International Journal of Bifurcation and Chaos, Vol. 14, No. 1 (2004) 309–319 c
World Scientific Publishing Company
ON THE SPATIAL ENTROPY OF TWO-DIMENSIONAL
GOLDEN MEAN
JONQ JUANG
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan [email protected]
SHIH-FENG SHIEH
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan [email protected]
Received August 12, 2002; Revised October 21, 2002
The aim of this paper is to derive a sharper lower bound for the spatial entropy of two-dimensional golden mean.
Keywords: Spatial entropy; subshift of finite type; cellular neural networks; two-dimensional golden mean.
1. Introduction
The dynamical properties of one-dimensional subshifts of finite type (Markov shifts) are well un-derstood. However, not much is known for a general theory of higher dimensional subshifts. For instance, the spatial entropy of subshifts of finite type is known to be the logarithm of the largest eigen-value of its corresponding transition matrix. On the other hand, very little is known on the spa-tial entropy of higher dimensional subshifts. Even the “trivially” looking problem of the spatial en-tropy of two-dimensional golden mean H = V = 1 1
1 0
remains open (see e.g. [Schmidt, 1990]). For the difficulties associated with higher dimen-sional Markov shifts, we refer to [Schmidt, 1990]. The two-dimensional golden mean problem cor-responds to fill Z2 lattice with {1, 2} with the following rules
∗ 1
1 ∗ 2 1 ,
where ∗ indicates no restriction on what 1 can be adjacent to. Such a pattern can also be generated by cellular neural networks (CNNs) (see e.g. [Chua
& Yang, 1988a, 1988b; Juang & Lin, 2000] and the work cited therein). More specifically, consider CNNs of the form dxij dt = −xij+ z + X |k|≤1, |l|≤1 ak,lf(xi+k,j+l), (i, j) ∈ Z , xi,j(0) = x0i,j.
Here the nonlinearity f is a piecewise-linear func-tion of the form
f(x) = 1
2(|x + 1| − |x − 1|) .
The numbers ak,l, |k| ≤ 1, |l| ≤ 1, k, l ∈ Z are
arranged in a 3 × 3 matrix form, which is called a space-invariant A-template A= a−1,1 a0,1 a1,1 a−1,0 a0,0 a1,0 a−1,−1 a0,−1 a1,−1 . Now, set A= 0 aε 0 aε aε aε
0 aε 0 .
309
By choosing z, a and ε approximately, say (z, a) ∈ [5, 1]ε, where aε < 0 (see Theorem 3.5 of [Juang
& Lin, 2000]), we see, via Lemma 4.1 of [Juang & Lin, 2000], that any positively saturated cell, de-noted by 1, can be adjacent to either positively saturated cell or negatively saturated cell, denoted by 2. Moreover, any negatively saturated cell must be adjacent to at least four positively saturated cells. These mosaic patterns are exactly generated by two-dimensional golden mean. It is easy to see that the entropy h of such problem satisfies the inequality
1
2 log 2 < h < log
1 +√5
2 ,
but the precise value of h is still not known (see e.g. [Markely & Paul, 1981a, 1981b]). In this pa-per, we will give a nontrivial lower bound of h. We also note that most of the discussion of higher di-mensional Markov shifts is restricted to examples of special nature (see e.g. [Baxter, 1982; Kaste-leyn, 1961; Lieb, 1967; Schmidt, 1990; Temperley & Lieb, 1971]). We conclude this introductory section by summarizing the organization of this paper. In Sec. 2, we recall some needed notations, definitions and known results. In Sec. 3, we define a class of Markov measures associated with a transition ma-trix A. Such class of the measures is then used to compute the measure theoretic entropy of the shift map σA. In Sec. 4, we combine the results from
Secs. 2 and 3 to get a nontrivial lower bound of the spatial entropy of two-dimensional gold mean.
