Cm;12 Cm;13Cm;14 Cm;21Cm;22 Cm;23Cm;24 Cm;31Cm;32 Cm;33Cm;34 Cm;41Cm;42 Cm;43Cm;44 (1.21) = Sm;11Sm;12Sm;21Sm;22 Sm;13Sm;14Sm;23Sm;24 Sm;31Sm;32Sm;41Sm;42 Sm;33Sm;34Sm;43Sm;44 , (1.22) where Cm;ij = ai1ai2 ai3ai4 ◦ ⊗ˆ B2;1B2;2 B2;3B2;4 m−2! 2×2 ! 2m−1×2m−1 ◦ E2m−2×2m−2⊗ a1j a2j a3j a4j 2m−1×2m−1 (1.23) is a 2m−1× 2m−1 matrix where E
k×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.
In this sense, Cm;ij is called as the connecting matrix in studying the
prim-itivity of An as in [6]. In Theorem 2.4, the entry of Cm;ij is shown to be
ai1i2ai2i3· · · aimim+1, with i1= i and im+1= j. Therefore, all admissible path of
A2from i to j with length m are arranged systematically in matrix Cm;ij. Now,
the recursive formula is stated as
A(k)m,n+1;α= 2m−1 X l=1 (Sm;α1)klA (l) m,n;1 2m−1 X l=1 (Sm;α2)klA (l) m,n;2 2m−1 X l=1 (Sm;α3)klA (l) m,n;3 2m−1 X l=1 (Sm;α4)klA (l) m,n;4 , (1.24)
for m ≥ 2, n ≥ 2, 1 ≤ k ≤ 2m−1 and 1 ≤ α ≤ 4. (1.24) is the generalization of
The recursive formula (1.24) immediately derives a lower bound of entropy. Indeed, for any positive integerK and diagonal periodic cycle β1β2· · · βKβK+1,
where βj∈ {1, 4} and βK+1= β1,
h(A2) ≥
1
mKlog ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβK+1). (1.25)
Equation (1.25) implies h(A2) > 0, indicating that spatial chaos occurs if a
di-agonal periodic cycle of β1β2· · · βKβ1 applies, with a maximum eigenvalue of
Sm;β1β2· · · Sm;βKβ1 greater than one. This represents a powerful method of
ver-ifing the positivity of spatial entropy, which is hard in examining the complexity of pattern-generation problems.
On the other hand, the subadditivity property of Γm,n(A2) is known to imply
h(A2) ≤
log Γm,n(A2)
mn (1.26)
as in [14]. Consequently, (1.8), (1.10) and (1.26) indicate an upper bound of entropy as
h(A2) ≤ log ρ(A n)
n , (1.27)
for any n ≥ 2.
However, the Perron-Frobenius theorem also implies
lim sup
m→∞
log tr(Am−1n )
m = log ρ(An), (1.28)
where tr(M ) denotes the trace of matrix M , see Appendix. Thus, (1.28) implies
h(A2) = lim sup m,n→∞
log tr(Am−1
n )
mn . (1.29)
In studying the double-limit of (1.29), for each fixed m ≥ 2, the n-limit in (1.29)
lim sup
n→∞
log tr(Am−1n )
n (1.30)
is first considered. (1.30) can be studied by introducing the following trace op-erator Tm= Cm;11Cm;22 Cm;33Cm;44 . (1.31)
Then, a recursive formula for tr(Am
n) can be verified tr(Amn) = Tn−2m trXm,2;1 trXm,2;4 , (1.32)
for n ≥ 2, where tr(Xm,n;α) = (trA (k) m,n;α)t1≤k≤2m−1 and |V | = l X j=1 vj for vector
V = (v1, · · · , vl)t, vj denote numbers or matrices with the same order.
Conse-quently, (1.29) and (1.32) yield
h(A2) = lim sup m→∞
log ρ(Tm)
m . (1.33)
Note that for a large class of A2, the limit sup in (1.28), (1.29), (1.30) and
(1.33) can be replaced by limit, for details see section 3.
Now, (1.33) can be applied to find the upper bounds of entropy. For example when A2is symmetric, then
h(A2) ≤ log ρ(T 2m)
2m , (1.34)
for any m ≥ 1. Since
Tn≤ Bn (1.35)
can be shown for any n ≥ 2. Generally, (1.33) gives a better approximation than (1.11).
In summary, this study yields lower-bound estimates of entropy like (1.25) by introducing reduction operators Cm, and the upper-bound estimates of
en-tropy like (1.34) by introducing trace operators Tm. This approach provides an
accurate and effective method to compute spatial entropy.
The rest of this paper is organized as follows. Section 2 derives the reduction operator Cm which can recursively reduce higher order elementary patterns to
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 discuss the theory for many symbols on larger lattices. The Appendices presents the Perron-Frobenius theorem and its related results.
2. Reduction Operators
2.1. Reduction operators and ordering matrix. This section derives reduction operators and investigates their properties. For clarity, two symbols on 2 × 2 lattice Z2×2 are examined first. Section 4 addresses more general situation.
Let A2and B2 be defined as in (1.1)∼(1.4). The column matrices fA2 and fB2
of A2and B2are defined by
f A2= a11 a21a12a22 a31 a41a32a42 a13 a23a14a24 a33 a43a34a44 = ˜ A2;1A˜2;2 ˜ A2;3A˜2;4 (2.1)
and f B2= b11 b21b12b22 b31 b41b32b42 b13 b23b14b24 b33 b43b34b44 = ˜ B2;1 B˜2;2 ˜ B2;3 B˜2;4 (2.2) , respectively.
For higher order matrices n ≥ 2, An, An+1 and An+1;α are defined as in
(1.5)∼(1.7).
For matrix multiplication, the indices of An;αare conveniently expressed as
An = An;11An;12 An;21An;22 . (2.3) Clearly, An;α= An;j1j2, where α = α(j1, j2) = 2(j1− 1) + j2. (2.4)
For m ≥ 2, the elementary pattern in the entries of Am
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 χ(Jm+1) = 1 + m X s=2 2m−l(js− 1). (2.5) Now, A(k)m,n;α= An;j1j2An;j2j3· · · An;jmjm+1, (2.6) where α = α(j1, jm+1) = 2(j1− 1) + jm+1 (2.7) and k = χ(Jm+1) (2.8)
is given in (2.5). Note that jm+1 does not involve in (2.5) and (2.8) but instead
in (2.7). Therefore Am n can be expressed as Amn = Am,n;1Am,n;2 Am,n;3Am,n;4 , (2.9) where Am,n;α= 2m−1 X k=1 A(k)m,n;α. (2.10) Furthermore, Xm,n;α= (A(k)m,n;α) t 1≤k≤2m−1. (2.11)
1 ≤ k ≤ 2m−1, X
m,n;α is a 2m−1 column-vector that consists of all elementary
patterns in Am,n;α. The ordering matrix Xm,n of Amn is now defined by
Xm,n= Xm,n;1
Xm,n;2
Xm,n;3Xm,n;4
. (2.12)
The following simplest example is studied first to illustrate the above concept. Example 2.1 Considering m = 2, it can easily be verified
A2n= A2 n;11+ An;12An;21 An;11An;12+ An;12An;22 An;21An;11+ An;22An;21 An;21An;12+ A2n;22 , (2.13) and A(1)2,n;1= A2n;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= A 2 n;22. . (2.14) Therefore, X2,n;1= A2 n;11 An;12An;21 , X2,n;2= An;11 An,12 An;12An;22 , X2,n;3= An;21An;11 An;22An;21 , X2,n;4= An;21An,12 An;22An;22 . . (2.15)
Applying (1.7), and by a straightforward computation,
X2,n+1;1 = A2 n+1;11 An+1;12An+1;21 (2.16) = b2 11A2n;1+ b12b13An;2An;3 b11b12An;1An;2+ b12b14An;2An;4 b13b11An;3An;1+ b14b13An;4An;3 b13b12An;3An;2+ b214A2n;4 b21b31A2n;1+ b22b33An;2An;3 b21b32An;1An;2+ b22b34An;2An;4 b23b31An;3An;1+ b24b33An;4An;3 b23b32An;3An;2+ b24b34A2n;4 Clearly, the j1j2 entries of A2n+1;11 and An+1;12An+1;21 in (2.16) consist of
entries of X2,n;αin (2.14) with α = α(j1, j2) in (2.4). Moreover, the terms (2.16)
can be rearranged in terms of X2,n;α by exchanging the second row in the first
matrix and the first row in the second matrix in (2.16) as follows. b2 11 b12b13 b21b31 b22b33 A2 n;1 An;2An;3 b11b12b12b14 b21b32b22b34 An;1An;2 An;2An;4 b13b11 b14b13 b23b31 b24b33 An;3An;1 An;4An;3 b13b12 b214 b23b32b24b34 An;3An;2 A2n;4 (2.17)
Applying (1.1), (1.2) and (2.1), (2.17) can be rewritten as a2 11 a12a21 a13a31 a14a41 A2 n;11 An;12An;21 a11a12a12a22 a13a32a14a42 An;11An;12 An;12An;22 a21a11 a22a21 a23a31 a24a41 An;21An;11 An;22An;21 a21a12 a222 a23a32a24a42 An;21An;12 A2n;22 = (B2;11◦A2;11˜ )X2,n;1(B2;11◦A2;12˜ )X2,n;2 (B2;21◦A2;11˜ )X2,n;3(B2;21◦A2;12˜ )X2,n;4 . (2.18)
Therefore, after the permutation of entries of X2,n+1;1 as in (2.17) or (2.18),
X2,n+1;1 can be represented by a 2 × 2 matrix
ˆ X2,n+1;1≡ P(X2,n+1;1) ≡ X2,n+1;1;1 X2,n+1;1;2 X2,n+1;1;3X2,n+1;1;4 , (2.19) where 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 (2.20) and S2;11= B2;11◦ ˜A2;11≡ C2;11, S2;12= B2;11◦ ˜A2;12≡ C2;12, S2;13= B2;21◦ ˜A2;11≡ C2;21, S2;14= B2;21◦ ˜A2;12≡ C2;22, . (2.21)
The above derivation indicates that X2,n+1;1 can be reduced to form X2,n;β
through multiplication by reduction matrices C2;αβ. This procedure can be
extneded to introduce the reduction operator Cm= [ Cm;αβ], for all m ≥ 2.