2. Preliminaries
To make the paper self-contained, we recall some definitions and results. Let N be a positive integer with N ≥ 2, let S = {1, 2, . . . , N}. Denote by Zd
the integer lattice on Rd where d ≥ 1 is a positive
integer representing the lattice dimension. The set of all functions u : Zd → S is denoted by SZd
. For α∈ Zd, we write u(α) as uα. The kth shift operator
on SZd
is defined by
(σku)α = uα+ek,
where α ∈ Zd and e
k = (0, . . . , 0, 1, 0, . . . , 0) is the
usual unit vector in the direction of the kth coordi-nate. For convenience we also write ΣN = SZ
d
. We define a metric d on ΣN as follows.
d(u, v) = X k∈Zd δ(uk, vk) 3|k| , (1) where δ(i, j) = 0 , if i = j , 1 , if i 6= j ,
and |k| = max{k1, k2, . . . , kd} for k =
(k1, k2, . . . , kd) ∈ Zd. The space ΣN with the shift
operators, (ΣN; σ1, . . . , σd), is called the symbol
space on N symbols, or the full N-shift space.
Definition 2.1. An N ×N matrix A = (aij) is said
to be a transition matrix if
(i) aij= 0 or 1 for all 1 ≤ i, j ≤ N,
(ii) P
iaij≥ 1 for all 1 ≤ j ≤ N,
(iii) P
jaij ≥ 1 for all 1 ≤ i ≤ N.
Definition 2.2. Given d transition matrices, Ak =
(ak ij)N×N, k = 1, . . . , d, let ΣA1,...,Ad = {u ∈ ΣN|a k uα,uα+ek = 1, for all α ∈ Zd, 1 ≤ k ≤ d} .
which determines all the admissible transitions be-tween symbols 1, . . . , N . Each element in ΣA1,...,Ad
is called a pattern. The shift operators σ1, . . . , σd
restricted on ΣA1,...,Ad are called the subshifts of
finite type for matrices A1, . . . , Ad.
We shall write ΣA1,...,Ad as Σd provided no
confusion arises. It is clear that Σd is closed with
respect to the metric defined in (1) and translation invariant, that is,
σk(Σd) = Σd
for all 1 ≤ k ≤ d. To measure the complexity of Σd, we compute the growth rate of the number of
patterns on a parallelepiped of size N1×N2×· · · Nd
on the lattice as N1, . . . , Nd go to infinity.
Definition 2.3. The spatial entropy h(Σd) is
defined by h(Σd) = lim N1,...,Nd→∞ log ΓN1,...,Nd(Σd) N1, N2· · · Nd . (2)
Here ΓN1,...,Nd(Σd) is the number of distinct
pat-terns that one observes among the elements of Σdby
restricting one’s observation to a parallelepiped of size N1×N2×· · · Ndon the lattice. The limit in (2) is
well-defined and exists (see e.g. [Chow et al., 1996]). Moreover, if Σdis replaced by U where U ⊂ ΣN and
satisfies σp1 1 (U) = σ p2 2 (U) = · · · = σ pd d (U) = U
for some (p1, p2, . . . , pd) ∈ Zd, the well-definedness
and existence of the limit in (2) remain true (see e.g. [Juang et al., 2002]).
Theorem 2.1 (see e.g. Theorem VIII-1.9 of [Robinson, 1993]). For d = 1, let A be a transition matrix on N symbols, so A is N × N. Then
h(ΣA) = log λ1
where λ1 is the dominant eigenvalue of A.
Definition 2.4. Let f : X → X be a continuous
map on the space X with metric d. For n a pos-itive integer and ε > 0, a set S ⊂ X is called (n, ε)-separated for f provided for every pair of dis-tinct points x, y ∈ S, there is at least one k with 0 ≤ k < n such that d(fk(x), fk(y)) > ε.
The number of different orbits of length n (as measured by ε) is defined by
r(n, ε, f ) = max{#(S)|S ⊂ X is a (n, ε) − separated set for f} ,
where #(S) is the number (cardinality) of elements in S. To measure the growth rate of r(n, ε, f ) as n increases, we define h(ε, f ) = lim sup n→∞ log r(n, ε, f ) n . If r(n, ε, f ) = enτ, then h(ε, f ) = τ . Thus, h(ε, f )
means the “exponent” of the manner in which r(n, ε, f ) grows with respect to n. Finally, we con-sider the way that h(ε, f ) varies as ε goes to zero, and define the topological entropy of f as
h(f ) = lim
ε→0h(ε, f ) .
We note that for 0 < ε1 < ε2, r(n, ε2, f) ≥
r(n, ε2, f), so h(ε, f ) increases as ε decreases and,
hence, the limit defining h(f ) exists. If f is C1
on a compact space, then it has been proven that h(f ) < ∞ (see e.g. [Bowen, 1971, 1988]).
The following theorem shows that h(σA) =
h(ΣA).
Theorem 2.2 (see e.g. Theorem VIII.1.9 of
[Robinson, 1993]). Let σ : ΣN → ΣN be the full
shift of N symbols (either one side of two). Assume X ⊂ ΣN is a closed invariant subset. Let Γn be the
number of words of length n in X, i.e. Γn= #{(s0, . . . , sn−1)|sj = xj, f or 0 ≤ j ≤ n for some x ∈ X} . Then h(σ|X) = lim sup n→∞ log Γn n .
We also need to recall two recursive formu-las, which was derived in [Juang et al., 2000] for computing the spatial entropy of two-dimensional golden mean. In the following, we first introduce some notations and concepts.
Given a transition matrix A = (ai,j)n×n. A
word ω = (ω0, ω1, . . . , ωk−1) of length k is called
admissible (allowable) if aωj−1,ωj = 1 for j =
1, 2, . . . , k −1. Let A be a transition matrix. The set of admissible words of length m whose first symbol is ω0 is to be denoted by ω(ω0, m; A). Set
ω(m; A) = set of all admissible words of length m
= [
1≤ω0≤n
ω(ω0, m; A) .
To save notation, the transition matrices A1and A2
introduced in Definition 2.2 will be denoted by H = (hi,j)N×N and V = (vi,j)N×N, respectively, called
horizontal and vertical transition matrices. Then Card(ω(m; H)) = Pn
i,j=1(Hm−1)i,j =: Nm. Here
H0 = identity matrix. Using these Nm symbols, we
may define a transition matrix TH,V(m) = (t(m)i,j ) of size Nm × Nm as follows. We begin with giving a
lex-icographic order for elements in ω(m; H). Specifi-cally, let s = (s1s2· · · sm) and p = (p1p2· · · pm) ∈
ω(m; H), and suppose that j is the smallest index for which sj 6= pj, then we define
s< p if sj < pj. (3)
With such ordering, the sets ω(m; H) and {1, 2, 3, . . . , Nm} can have an association that is one
to one, onto and order preserving.
Definition 2.5. If s and p in ω(m; H) are
associ-ated with positive integers k and l, where 1 ≤ k, l≤ Nm respectively, then we define the (k, l)-entry
or (s, p)-entry of TH,V(m) as t(m)s,p = t(m)k,l = vs1,p1· vs2,p2· · · vsm,pm := m Y i=1 vsi,pi, (4) i.e. t(m)k,l = 1 provided that for all 1 ≤ i ≤ m, the words si
pi
are admissible with respect to V. Otherwise, t(m)k,l = 0. For convenience, we shall use t(m)s,p to denote t(m)k,l . We shall call TH,V(m) the
m-transition matrices with respect to the horizontal and vertical transition matrices H and V, or for short, the m-transition matrix. If we start out with Int. J. Bifurcation Chaos 2004.14:309-319. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
a lexicographic order for elements in ω(m; V), we shall obtain the so-called m-transition matrix TV,H(m) with respect to V and H.
The relationship between m-transition matrix TH,V(m) and h(ΣH,V) is given in the following.
Proposition 2.1 (Proposition 2.1 of [Juang et al.,
2000]). Let TH,V(m) be the m-transition matrix with respect to H and V. Let ρ(TH,V(m)) be the maximal eigenvalue of TH,V(m) = (t(m)s,p), where t(m)s,p are given
in (4 ). Then h(Σ2) = lim m→∞ log ρ(TH,V(m)) m , (5) where Σ2= ΣH,V.
The following recursive formula for construct-ing TH,V(m) can also be found in [Juang et al., 2000]. Note first that TH,V(m) can be written as the following block structure
TH,V(m) = (Ti,j(m)) , 1 ≤ i, j ≤ n , (6) where Ti,j(m) is a matrix of size Card(ω(i, m; H))× Card(ω(j, m; H)). Let 1 ≤ k ≤ Card(ω(i, m; H)) and 1 ≤ l ≤ Card(ω(j, m; H)). Via the lexi-cographic order defined in (3), there exist s ∈ ω(i, m; H) and p ∈ ω(j, m; H) whose associated numbers are k and l, respectively. Then the (k, l)-entry, or simply (s, p)-l)-entry, of the matrix Ti,j(m) is 1 provided that for all 1 ≤ r ≤ m, spr
r
is an ad-missible word of size two with respect to vertical transition matrix V. Otherwise, the entry is zero. We are now ready to state the following result.