Before introducing Cm, 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
M ⊗ N = (MijN). (2.22)
For any n ≥ 1,
⊗Nn= N ⊗ N ⊗ · · · ⊗ N, n-times in N.
Next, for any two n × n matrices
C = (Cij) and D = (Dij)
where Cij and Dij are numbers or matrices, the Hadamard product C ◦ D is
defined by
where the product Cij · Dij of Cij and Dij 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
M2= m11m12m21m22 m13m14m23m24 m31m32m41m42 m33m34m43m44 = M2;1M2;2 M2;3M2;4 (2.24)
and any 2 × 2 matrix
N = NN1 N2
3 N4
, (2.25)
where mij are numbers and Nk are numbers or matrices, for 1 ≤ i, j, k ≤ 4,
define M2⊗N =ˆ m11N1m12N2m21N1m22N2 m13N3m14N4m23N3m24N4 m31N1m32N2m41N1m42N2 m33N3m34N4m43N3m44N4 . (2.26)
Furthermore, for n ≥ 1, the n + 1 th order of transition matrix of M2 is defined
by
Mn+1≡ ˆ⊗Mn2 = M2⊗Mˆ 2⊗ · · · ˆˆ ⊗M2,
n-times in M2. More precisely,
Mn+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 ) = m11Mn;1m12Mn;2m21Mn;1m22Mn;2 m13Mn;3m14Mn;4m23Mn;3m24Mn;4 m31Mn;1m32Mn;2m41Mn;1m42Mn;2 m33Mn;3m34Mn;4m43Mn;3m44Mn;4 = Mn+1;1Mn+1;2 Mn+1;3Mn+1;4 , (2.27) where Mn= ˆ⊗Mn−12 = Mn;1Mn;2 Mn;3Mn;4 . Here, the following convention is adopted,
ˆ ⊗M0
2= E4×4.
Definition 2.3 For m ≥ 2, define
Cm= Cm;11Cm;12 Cm;13Cm;14 Cm;21Cm;22 Cm;23Cm;24 Cm;31Cm;32 Cm;33Cm;34 Cm;41Cm;42 Cm;43Cm;44 = Sm;11Sm;12Sm;21Sm;22 Sm;13Sm;14Sm;23Sm;24 Sm;31Sm;32Sm;41Sm;42 Sm;33Sm;34Sm;43Sm;44 , (2.28) where Cm;αβ= aα1 aα2 aα3 aα4 ◦ ⊗ˆ B2;1B2;2 B2;3B2;4 m−2! 2×2 ! 2m−1×2m−1 ◦ E2m−2×2m−2⊗ a1βa2β a3βa4β 2m−1×2m−1 . (2.29)
Similarly, for B2, define Um= Um;11Um;12Um;13Um;14 Um;21Um;22Um;23Um;24 Um;31Um;32Um;33Um;34 Um;41Um;42Um;43Um;44 = Wm;11Wm;12Wm;21Wm;22 Wm;13Wm;14Wm;23Wm;24 Wm;31Wm;32Wm;41Wm;42 Wm;33Wm;34Wm;43Wm;44 , (2.30) where Um;αβ= bα1 bα2 bα3 bα4 ◦ ⊗ˆ A2;1 A2;2 A2;3 A2;4 m−2! 2×2 ! 2m−1×2m−1 ◦ E2m−2×2m−2⊗ b1βb2β b3βb4β 2m−1×2m−1 . (2.31) Sm= [Sm;αβ] and Wm= [Wm;αβ].
Now Rm+1can be found from Cm by a recursive formula, as in (1.7).
Theorem 2.4 For any m ≥ 2 and 1 ≤ α, β ≤ 4, Cm+1;αβ= aα1 Cm;1β aα2Cm;2β aα3Cm;3β aα4Cm;4β , (2.32) and Um+1;αβ= bα1 Um;1β bα2Um;2β bα3Um;3β bα4Um;4β . (2.33) Proof. By (2.27), ˆ ⊗Bm−1 2 = B2⊗( ˆˆ ⊗Bm−22 ) = B2;1◦ ( ˆ⊗Bm−22 ) B2;2◦ ( ˆ⊗Bm−22 ) B2;3◦ ( ˆ⊗Bm−22 ) B2;4◦ ( ˆ⊗Bm−22 ) . Therefore, Cm+1;αβ= (B2;α◦ ( ˆ⊗Bm−12 )) ◦ (E2m−1×2m−1⊗ ˜A2;β) = aα1(B2;1◦ ˆ⊗B m−2 2 ) aα2(B2;2◦ ˆ⊗Bm−22 ) aα3(B2;3◦ ˆ⊗Bm−22 ) aα4(B2;4◦ ˆ⊗Bm−22 ) ◦ (E2m−1×2m−1⊗ ˜A2;β) = aα1[(B2;1◦ ˆ⊗B m−2 2 ) ◦ (E2m−2×2m−2 ⊗ ˜A2;β)] aα2[(B2;2◦ ˆ⊗Bm−22 ) ◦ (E2m−2×2m−2⊗ ˜A2;β) aα3[(B2;3◦ ˆ⊗Bm−22 ) ◦ (E2m−2×2m−2 ⊗ ˜A2;β)] aα4[(B2;4◦ ˆ⊗Bm−22 ) ◦ (E2m−2×2m−2⊗ ˜A2;β)] = aα1Cm;1β aα2Cm;2β aα3Cm;3β aα4Cm;4β .
A similar result also holds for Um;αβ; the details are omitted here. The proof is
complete.
Note that (2.32) implies that the entry of Rm;ij is ai1i2ai2i3· · · aimim+1 with
i1 = i and im+1 = j. Rm;ij is consist of all words of length m starting from i
and ending at j.