Theorem 2.3. Let TH,V(m+1) and TH,V(m) be,
respectively, (m + 1)- and m-transition matrix with respect to horizontal and vertical transition matrices H= (hi,j) and V = (vi,j). Let α(i) = {q ∈ N : 1 ≤
q ≤ n, hi,q = 1} and Card(α(i)) = αi. Moreover,
we set α(i) = {i1, i2, . . . , iαi} with the following
or-der i1 ≤ i2 ≤ · · · ≤ iαi. Then T (m) H,V can be defined recursively as follows: TH,V(1) = V , and TH,V(m+1) = (Tk,l(m+1))n×n, 1 ≤ k, l ≤ n . (7a)
Here the block matrices Tk,l(m+1) are of following form Tk,l(m+1) = vk,l Tk(m)1,l1 Tk(m)1,l2 · · · Tk(m)1,l αl Tk(m)2,l1 Tk(m)2,l2 · · · Tk(m)2,l αl .. . ... . .. ... Tk(m) αk,l1 T (m) kαk,l2 · · · T (m) kαk,lαl , (7b) where ki ∈ α(k), li ∈ α(l), Tk,l(m+1) and Tk(m)v,lq,
1 ≤ v ≤ αk and 1 ≤ q ≤ αi, are defined as
in (6 ).
We next recall some basic definitions and well-known results from ergodic theory. Let (X, B, m) be a measure space. Here B denotes the σ-algebra of all measurable sets in X and m denotes the measure on X. Let f : X → X be a measurable function. f is said to be measure preserving with respect to the measure m if m(S) = m(f−1(S)) for all S ∈ B. Here m is called an invariant measure for f .
Definition 2.6. Let f be measure preserving on
(X, B, m). A set S ∈ B is called f-invariant if f−1(S) = S. f is said to be ergodic if every f -invariant set has measure 0 or full measure.
We are now ready to state a well-known theo-rem in ergodic theory.
Theorem 2.4 (Birkhoff Ergodic Theorem (see
e.g. [Mane, 1983]). Let f be measure preserving on (X, B, m) and g be in L1(X).
(1) There exists an integrable function g∗ such that lim n→∞ 1 n n−1 X k=0 g(fk(x)) = g∗(x) (8) for almost every point x ∈ X.
(2) For all k ∈ N, g∗(fk(x)) = g∗(x) a.e . (3) If m(X) = 1, then Z X gdm= Z X g∗dm . (9)
Here the left side of (8) is called the ergodic av-erage. In the following corollary we see that under the condition of ergodicity, the ergodic average is equal to the “Riemann sum” of R
Xgdm.
Theorem 2.5 (see e.g. [Mane, 1983]). If f is er-godic and m(X) = 1, lim n→∞ 1 n n−1 X k=0 g(fk(x)) = Z X gdm a.e . (10)
The next part of our preliminaries is about the measure theoretic entropy.
Definition 2.7. Let (X, B, m) be a measure space
and P be a partition of X, the entropy of partition P is defined to be
H(P) = −X
P∈P
m(P ) log m(P ) .
Let f : X → X be measure preserving. The entropy of f with respect to P is defined by
h(f, P) = lim n→∞ 1 nH n−1 _ j=0 f−j(P) . (11) Here the notation Wn−1
j=0f−j(P) denotes the
parti-tion whose elements are of the form A0∩ · · · ∩ An−1
for Ai ∈ f−j(P), i = 0, . . . , n − 1, satisfying
m(A0 ∩ · · · ∩ An−1) 6= 0. The measure theoretic
entropy of f is then given by hm(f ) = sup
P: partition
h(f, P) .
Proposition 2.2 (Proposition IV.3.2 of [Mane,
1983]). The limit in (11) is well defined and exists. Let A be an n×n transition matrix. P = (pij) ∈
Mn×n(R) is said to be a stochastic matrix associated
with A if
1. pij= 0 if and only if aij= 0 for 1 ≤ i, j ≤ n.