In this sense, Cm;ij is called a connecting matrix in studying the primitivity
of An as in [6]. To show that Cmis a reduction operator, results like (2.20) must
using (1.7), A(k)m,n+1;α = An+1;j1j2An+1;j2j3· · · An+1,jmjm+1 = m Y i=1 bαi1An;11bαi2An;12 bαi3An;21bαi4An;22 (2.34)
where αi= α(ji, ji+1), for 1 ≤ i ≤ m. After executing m-times matrix
multipli-cations in (2.34), A(k)m,n+1;α= " A(k)m,n+1;α;1A(k)m,n+1;α;2 A(k)m,n+1;α;3A(k)m,n+1;α;4 # (2.35) where A(k)m,n+1;α;β= 2m−1 X l=1 C(m; α, β; k, l)A(l)m,n;β (2.36)
is a linear combination of A(l)m,n;β with the coefficients C(m; α, β; k, l) which are products of bαlj, 1 ≤ l ≤ m. Note that Amn+1= Am,n+1;1 Am,n+1;2 Am,n+1;3 Am,n+1;4 (2.37) = 2m−1 X k=1 A(k)m,n+1;1 2m−1 X k=1 A(k)m,n+1;2 2m−1 X k=1 A(k)m,n+1;3 2m−1 X k=1 A(k)m,n+1;4 = P2m−1 k=1 A (k) m,n+1;1;1 P2m−1 k=1 A (k) m,n+1;1;2 P2m−1 k=1 A (k) m,n+1;2;1 P2m−1 k=1 A (k) m,n+1;2;2 P2m−1 k=1 A (k) m,n+1;1;3 P2m−1 k=1 A (k) m,n+1;1;4 P2m−1 k=1 A (k) m,n+1;2;3 P2m−1 k=1 A (k) m,n+1;2;4 P2m−1 k=1 A (k) m,n+1;3;1 P2m−1 k=1 A (k) m,n+1;3;2 P2m−1 k=1 A (k) m,n+1;4;1 P2m−1 k=1 A (k) m,n+1;4;2 P2m−1 k=1 A (k) m,n+1;3;3 P2m−1 k=1 A (k) m,n+1;3;4 P2m−1 k=1 A (k) m,n+1;4;3 P2m−1 k=1 A (k) m,n+1;4;4 Now, Xm,n+1;α;β is defined as Xm,n+1;α;β = (A (k) m,n+1;α;β) t. (2.38)
As in (2.17), the entries of Xm,n+1;α are rearranged into a new matrix
ˆ Xm,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 . (2.39) The ordering matrix Xm,nallows the elementary patterns to be tracked during
the reduction from Am
n+1to Amn. This careful book-keeping provides a
systemat-ical way to generate the admissible patterns and also lower-bound estimates of spatial entropy later.
From (2.36) and (2.38),
Xm,n+1;α;β = C(m; α, β)Xm,n;β (2.40)
where
C(m; α, β) = (C(m; α, β; k, l)), 1 ≤ k, l ≤ 2m−1, is a 2m−1× 2m−1
matrix. Now, C(m; α, β) = Sm;αβmust 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,
Xm,n+1;α;β= Sm;αβXm,n;β, (2.41)
or equivalently, the recursive formula (1.24) holds, i.e.,
A(k)m,n+1;α= 2m−1 X l=1 (Sm;α1)klA (l) m,n;1 2m−1 X l=1 (Sm;α2)klA (l) m,n;2 2m−1 X l=1 (Sm;α3)klA (l) m,n;3 2m−1 X l=1 (Sm;α4)klA (l) m,n;4 . (2.42) Moreover, for n = 1, A(k)m,2;α= 2m−1 X l=1 (Sm;α1)kl 2m−1 X l=1 (Sm;α2)kl 2m−1 X l=1 (Sm;α3)kl 2m−1 X l=1 (Sm;α4)kl (2.43)
for any 1 ≤ k ≤ 2m−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 α = 2, 3 and 4 can also be proved analogously; the details are omitted.
Now, assume that (2.41) holds for m; the goal is to show that it also holds for m + 1. Since Am+1n+1 = An+1· Amn+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 , (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 then (An+1;1P(Xm,n+1;1), An+1;4P(Xm,n+1;4))t = b11An;1b12An;2 b13An;3b14An;4 Sm;11Xm,n;1Sm;12Xm,n;2 Sm;13Xm,n;3Sm;14Xm,n;4 b21An;1b22An;2 b23An;3b24An;4 Sm;31Xm,n;1Sm;32Xm,n;2 Sm;33Xm,n;3Sm;34Xm,n;4 = b11Sm;11An;1Xm,n;1+ b12Sm;13An;2Xm,n;3b11Sm;12An;1Xm,n;2+ b12Sm;14An;2Xm,n;4 b13Sm;11An;3Xm,n;1+ b14Sm;13An;4Xm,n;3b13Sm;12An;3Xm,n;2+ b14Sm;14An;4Xm,n;4 b21Sm;31An;1Xm,n;1+ b22Sm;33An;2Xm,n;3b21Sm;32An;1Xm,n;2+ b22Sm;34An;2Xm,n;4 b23Sm;31An;3Xm,n;1+ b24Sm;33An;3Xm,n;3b23Sm;32An;3Xm,n;3+ b24Sm;34An;4Xm,n;4
Hence Xm+1,n+1;1 can be represented by a matrix
ˆ Xm+1,n+1;1≡ P(Xm+1,n+1;1) ≡ Xm+1,n+1;1,1Xm+1,n+1;1,2 Xm+1,n+1;1,3Xm+1,n+1;1,4 = b11Sm;11b12Sm;13 b21Sm;31b22Sm;33 An;1Xm,n;1 An;2Xm,n;3 b11Sm;12b12Sm;14 b21Sm;32b22Sm;34 An;1Xm,n;2 An;2Xm,n;4 b13Sm;11b14Sm;13 b23Sm;31b24Sm;33 An;3Xm,n;1 An;4Xm,n;3 b13Sm;12b14Sm;14 b23Sm;32b24Sm;34 An;3Xm,n;2 An;4Xm,n;4
Once again, (1.1), (1.2) and (2.1) can be used to recast the matrix ˆXm+1,n+1;1
as a11Cm;11a12Cm;21 a13Cm;31a14Cm;41 Xm+1,n;1 a11 Sm;12a12Sm;22 a13Sm;32a14Sm;42 Xm+1,n;2 a21Cm;11a22Cm;21 a23Cm;31a24Cm;41 Xm+1,n;3 a21Cm;12a22Cm;22 a23Cm;32a24Cm;42 Xm+1,n;4
According to Theorem 2.4, the above matrix becomes
= Cm+1;11Xm+1,n;1Cm+1;12Xm+1,n;2 Cm+1;21Xm+1,n;3Cm+1;22Xm+1,n;4 = Sm+1;11Xm+1,n;1Sm+1;12Xm+1,n;2 Sm+1;13Xm+1,n;3Sm+1;14Xm+1,n;4 .
The cases of 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
A1≡ A1;11 A1;12 A1;21 A1;22 ≡ A1;1 A1;2 A1;3 A1;4 ≡ 1 1 1 1 (2.44)
is introduced, then the previous argument also hold for n = 1. Thus, (2.43)
For any positive integer p ≥ 2, repeatedly applying Theorem 2.5 p times permits the elementary patterns of Am
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 Am
n+p is first studied.
For any p ≥ 2 and 1 ≤ q ≤ p − 1, define A(k)m,n+p;α;β 1;β2;··· ;βq= " A(k)m,n+p;α;β 1;β2;··· ;βq;1A (k) m,n+p;α;β1;β2;··· ;βq;2 A(k)m,n+p;α;β 1;β2;··· ;βq;3A (k) m,n+p;α;β1;β2;··· ;βq;4 # . (2.45) Then A(k)m,n+p;α;β 1;β2;··· ;βp= 2m−1 X l1=1 · · · 2m−1 X lp=1 ( p Y i=1
C(m; βi−1, βi; li−1, li))A (lp)
m,n;βp, (2.46)
where β0 = α and l0 = k can be easily verified. Therefore, for any p ≥ 1, a
generalization for (2.37) can be found for Amn+pas a 2
p+1× 2p+1 matrix Amn+p=Am,n+p;α;β1;β2··· ;βp (2.47) where Am,n+p;α;β1;β2··· ;βp= 2m−1 X k=1 A(k)m,n;α;β 1;β2··· ;βp. (2.48)
In particular, if α; β1, β2· · · , βp ∈ {1, 4}, then Am,n+p;α;β1;β2··· ;βp lies on the
diagonal of Am n+pin (2.47). Now, define Xm,n+p;α;β1;β2;··· ;βp = (A (k) m,n+p;α;β1;β2;··· ;βp) t. (2.49)
Therefore, Theorem 2.5 can be generalized to Theorem 2.6 For any m ≥ 2, n ≥ 2 and p ≥ 1,
Xm,n+p;α;β1;β2···βp= Sm;αβ1Sm;β1β2· · · Sm;βp−1βpXm,n;βp (2.50)
where α, βi∈ {1, 2, 3, 4} and 1 ≤ i ≤ p.
Proof. From (2.46), (2.40) and (2.42),
A(k)m,n+p;α;β 1;β2;··· ;βp= 2m−1 X l1=1 · · · 2m−1 X lp=1 ( p Y i=1
C(m; βi−1, βi; li−1, li))A (lp) m,n;βp = 2m−1 X l1=1 · · · 2m−1 X lp=1 ( p Y i=1 (Sm;βi−1,βi)li−1,li)A (lp) m,n;βp = 2m−1 X l1=1 · · · 2m−1 X lp=1 (Sm;β0,β1)l0,l1(Sm;β1,β2)l1,l2· · · (Sm;βp−1,βp)lp−1,lpA (lp) m,n;βp = 2m−1 X lp=1 (Sm;β0,β1Sm;β1,β2· · · Sm;βp−1,βp)l0,lpA (lp) m,n;βp = 2m−1 X lp=1 (Sm;α,β1Sm;β1,β2· · · Sm;βp−1,βp)k,lpA (lp) m,n;βp
is derived. By (2.49), then Xm,n+p;α;β1;β2;··· ;βp = (A (k) m,n+p;α;β1;β2;··· ;βp) t = Sm;α,β1Sm;β1,β2· · · Sm;βp−1,βp)k,lp(A (lp) m,n;βp) t = Sm;α,β1Sm;β1,β2· · · Sm;βp−1,βp)k,lpXm,n;βp.
The proof is complete.
2.2. Lower bound of entropy. In this subsection, the reduction 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 property in m and n:
Γm1+m2,n≤ Γm1,nΓm2,n, (2.51) and Γm,n1+n2 ≤ Γm,n1Γm,n2, (2.52) or equivalently, |Am1+m2 n | ≤ |A m1 n |A m2 n | (2.53) and |Am n1+n2| ≤ |A m n1||A m n2|, (2.54)
for positive integers m, n, m1, n1, m2and n2. Here
A1= 1 11 1
(2.55) is applied.
The subadditivity property implies lim sup m,n→∞ log |Amn| mn ≤ log |Ap−1 q | pq (2.56)
for any p and q ≥ 2. Therefore,
h(A2) = lim m,n→∞
log |Am n|
mn exists, and equals
inf
p,q≥2
log |Ap−1 q |
pq . (2.57)
In particular, h(A2) has an upper bound
h(A2) ≤
log |Ap−1q |
pq (2.58)
Similarly, when A2 is horizontal (or vertical) transition matrix for any m ≥ 1 and q ≥ 2, lim sup n→∞ log |Am n| n ≤ log |Amq | q . (2.59)
Hence, the spatial entropy hm(A2) on infinite lattice Zm+1×∞ (or Z∞×m+1)
exists and hm(A2) ≡ lim n→∞ log |Amn| n = infq≥2 log |Am q | q . (2.60)
For the proof of the above results, see [14]. Furthermore, by Perron-Frobenius theorem,
lim
m→∞
log |Am n|
m = log ρ(An). (2.61)
Therefore, for any n ≥ 2
h(A2) ≤ log ρ(A n)
n . (2.62)
For the proof of (2.61), see [4], [29]. The following notation is adopted.
Definition 2.7 Let X = (X1, · · · , XM)t, where Xk are N × N matrices. Define
the summation of Xk by |X| = N X k=1 Xk. (2.63) If M = [Mij] is a M × M matrix, then |MX| = M X i=1 M X j=1 MijXj. (2.64)
Note that, (2.63) implies
|Xm,n;α| = 2m−1
X
k=1
A(k)m,n;α= Am,n;α. (2.65)
As usual, the set of all matrices with the same order can be partially ordered. Definition 2.8 Let M = [Mij] and N = [Nij] be two M × M matrices, M ≥ N
if Mij ≥ Nij for all 1 ≤ i, j ≤ M .
Note that, if A2≥ A02 then An≥ A0n for all n ≥ 2. Therefore, h(A2) ≥ h(A02).
Hence, the spatial entropy as a function of A2 is monotonic with respect to the
Definition 2.9 A K + 1 multiple index
BK≡ (β1β2· · · βKβK+1) (2.66)
is called a (periodic) cycle if
βK+1= β1. (2.67)
It is called a diagonal cycle if (2.67) holds and
βk ∈ {1, 4} (2.68)
for each 1 ≤ k ≤ K + 1.
For a diagonal cycle (2.66), denote ¯ βK = β1; β2; · · · ; βK (2.69) and ¯ βK n = ¯βK; ¯βK; · · · ; ¯βK. (n times) (2.70)
First, prove the following Lemma.
Lemma 2.10 Let m ≥ 2, K ≥ 1, BK be a diagonal cycle. Then, for any n ≥ 1,
ρ(AmnK+2) ≥ ρ(|(Sm;β1β2Sm;β2β3· · · Sm;βKβK+1)
nX
m,2;β1|) (2.71)
Proof. Since BK is a periodic cycle, Theorem 2.6 implies
Xm,nK+2; ¯β
Kn = (Sm;β1β2Sm;β2β3· · · Sm;βKβK+1)
nX
m,2;β1. (2.72)
Furthermore BK is diagonal, and |Xm,nK+2; ¯βkn| = Am,nK+2; ¯βkn lies on the
di-agonal part as in (2.47) with n + p = nK + 2, therefore
ρ(AmnK+2) ≥ ρ(|Xm,nK+2; ¯βKn|). (2.73)
Therefore, (2.71) follows from (2.72) and (2.73).
The proof is complete.
The following lemma is valuable in studying maximum eigenvalue of (Sm;β1β2· · · Sm;βKβK+1)
nX
m,2;β1 in (2.71).
Lemma 2.11 For any m ≥ 2, 1 ≤ k ≤ 2m−1 and α ∈ {1, 4}, if
tr(A(k)m,2;α) = 0, (2.74)
then for all 1 ≤ l ≤ 2m−1,
(Sm,α1)kl= 0 and (Sm;α4)kl= 0, (2.75)
i.e. the k-th rows of matrices Sm;α1 and Sm;α4 are zeros. Furthermore, for any
diagonal cycle BK, let U = (u1, u2, · · · , u2m−1) be an eigenvector of
Sm;β1β2Sm;β2β3· · · Sm;βKβ1, if uk6= 0 for some 1 ≤ k ≤ 2
m−1, then
Proof. Since A(k)m,2;α can be expressed as in (2.43). Therefore, tr(A(k)m,2;α) = 0 if and only if (2.75) holds for all 1 ≤ l ≤ 2m−1. The second part of Lemma follows easily from the first part.
The proof is complete.
By Lemma 2.10 and Lemma 2.11, the lower bound of entropy can be obtained as follows.
Theorem 2.12 Let β1β2· · · βKβ1 be a diagonal cycle. Then for any m ≥ 2,
h(A2) ≥ 1 mKlog ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβ1). (2.77) and h(A2) ≥ 1 mKlog ρ(Wm;β1β2Wm;β2β3· · · Wm;βKβ1). (2.78)
In particular, if a diagonal cycle β1β2· · · βKβ1 exists and m ≥ 2 such that
ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβ1) > 1,
or
ρ(Wm;β1β2Wm;β2β3· · · Wm;βKβ1) > 1
then h(A2) > 0.
Proof. First, show that h(A2) ≥ 1 mKlim supn→∞ (log ρ(|(Sm;β1β2Sm;β2β3· · · Sm;βKβ1) nX m,2;β1|). (2.79)
Indeed, from (2.62) and (2.71), h(A2) = lim n→∞ 1 nK + 2log ρ(AnK+2) = lim n→∞ 1 m(nK + 2)log ρ(A m nK+2) ≥ 1 mKlim supn→∞ (log ρ(|(Sm;β1β2· · · Sm;βKβ1) nX m,2;β1|).
Now, it remains to show lim sup n→∞ 1 n(log ρ(|(Sm;β1β2· · · Sm;βKβ1) nX m,2;β1|) = log ρ(Sm;β1β2· · · Sm;βKβ1). (2.80) Since Xm,2;β1= (A (k) m,2;β1) t, if tr(A(k)
m,2;β1) = 0 then Lemma 2.11 implies the
k-th row of Sm;β1β2is zero which also implies that the k-th row of (Sm;β1β2· · · Sm;βKβ1)
n
is also zero for any n ≥ 1. If tr(A(k)m,2;β
1) = 0 for all 1 ≤ k ≤ 2
m−1, then S
m;β1β2 ≡ 0. (2.80) holds
trivially.