2. 0 ≤ pij≤ 1 for all 1 ≤ i, j ≤ n.
3. P
jpij = 1.
Clearly, there exists a left eigenvector q = (q1, . . . , qn)T satisfying the following:
qTP= qT, (12a) and n X i=1 qi= 1 . (12b)
We define a Markov measure µ = µP,q
associ-ated with (P, q) by
µ(C(i0, i1, . . . , ik)) = qi0pi0,i1· · · pik−1,ik, (13)
where C(i0, i1, . . . , ik) = {(j0, j1, . . .) ∈ ΣA|j0 =
i0, . . . , jk = ik} is called a cylinder.
Proposition 2.3 (see e.g. Theorem I-10.1 of
[Mane, 1983]). µ = µP,q is an invariant measure
of the Markov shift σA.
Theorem 2.6. Let A be an n × n transition
ma-trix and µP,q = µ be the invariant Markov measure
defined by (P, q) associated with A. Then
(i) (see e.g. p. 221 of [Mane, 1983 ]) hµ(σA) =
−P
ijqipij log pij.
(ii) [Parry, 1964] If σA is topological mixing, then
for any invariant measure µ0,
hµ0(σA) ≤ log λ1
where λ1 is the dominant eigenvalue of A.
Moreover, there is a unique measure such that the equality attains.
It has been shown in Theorem 2.1 that htop(σA) = log λ1. Theorem 2.6 states that for
topological mixing Markov shifts, the topological entropy is the maximal of measure theoretic en-tropy. This is also true for a general class of maps [Misiurewicz, 1976].
3. Shift Map and Entropy
Let A be an n × n transition matrix, and let P be a set of vectors satisfying the following
P =
x= (x1, x2, . . . , xn)T :
xi>0 for all i and n X i=1 xi = 1 . (14) Given x ∈ P, we set si:= (Ax)i xi ,
where (Ax)i is the ith-component of vector Ax.
Since diag(s−11 , . . . , s−1n )Ax = x, there exists a left eigenvector y satisfying the following.
yTdiag(s−11 , . . . , s−1n )A = yT (15a) and
yTx= 1 . (15b)
We note that if, in addition, A is symmetric, then yT = x
TA
xTAx.
Now, if we set
Px= (diag Ax)−1Adiag x , (16a)
where diag Ax = diag((Ax)1, . . . ,(Ax)n) and
diag x is also defined similarly, and qTx= yTdiag x = x TA(diag x) xTAx if A is symmetric . (16b) Clearly, Px is a stochastic matrix associated with
A and qx is the left eigenvector of Px satisfying
(12). We are now ready to state the main result of this section.
Theorem 3.1. Let A be an n × n transition
ma-trix which is irreducible. Let x ∈ P, and Px and
qxT are defined as in (16a) and (16b), respectively.
Let µPx,qx = µx be the Markov measure given as in
(13). Then (1)
hµx = y
T log diag(s
1, s2, . . . , sn)x . (17a)
If, in addition, A is symmetric, then hµx =
xTA log diag(s1, s2, . . . , sn)x
xTAx . (17b)
(2)
hµx ≤ log λ for any x ∈ P . (18)
Here λ is the maximal eigenvalue of A. The equality can be achieved by choosing x to be the left eigenvector of A associated to eigenvalue λ with Pn
i=1xi = 1.
Proof. We first prove (17). Let P = (pij), and so
(pij) = ((Ax)xj iaij). Set ˜P = (pij log pij), and e =
(1, . . . , 1)T, it follows from (16a), (16b) and
Theo-rem 2.6(i) that
hµP,q(σA) = −q TPe˜ = −yT(diag x) ˜Pe. (19) Now, ˜ Pe= xj (Ax)i aij log xj (Ax)i aij n×n e = (diag Ax)−1 aij log xj (Ax)i aij n×n diag(x1, . . . , xn)e = (diag Ax)−1 aij log xj (Ax)i aij n×n x. (20)
Moreover, we have that − aij log xj (Ax)i aij n×n
= −(aij log aij)n×n+ (aij log(Ax)i)n×n
−(aij log xj)n×n. (21)
Since either aij = 0 or aij = 1, we see that
aij log aij = 0. We also note that
(aij log(Ax)i)n×n= log(diag Ax)A
and
(aij log xj)n×n= A log diag x .