Now, assume that 1 ≤ k0≤ 2m−1exists such that tr(A(k0)
m,2;β1) > 0. Define ˆ X = (A(km,2;β0) 1) t= ( ˆX 1, · · · , ˆXM), (2.81)
where tr(A(k
0)
m,2;β1) > 0 for 1 ≤ k
0 ≤ M ≤ 2m−1. Then ρ( ˆX
j) > 0 for 1 ≤ j ≤ M .
Let M be the M × M sub-matrix of Sm;β1β2· · · Sm;βKβ1 which k-th rows and
k-th columns have been removed whenever tr(A(k)m,2;β
1) = 0 for 1 ≤ k ≤ 2 m−1. Clearly, |(Sm;β1β2· · · Sm;βKβ1) nX m,2;β1| = |M nX|,ˆ (2.82) and ρ(Sm;β1β2· · · Sm;βKβ1) = ρ(M). (2.83)
The proof of (2.80) proceeds into three steps according to (i) M is primitive,
(ii) M is irreducible, and (iii) M is reducible.
(i) M is primitive. Then by Perron-Frobenius Theorem (see Appendices) the maximum eigenvalue ρ(M) of M is unique with maximum modulus, i.e.
ρ(M) = λ1> |λj|, (2.84)
for all 2 ≤ j ≤ M , where λj are eigenvalues of M. Moreover, a positive
eigenvector V1= (v1, v2, · · · , vM)tis associated with λ1.
By Jordan canonical form theorem, a non-singular matrix P = [Pij]M ×M
exists such that the real Jordan canonical form of M is
ˆ M ≡ PMP−1= λ1 0 · · · 0 0 Jn2 .. . . .. 0 Jnq , (2.85)
where Jnk, 2 ≤ k ≤ q are real Jordan block and the associated eigenvalue λk
of Jnk satisfies (2.84). Moreover, P can be chosen such that
M X i=1 Pij= 1 (2.86) and P1j> 0 (2.87)
for all 1 ≤ j ≤ M (see Appendices). Therefore, by (2.86) |MnX| = |PMˆ nX| = |PMˆ n P−1PX|ˆ = |(PMP−1)n PX| = | ˆˆ MnPX|ˆ = λ1n{ M X j=1 P1jXˆj+ M X j=1 qn,jXˆj} where lim n→∞qn,j= 0, (2.88) for all 1 ≤ j ≤ M , by (2.84). Hence, by (2.79), (2.87) and (2.88), lim n→∞ log ρ(|MnX|)ˆ n = log λ1. (2.89)
(ii) M is irreducible.
If M is irreducible but imprimitive, then there exists k ≥ 2 exists, such that λ1= |λ2| = · · · = |λk| > |λj|
for all j > k. Then, by applying a permutation, M can be expressed as
M = 0 M12 0 · · · 0 0 0 M23· · · 0 .. . 0 · · · 0 Mk−1,k Mk1 0 · · · 0 , (2.90) and, Mk= M1 0 · · · 0 0 M2· · · 0 .. . . .. ... 0 · · · 0 Mk , (2.91)
where Mj= Mj,j+1Mj+1,j+2· · · Mj−1,j is primitive with the maximum
eigen-value λk
1 (see Theorem A.2 of Appendices). Hence, by the same argument as
in (i) lim n→∞ log ρ(|MnkX|)ˆ n = λ k 1, (2.80) follows. (iii) M is reducible.
In this case, by applying a permutation, M can be expressed as a block upper triangular matrix: M = M11M12· · · M1k 0 M22· · · M2k 0 0 . .. .. . 0 0 · · · 0 Mkk , (2.92)
where Mii is either irreducible or zero. Furthermore,
σ(M) =
k
[
j=1
σ(Mjj),
where σ(M) and σ(Mjj) are the set of eigenvalues of M and Mjj, respectively
(see Theorem A.3 of Appendices). In particular, 1 ≤ j ≤ k exists, such that ρ(Mjj) = ρ(M) = λ1. (2.93)
Therefore, applying (2.83), (2.93) and the same argument as in (ii), (2.80) follows.
Definition 2.13 Let D denote the set of all diagonal cycle, i.e., D = {β1β2· · · βKβK+1|β1β2· · · βKβK+1 satisfies (2.67) and (2.68)}, define h∗(A2) = sup m≥2,β1β2···βK∈D 1 mKlog ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβ1). (2.94) and h0∗(A2) = sup m≥2, β1···βK∈D 1 mK log ρ(Wm;β1β2Wm;β2β3· · · Wm;βKβ1). (2.95)
Then Theorem 2.12 implies
h(A2) ≥ h∗(A2) and h(A2) ≥ h0∗(A2). (2.96)
Knowing whether equality holds for all A2is of interest, since h∗(A2) and h0∗(A2)
are more manageable than h(A2). However, a class of A2 has been found where
equality (2.96) holds; details can be found in Example 2.14. of the next subsec-tion.
2.3. Examples of Transition matrices with positive entropy. In this subsection, various examples are studied to elucidate the power of Theorem 2.12 in verify-ing the positivity of entropies. First, Golden-Mean type transition matrices are studied.
Example 2.14(a) Golden-Mean
When two symbols on two-cell horizontal lattice Z2×1 and vertical lattice
Z1×2 are considered and both transition matrices are given by golden-mean,
i.e., H1= V1= 1 1 1 0 . Then the (horizontal) transition matrix A2 on Z2×2 is
A2= 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 , (2.97) as in [40]. Verifying B2= eA2= eB2= A2. (2.98)
is also easy. Furthermore, for any n ≥ 2,
An+1= An+1Bn+1 Cn+1 0 = An Bn An0 Cn 0 Cn0 An Bn 0 0 0 0 0 0 , (2.99)
where
An+1= An
Bn
Cn 0
with Cn= Bnt and Ant= An, i.e., An are symmetric for all n ≥ 2.
Moreover, the following two properties hold: (i) For any m ≥ 2,
Cm;11= Am−1, (2.100) where A1≡ a11 a11a12a21 a13a31a14a41 , (2.101) and
(ii) for any m ≥ 2,
log ρ(Am−1) m ≤ h(A2) ≤ log ρ(Am) m . (2.102) Therefore, h(A2) = h∗(A2) > 0. (2.103)
The numerical results appear in Example 3.12. (b) Simplified Golden-Mean. Consider A2= 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 , (2.104)
(2.104) cannot be generated from one-dimensional transitional matrices H1
and V1 as in Golden-Mean (2.97). Equation (2.104) is obtained by letting
a23= a32= 0 in Golden-Mean (2.97). It is easy to verify (2.98) and for any
n ≥ 2, An+1= An An−1 0 0 0 An−10 0 0 0 0 0 0 . (2.105)
Furthermore, properties (i) and (ii) and (2.103) hold as in (a). (c) Generally, if A2 satisfies the following three conditions
(C1) B2= A2,
(C2) a1j = 1 if A2;j6= 0 for 1 ≤ j ≤ 4,
(C3) eA2;1 ≥ A2;j for 1 ≤ j ≤ 4,
then properties (i) and (ii) and (2.103) hold. The matrices A2 which satisfy
(C1), (C2) and (C3) can be listed as 1 1 1 0 1 0 a230 1 a32 0 0 0 0 0 0 , (2.106)
and 1 1 1 1 1 1 a23a24 1 a32 1 a34 1 a34a43a44 , (2.107)
where aij is either 0 or 1 in (2.106) and (2.107).
Note that if (C2) and (C3) are replaced by (C2)0 a4j = 1 if A2;j6= 0 for 1 ≤ j ≤ 4,
(C3)0 Ag2;4 ≥ A2;j for 1 ≤ j ≤ 4,
then for any m ≥ 2,
Cm;44= Am−1 (2.108) with A1= a41a14 a42a24 a43a34 a44a44 , (2.109)
and property (ii) and equation (2.103) hold.
In Example 2.14, the diagonal parts A2;1 or A2;4 are dominant. In this case,
only Cm;11or Cm;44is required in applying Theorem 2.12. On the contrary, when
A2;1and A2;4are no longer dominant as in the following examples, A2;2and A2;3
can complement each other to establish positive entropy. Example 2.15(a) Consider
A2= 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 , (2.110)
which can be verified that (2.98) holds and
C2;11= 0 1 1 0 , C2;22= 1 0 1 0 C2;33= 1 10 0 , C2;44= 0 00 0 Therefore, S2;14S2;41= 1 11 1 and h(A2) ≥ log 2 4 . (b) Consider A2= 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0 . (2.111)
Then verifying B2= 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 , eB2= 0 1 1 0 1 0 0 1 1 1 0 1 1 1 1 0 , and eA2= 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 . is simple. Furthermore, C2;11= 0 1 1 0 , C2;22= 1 0 0 1 C2;33= 1 0 0 1 , C2;44= 0 1 1 0 and U2;11= 0 11 0 , U2;22= 1 00 1 , U2;33= 1 01 1 , U2;44= 0 11 0 .