Substituting (21) into (20), we get that − ˜Pe= (diag Ax)−1 log(diag Ax)Ax
− (diag Ax)−1A(log diag x)x . (22)
Here log A = (log aij). To further simplify (19), we
note that
yT diag x(diag Ax)−1A= yT (23) and
yT diag x(diag Ax)−1 log(diag Ax)Ax
= yT log(diag Ax)(diag x)(diag Ax)−1Ax = yT log(diag Ax)(diag x)e
= yT log(diag Ax)x . (24)
It then follows from (22)–(24) and (19) becomes hµP,q(σA) = y
T log(diag Ax)x − yT(log diag x)x
= yT log(diag(s1, . . . , sn))x .
The inequality in (18) is a direct consequence of Theorem 2.6(ii). A direct calculation would yield Int. J. Bifurcation Chaos 2004.14:309-319. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
the last assertion of the theorem. We thus complete
the proof of the theorem.
4. Two-Dimensional Golden Mean
In this section, we study the two-dimensional golden mean, that is, the two-dimensional subshifts of finite type with H = V = 1 1
1 0
. Recall in Sec. 2 that T(m)H,V, (resp. T(m)H,V) represents the m-transition ma-trices with respect to H and V (resp. V and H). Since H = V, we see that T(m)V,H = T(m)H,V(:= T(m)). Applying Theorem 2.3, T(m) can be written recur-sively as follows: T(1)= 1 1 1 0 = T(0)1,1 T(0)1,2 T(0)2,1 T(0)2,2 and T(m+1) = T(m)1,1 T(m)1,2 T(m)2,1 0 T(m)1,1 T(m)2,1 T(m)1,1 T(m)1,2 0 . (25)
Let an be the size of T(n), then an satisfies the
fol-lowing recursive formula.
an+1 = an+ an−1 (26a)
and
a1 = 2, a2 = 3 . (26b)
Proposition 4.1. For each n ≥ 1, T(n) is
sym-metric and irreducible.
Proof. It is easy to see that T(n) is symmetric for
all n ≥ 1. We next prove that each T(n) is even-tually positive. Since T(1) and T(2) are eventually
positive, we assume that T(n−1) and T(n) are even-tually positive for some n. Then there exists m > 0 such that
(T(n))m > En and (T(n−1))m> En−1.
Here En = (1)an×an. We observe in (25) that
T(n+1)11 = T(n), thus the matrix multiplication gives
(T(n+1))m+1> En En T(n)11 T(n)21 ! T(n)11 T(n)12 En En−1 >0 .
An inductive argument then leads to the assertion of the proposition. Letting en = (1, . . . , 1)T ∈ Ran, then hµen as defined in (17b) becomes hµen = λn= eTnT(n) log(diag eT nT(n))en eT nT(n)en . (27a) Moreover, if we let eT nT(n) =: v(n) = (v (n) i ) ∈ Ran and sn = Pan
i,j=1(T(n))ij be the sum taken over all
entries of T(n). Then λn= an X i=1 vi(n) log vi(n) sn . (27b)
We remark that sn satisfy the following recursive
formulas:
sn+1 = 2sn+ sn−1 (28a)
and
s1 = 3, s2 = 7 . (28b)
Applying Theorem 3.1 and (27), and Proposi-tion 2.1 we obtain the following lower bound for h(ΣH,V) of two-dimensional golden mean.
Theorem 4.1 h(ΣH,V) ≥ lim sup n→∞ an X i=1 v(n)i log v(n)i nsn . (29) The remainder of the section is to compute the sum of the infinite series as given in (29).
We first observe that v(1) = (2, 1)
v(2) = (3, 2, 2)
v(3) = (5, 3, 4, 3, 2)
v(4) = (8, 5, 6, 6, 4, 5, 3, 4)
v(5) = (13, 8, 10, 9, 6, 10, 6, 8, 8, 5, 6, 6, 4). Int. J. Bifurcation Chaos 2004.14:309-319. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
To derive a recursive formula for v(n), we first write v(n) as
v(n) = (u(n+1), u(n)) .