Now, for any diagonal cycle, β1· · · βKβ1, ρ(S2;β1β2· · · S2;βKβ1) = 1 which cannot
establish h(A2) > 0. However, W2;11W2;14W2;41 = U2;11U2;22U2;33= 1 11 0 which implies h(A2) ≥ 1 6log g, where g = 1 + √ 5 2 (2.112)
is the golden mean and is a root of λ2− λ − 1 = 0.
This example demonstrates asymmetry of A2 and B2 in applying Theorem
2.12 to verify the positivity of entropy. Generally, both Cmand Umare checked
for completeness. Example 2.16 Consider A2= 1 1 1 1 0 0 0 1 0 0 0 1 1 0 0 0 . (2.113)
Then it is easy to check that
W2;11W2;14W2;41= 2 00 0 , S3;44= G 00 0 , and S4;44 = G 0 0 0 0 e10 0 0 0 0 0 0 0 0 0 ,
where G = 1 1 1 0 and e1= 1 0 0 0 . (2.114) Therefore, h(A2) ≥ max{ 1 6log 2, 1 3log g, 1 4log g} = 1 3log g. Example 2.17 Consider A2= 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 . (2.115) Then B2= 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 = ˜A2 and ˜B2= A2. Therefore C2,11= 0 11 1 ≡ G0. Furthermore, C4;11= G0⊗ e1⊗ G0 and C2m;11= G0⊗ (G0⊗ e1)m−1
can be proved, and which implies log ρ(C2m;11) 2m = 1 2log g. (2.116) for all m ≥ 1. 3. Trace operators
3.1. Trace operator Tm. The preceding section introduces reduction operators
Cm, which can be applied to find lower bounds of spatial entropy. This section
studies the diagonal part of Cm, which can be used to investigate the trace of
Amn and then to find the upper bound of spatial entropy. When A2is symmetric,
the method is very powerful.
The trace operator is defined first.
Definition 3.1 For m ≥ 2, the m-th order trace operator Tm of A2 is defined
by Tm= Cm;11Cm;22 Cm;33Cm;44 = Sm;11Sm;14 Sm;41Sm;44 , (3.1)
where Cm;ij is given in (1.23) or (2.29).
Similarly, the m-th order trace operator T0mof B2 is defined by
T0m= Um;11Um;22 Um;33Um;44 = Wm;11Wm;14 Wm;41Wm;44 (3.2) where Um;ij is given in (2.31).
Theorem 3.2 For any m ≥ 2, Tm= (Bm)2m×2m◦ E2m−2×2m−2⊗ a11a21 a31a41 E2m−2×2m−2⊗ a12a22 a32a42 E2m−2×2m−2⊗ a13 a23 a33a43 E2m−2×2m−2⊗ a14 a24 a34a44 (3.3) and T0m= (Am)2m×2m◦ E2m−2×2m−2⊗ b11b21 b31b41 E2m−2×2m−2⊗ b12b22 b32b42 E2m−2×2m−2⊗ b13b23 b33b43 E2m−2×2m−2⊗ b14b24 b34b44 . (3.4) In particular, Tm≤ Bm and T0m≤ Am. (3.5) Proof. By (3.1) and (2.29), Tm= (Bm)2m×2m◦ E2m−2×2m−2⊗ a11a21 a31a41 E2m−2×2m−2⊗ a12a22 a32a42 E2m−2×2m−2⊗ a13 a23 a33a43 E2m−2×2m−2⊗ a14 a24 a34a44 .
A similar result also holds for T0m. Hence, (3.5) follows immediately.
The proof is complete.
Notably, the trace operator Tm(or T0m) keep all periodic words ai1i2ai2i3· · · aimim+1
(bi1i2bi2i3· · · bimim+1) with im+1 = i1 of length m and preserve them in a
sys-tematically way as Am(or Bm).
The traces of elementary patterns are defined accordingly. Definition 3.3 For m, n ≥ 2 and 1 ≤ α ≤ 4, define
t(k)m,n;α= tr(A(k)m,n;α), (3.6) tr(Xm,n;α) = (t(k)m,n;α)1≤k≤2m−1, (3.7)
and
tm,n= (tr(Xm,n;1), tr(Xm,n;4))t, (3.8)
which are 2m−1 and 2m vectors, respectively. Note that tr(Amn) = tr( P2m−1 k=1 A (k) m,n;1+ P2m−1 k=1 A (k) m,n;4) = |tr(Xm,n;1)| + |tr(Xm,n;4)| = |tm,n|. (3.9)
Proposition 3.4 For m ≥ 2 and n ≥ 2,
tm,n+1= Tmtm,n (3.10)
Proof. By Theorem 2.4, it is easy to see tr(Xm,n+1;1) tr(Xm,n+1;4) = Cm;11tr(Xm,n;1) + Cm;22tr(Xm,n;4) Cm;33tr(Xm,n;1) + Cm;44tr(Xm,n;4) . Then, (3.10) follows immediately.
The proof is complete.
Repeatedly applying Proposition 3.4 yields the following result. Theorem 3.5 For m ≥ 2 and n ≥ 1,
tr(Amn+2) = |T n mtm,2| (3.11) ≡ X βk∈{1,4} |Sm;β1β2Sm;β2β3· · · Sm;βnβn+1tr(Xm,2;βn+1)|. (3.12) Proof. tr(Am n) = 2m−1 X k=1 tr(A(k)m,n;1;1) + 2m−1 X k=1 tr(A(k)m,n;1;4) + 2m−1 X k=1 tr(A(k)m,n;4;1) + 2m−1 X k=1 tr(A(k)m,n;4;4) = |tr(Xm,n;1;1)| + |tr(Xm,n;1;4)| + |tr(Xm,n;4;1)| + |tr(Xm,n;4;4)| = |tr(Sm;11Xm,n−1;1)| + |tr(Sm;14Xm,n−1;4)| + |tr(Sm;41Xm,n−1;1)| + |tr(Sm;44Xm,n−1;4)| = |Tmtm,n−1|,
here Theorem 2.4 is used. Reduction on n, yields
tr(Amn) = |T n−2 m tm,2|.
Finally, (3.12) follows from (3.1) and (3.6).
The proof is complete.
To show (1.33), the following lemma is needed.
Lemma 3.6 Let Um be a nonnegative eigenvector of Tm with respect to the
maximum eigenvalue ρ(Tm). If ρ(Tm) > 0, then
hUm, tm,2i > 0,
where h , i denotes the standard inner product of R2m
.
Proof. Let Um= (u1, · · · , uM, u01, · · · , u0M) be a nonnegative eigenvector of Tm,
where M = 2m−1. Since ρ(Tm) > 0, by Lemma 2.11, if uk > 0 (or u0l> 0) then
tr(A(k)m,2;1) > 0 (or tr(A(l)m,2;4) > 0). The result follows by (3.9).
Now, (1.33) can be proved.
Theorem 3.7 For any m ≥ 2, lim sup n→∞ log tr(Am n) n = log ρ(Tm). (3.13) and,
h(A2) = lim sup m→∞
log ρ(Tm)
m . (3.14)
Furthermore, if An are primitive for all n ≥ 2, then limsup in (3.13) and (3.14)
can be replaced by lim, i.e., lim n→∞ log tr(Amn) n = log ρ(Tm) (3.15) and h(A2) = lim m→∞ log ρ(Tm) m . (3.16)
Proof. By Perron-Frobenius theorem, for all n ≥ 2, we have lim sup
m→∞
log tr(Am n)
m = log ρ(An). (3.17)
Therefore, by (3.17) and Theorem 3.5, we have h(A2) = lim n→∞ log ρ(An) n = lim supn,m→∞ log tr(Amn) mn = lim supn,m→∞ log |Tnmtm,2| mn .
By Lemma 3.6 and using an argument as in proving Theorem 2.12, lim sup n→∞ 1 nlog |T n mtm,2| = log ρ(Tm) (3.18)
can be shown, and (3.13) and (3.14) follow immediately.
When An are primitive for all n ≥ 2, by Theorem A.5. of Appendices, (3.15)
and (3.16) follows.
The proof is complete.
Now show that the symmetry of A2 can be inherited by its higher order
ma-trices.
Proposition 3.8 If A2is symmetric, then An is also symmetric for each n ≥ 3.
Proof. The proposition is proven by induction on n. Let M = MM1M2
3M4
be a square matrix and Mi, 1 ≤ i ≤ 4, be all square
matrices. Then the transpose matrix Mtof M is Mt= M1 tM 3t M2tM4t .
Therefore, M is symmetric if and only if
M1t= M1, M3t= M2 and M4t= M4.
In particular, A2is symmetric, if and only if
At2;1= A2;1, At2;3= A2;2 and At2;4 = A2;4. (3.19)
Now, assume that An is symmetric, i.e.,
Atn;1= An;1, Atn;3= An;2 and Atn;4= An;4. (3.20) Since An+1;α= [A2;α]2×2◦ An;1An;2 An;3An;4 , (3.19) and (3.20) imply Atn+1;1= An+1;1, Atn+1;3= An+1;2 and Atn+1;4= An+1;4. Hence An+1is symmetric.
The proof is complete.
Now, an upper estimates of spatial entropy h(A2) can be produced when A2 is
symmetric.
Theorem 3.9 If A2 is symmetric then for any m ≥ 1, we have
h(A2) ≤ log ρ(T 2m)
2m . (3.21)
Proof. By Proposition 3.8, A2m
n is symmetric for any m ≥ 1. The symmetry of
A2mn implies that all eigenvalue of A 2m
n are non-negative. Hence,
ρ(An)2m= ρ(A2mn ) ≤ tr(A 2m
n ). (3.22)
On the other hand, the subadditivity of (2.58) implies h(A2) ≤ log |A
2mk n |
(2mk + 1)n. (3.23)
Therefore, (3.22), (3.23) and (3.11) imply h(A2) ≤ lim n,k→∞ log |A2mk n | (2mk + 1)n = limn→∞ log ρ(A2m n ) 2mn ≤ lim n→∞ log tr(A2m n ) 2mn = limn→∞ log |Tn 2mtm,2| 2mn ≤ log ρ(T2m) 2m .
The proof is complete.
Furthermore Tmand T0mgive better estimate than An or Bn whenever
h(A2) ≤ log ρ(T m)
n (3.24)
Remark 3.10 (i) The problem where An are primitive for all n ≥ 2 has been
investigated in [6]. In [6], various sufficient conditions have been found to ensure that An are primitive for all n ≥ 2. Note that the ingredient of having
limit in (3.15) and (3.16) instead of limsup in (3.13) and (3.14) is that An
has unique maximum eigenvalue with maximum modulus. Therefore, An may
be imprimitive but (3.15) and (3.16) still hold. For example, Golden-Mean and simplified Golden-Mean in Example 2.14 are imprimitive but (3.15) and (3.16) still hold. Since the remaining matrices of these Anare primitive if the
rows and columns with zero entries of An are removed.
(ii) In general, limsup cannot be replaced by limit. For example, consider A2= 0 JJ 2 2 0 (3.25) where J2= 0 1 1 0
. Then, for any n ≥ 3,
An = 0 Jn Jn 0 , where Jn= 0 Jn−1 Jn−1 0 . Furthermore, for any m ≥ 1,
A2mn = I2n×2n and A2m+1n = An. Therefore, tr(A2mn ) = 2 n and tr(A2m+1n ) = 0.
Hence (3.17) holds only for limsup. A further computation shows that
T2m+1= 0 and T2m= ⊗(e4 ⊗ e1)m−1⊗ e4 ⊗(e3⊗ e2)m−1⊗ e3 ⊗(e2⊗ e3)m−1⊗ e2 ⊗(e1⊗ e4)m−1⊗ e1 for all m ≥ 1. Therefore, ρ(T2m+1) = 0 and ρ(T2m) = ρ e4 e3 e2e1 = 2. Hence (3.14) holds only for limsup. Unlike (2.62) this example also demonstrates that (3.24) does not hold for any n = 2m + 1. This phenomenon is a disadvantage in the upper estimate of entropy from replacing An with Tn.
Example 3.11 Consider A2= 1 1 1 1 0 0 0 1 0 0 0 1 1 0 0 0
which has been studied as in Example 2.16. Now A2is asymmetric. Furthermore,
for all n ≥ 2 can be verified. Hence, (3.22) and then (3.21) fail when m = 1. On the other hand,
C4;44= G 0 0 0 0 e10 0 0 0 0 0 0 0 0 0 , where G = 1 1 1 0 , e1 = 1 0 0 0 and 0 = 0 0 0 0 . Hence tr(A4 n) grows at least
exponentially with exponent ρ(G) = g = 1 + √
5
2 , the golden-mean. Whether or not (3.21) holds for some m ≥ 2 is of interest.
Example 3.12 Numerically results of Golden-Mean and Simplified Golden-Mean.
4. More symbols on larger lattice
As mentioned in the introduction, many physical and engineering problems in-volve many symbols (more than two) and larger lattices. Therefore, the results found in the previous sections must be extended to any finite number of sym-bols p ≥ 2 on any finite square lattice Z2l×2l, l≥1. The results are only outlined,
and the details are left to the interested readers. The proofs of the theorems are omitted for brevity.
For fixed p ≥ 2 and l ≥ 1, denote by
q = pl2. (4.1)
The horizontal and vertical transition matrices are given by
A2= a1,1 a1,2 · · · a1,q2 a2,1 a2,2 · · · a2,q2 .. . ... . .. ... aq2,1aq2,2· · · aq2,q2 (4.2) and B2= b1,1 b1,2 · · · b1,q2 b2,1 b2,2 · · · b2,q2 .. . ... . .. ... bq2,1bq2,2· · · bq2,q2 , (4.3) respectively.
Now, A2 and B2are related to each other by
A2= A2;1 A2;2 · · · A2;q A2;q+1 A2;q+2· · · A2;2q .. . ... . .. ... A2;q(q−1)+1 · · · A2;q2 (4.4)
where A2;α= bα,1 bα,2 · · · bα,q bα,q+1 bα,q+2 · · · bα,2q .. . ... . .. ... bα,q(q−1)+1bα,q(q−1)+2· · · bα,q2 , (4.5) and B2= B2;1 B2;2 · · · B2;q B2;q+1 B2;q+2 · · · B2;2q .. . ... . .. ... B2;q(q−1)+1 · · · B2;q2 (4.6) where B2;α= aα,1 aα,2 · · · aα,q aα,q+1 aα,q+2 · · · aα,2q .. . ... . .. ... aα,q(q−1)+1aα,q(q−1)+2· · · aα,q2 , (4.7)
respectively, where α ∈ {1, 2, 3, 4}. The column matrices fA2and fB2of A2and B2
are defined as in (2.1) and (2.2). For higher order transition matrices An, n ≥ 3,
are defined by An = An;1 An;2 · · · An;q An;q+1 An;q+2 · · · An;2q .. . ... . .. ... An;q(q−1)+1An;(q−1)q+2 · · · An;q2 (4.8) where An;α= bα,1An−1;1 bα,2An−1;2 · · · bα,qAn−1;q bα,q+1An−1;q+1 bα,q+2An−1;q+2 · · · bα,2qAn−1;2q .. . ... . .. ... bα,q(q−1)+1An−1;q(q−1)+1bα,q(q−1)+2An−1;q(q−1)+2· · · bα,q2An;q2 . (4.9) Rewriting the indices of An;α as follows, facilitates matrix multiplication.
An = An;11An;12· · · An;1q An;21An;22· · · An;2q .. . · · · . .. · · · An;q1An;q2· · · An;qq . (4.10) Clearly, An;α= An;j1j2, where α = α(j1, j2) = q(j1− 1) + j2. (4.11)
For m ≥ 2, the elementary pattern in the entries of Amn is given by
An;j1j2An;j2j3· · · An;jmjm+1,
A lexicographic order for multiple indices Jm+1= (j1j2· · · jmjm+1) is introduced by χ(Jm+1) = 1 + m X l=2 qm−l(jl− 1). (4.12) Specify A(k)m,n;α= An;j1j2An;j2j3· · · An;jmjm+1,
where α = α(j1, jm+1) satisfies (4.11) and k = χ(Jm+1) is as given in (4.12).
With this arrangement, Am
n can be written as Amn = Am,n;1 Am,n;2 · · · Am,n;q Am,n;q+1 Am,n;q+2 · · · Am,n;2q .. . ... . .. ... Am,n;q(q−1)+1Am,n;q(q−1)+2· · · Am,n;q2 , where Am,n;α= qm−1 X k=1 A(k)m,n;α. Moreover, Xm,n;α= (A (k)
m,n;α)t, where 1 ≤ k ≤ qm−1and Xm,n;αis a qm−1-vector
that is composed of all elementary patterns in Am,n;α. The ordering matrix Xm,n
of Am n is now defined by Xm,n= Xm,n;1 Xm,n;2 · · · Xm,n;q Xm,n;q+1 Xm,n;q+2 · · · Xm,n;2q .. . ... . .. ... Xm,n;q(q−1)+1Xm,n;q(q−1)+2· · · Xm,n;q2 .
And Xm,n+1;βcan be reduced to form X2,n;βby a multiplication with reduction
matrices Cm;αβ. The reduction operator Rmis defined as follows.
Definition 4.1 For m ≥ 2, define
Cm= Cm;1,1 Cm;1,2 · · · Cm;1,q2 Cm;2,1 Cm;2,2 · · · Cm;2,q2 .. . ... . .. ... Cm;q2,1Cm;q2,2· · · Cm;q2,q2 = Sm;1,1 Sm;1,2 · · · Sm;1,q Sm;1,q+1 Sm;1,q+2 · · · Sm;1,2q .. . ... . .. ... Sm;1,q(q−1)+1Sm;1,q(q−1)+2· · · Sm;1,q2 · · · Sm;q,1 Sm;q,2 · · · Sm;q,q Sm;q,q+1 Sm;q,q+2 · · · Sm;q,2q .. . ... . .. ... Sm;q,q(q−1)+1Sm;q,q(q−1)+2· · · Sm;q,q2 .. . . .. ... Sm;q(q−1)+1,1 Sm;q(q−1)+1,2 · · · Sm;q(q−1)+1,q Sm;q(q−1)+1,q+1 Sm;q(q−1)+1,q+2 · · · Sm;q(q−1)+1,2q .. . ... . .. ... Sm;q(q−1)+1,q(q−1)+1 Sm;q(q−1)+2,q(q−1)+1 · · · Sm;q(q−1)+1,q2 · · · Sm;q2,1 Sm;q2,2 · · · Sm;q2,q Sm;q2,q+1 Sm;q2,q+2 · · · Sm;q2,2q .. . ... . .. ... Sm;q2,q(q−1)+1 Sm;q2,q(q−1)+2 · · · Sm;q2,q2 (4.13)
where
Cm;αβ= (B2;α◦ ˆ⊗ m−2
B2) ◦ (Eqm−2×qm−2⊗ ˜A2;β). (4.14)
Like Theorem 2.4, Cm+1;αβ can be obtained in terms of Cm;γβ.
Theorem 4.2 For any m ≥ 2 and 1 ≤ α, β ≤ q2
Cm+1;αβ= aα;1Cm;1,β aα;2Cm;2,β · · · aα;qCm;q,β aα;q+1Cm;q+1,β aα;q+2Cm;q+2,β · · · aα;2qCm;2q,β .. . ... . .. ... aα;q(q−1)+1Cm;q(q−1)+1,βaα;q(q−1)+2Cm;q(q−1)+2,β · · · aα;q2Cm;q2,β . Denote by 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;q A(k)m,n+1;α;q+1 A(k)m,n+1;α;q+2 · · · A(k)m,n+1;α;2q .. . ... · · · ... A(k)m,n+1;α;q(q−1)+1A(k)m,n+1;α;q(q−1)+2· · · A(k)m,n+1;α;q2 and Xm,n+1;α;β = (A (k) m,n+1;α;β) t where A(k) m,n+1;α;β is a linear combination of
A(l)m,n;γ. Now, Theorem 2.5 can be generalized to the following Theorem.
Theorem 4.3 For any m ≥ 2 and n ≥ 2, let Sm;αβ be as given in (4.13) and
(4.14). Then Xm,n+1;α;β= Sm;αβXm,n;β.
Appendices.
In appendices, some results in matrix theory are recalled, in particular, Perron-Frobenius Theorem, Jordan canonical form theorem and related results. For the references, see [25], [26], [44].
The following notions are required:
Definition A.1 A m × m matrix M = [Mij] is nonnegative if Mij ≥ 0 for
all 1 ≤ i, j ≤ m, and is positive if Mij > 0 for all 1 ≤ i, j ≤ m. M is called
irreducible (or strongly connected) if for each 1 ≤ i, j ≤ m, a k = k(i, j) ≥ 1 exists such that (Mk)
ij > 0. M is primitive (or essential positive) if it is
nonnegative and irreducible with only one eigenvalue of maximum modulus. M is known to be primitive if and only if there exists k ≥ 1 such that Mk > 0, i.e., M is essential positive.
First Perron-Frobenius Theorem is recalled.
Perron-Frobenius Theorem Suppose that M is a nonnegative and irreducible m × m matrix. Then a real positive eigenvalue ρ(M) = λ1 exists which is
(al-gebraic) simple with maximum modulus, i.e., if λ is any eigenvalue of M, then ρ(M) ≥ |λ|. Then eigenvector U of ρ(M) can be chosen with all entries strictly positive.
Furthermore, if M is primitive with Ml> 0, then (ρ(M)−1M)n → L as n → ∞,
where L = U Vt
, MU = ρ(M)U and Mt
V = ρ(M)V, U > 0, V > 0 and UtV = 1. In particular, if X is any unit M vector, then
MnX ||MnX|| →
U
||U || as n → ∞ and positive constants c1 and c2 exist such that
c1ρ(M)n≤ ||MnX|| ≤ c2ρ(M)n (A.1)
for all n ≥ l, where ||V || =
m
X
j=1
|vj| for vector V = (v1, · · · , vm).
If M ≥ 0 is irreducible but imprimitive, then the following results hold.
Theorem A.2 Suppose that M ≥ 0 is irreducible and the set S = {λ1 =
ρ(M), λ1, λ2, · · · , λk} of eigenvalues of maximum modulus has exactly k
dis-tinct elements λ ∈ S is algebraic simple and
S = {e2πil/kρ(M) : l = 0, 1, · · · , k − 1} (A.2) Furthermore, there is a permutation matrix P exists such that
PMPt= 0 M12 0 · · · 0 0 0 M23· · · 0 .. . 0 · · · 0 Mk−1,k Mk1 0 · · · 0 0 , (A.3)
where Mij denote square matrix with the same order m/k, and
Mk = diag(N1, N2, · · · , Nk), (A.4)
where
Nj= Mj,j+1· · · Mk−1,kMk,1· · · Mj−1,j (A.5)
is primitive for each 1 ≤ j ≤ k.
The reducible matrix can be transformed into irreducible normal form as follows. Theorem A.3 Assume that M ≥ 0 and is reducible. Then a permutation matrix P exists such that M can be transformed into a block upper triangular form
PMPt= M1M12· · · M1r 0 M2 · · · M2r 0 0 . .. 0 0 0 Mr , (A.6)
where Mj are square matrices that is either irreducible or zero. The eigenvalues
σ(M) of M satisfies σ(M) = C [ j=1 σ(Mj). (A.7)
The Jordan canonical form theorem is also needed.
Theorem A.4 Let M be a m × m real matrix. A non-singular matrix P exists such that PMPt= Jn1(λ1) 0 Jn2(λ2) . .. 0 Jnk(λk) , (A.8)
and n1+ n2+ · · · + nk= n where the Jordan block J (λ) is
J (λ) = λ 1 0 . .. . .. . .. 1 0 λ .
Note that the non-singular matrix P can be chosen such that
m
X
i=1
Pij = 1. (A.9)
From Perron-Frobenius Theorem, the following results are easily verified. Theorem A.5 If M ≥ 0 is primitive then
lim n→∞ log |Mn| n = limn→∞ log tr(Mn) n = log ρ(M). (A.10)
If M ≥ 0 is irreducible and imprimitive of the form (A.3), then
lim sup n→∞ log |Mn| n = limn→∞ log tr(Mnk) nk = log ρ(M), (A.11) and lim sup n→∞ log tr(Mn) n = limn→∞ log tr(Mnk) nk = log ρ(M). (A.12) If M ≥ 0 is reducible and of the form (A.6), then
lim sup n→∞ log |Mn| n = lim supn→∞ log tr(Mn) n = max{(ρ(M1), · · · , ρ(Mr))}. (A.13) For the proofs involving |Mn|, see, for example, [44]. The proofs involving tr(Mn)
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