Here u(n+1) and u(n) are row vectors whose
dimen-sions are 1 ×an−1and 1 ×an−2, respectively. For
in-stance, v(4)= (u(5), u(4)), where u(5) = (8, 5, 6, 6, 4)
and u(4) = (5, 3, 4). Clearly, u(n+1) can be recur-sively defined as u(n+1)= (u(n)+ v(n−2),2u(n−1)) = (u(n)+ (u(n−1), u(n−2)), 2u(n−1)) (30) with u(1) = 1 , u(2) = 2 , u(3)= (3, 2) . For example, u(6)= (u(5)+ (u(4), u(3)), 2u(4)) = ((8, 5, 6, 6, 4) + (5, 3, 4, 3, 2), 2(5, 3, 4)) = (13, 8, 10, 9, 6, 10, 6, 8) .
We are ready to state the following useful proposi-tion.
Proposition 4.2.
u(n)= (an−1, an−2u(1), . . . , an−i−1u(i), . . . , a1u(n−2)) .
(31)
Proof. Let n = 3, we see that 31 is clearly
satis-fied. Suppose 31 is true for k = 3, . . . , n. Then u(n)+ (u(n−1), u(n−2)) = (an−1, an−2u(1), . . . , an−i−1u(i), . . . , a2u(n−3), a1u(n−2)) + (an−2, an−3u(1), . . . , an−i−2u(i), . . . , a1u(n−3), u(n−2)) = (an, an−1u(1), . . . , an−iu(i), . . . , a3u(n−3),(1 + a1)u(n−2)) = (an, an−1u(1), . . . , an−iu(i), . . . , a3u(n−3), a2u(n−2)) . Thus, u(n+1)= (u(n)+ (u(n−1), u(n−2)), 2u(n−1)) = (an−1, an−2u(1), . . . , an−i−1u(i), . . . , a2u(n−2), a1u(n−1)) . To investigate P iv (n) i log v (n) i , we define L : RN → R, as L(x) = N X i=1 xi log xi,
where x = (x1, . . . , xN)T. Clearly, for any c ∈ R, we
have that L(cx) = N X i=1 xi ! clog c + cL(x) . (32) Let u(n) = (u(n)1 , u(n)2 , . . . , u(n)an−1). We set αn= an−1 X i=1 u(n)i , (33a) βn= anlog an, (33b) Ln= L(u(n)) , (33c) pn= n−2 X i=1 an−i−1Li, (33d) qn= βn−1+ n−2 X i=1 βn−i−1αi. (33e)
Applying (32) and (33a), we have that L(u(n)) = an−1 log an−1+
n−2
X
i=1
L(an−i−1u(i))
= pn+ qn. (34)
Proposition 4.3. αn, pn and qn satisfy the
follow-ing recursive formulas
(i) αn+1= 2αn+ αn−1, α1 = 1 , α2 = 2 , (35a)
(ii) pn+1 = pn+ 3pn−1+ pn−2+ 2qn−1
+ qn−2, (35b)
(iii) qn+1 = 2qn+ qn−1
+(βn− βn−1− βn−2) . (35c)
Proof. Clearly, α1 = 1, α2 = 2, α3 = 5. Hence
α3 = 2α2 + α1. Assume that αk+1 = 2αk + αk−1
holds for 2 ≤ k ≤ n − 1. Then αn+1 =
an
X
i=1
u(n)i = αn+ 3αn−1+ αn−2. (36)
We have used (30) to justify the last equality in (36). Using the inductive hypothesis, we get that αn+1= 2αn+ αn−1. The proof of the first assertion
of the proposition is thus complete. To prove (ii), we see that pn+1− pn = (an−1L1+ . . . + a3Ln−3+ a2Ln−2+ a1Ln−1) −(an−2L1+ . . . + a2Ln−3+ a1Ln−2) = an−3L1+ . . . + a1Ln−3+ Ln−2+ 2Ln−1 = pn−1+ Ln−2+ 2Ln−1 = pn−1+ pn−2+ qn−2+ 2(pn−1+ qn−1) = 3pn−1+ pn−2+ 2qn−1+ qn−2.
This proves (ii). To prove (iii), we note that 2qn+ qn−1= 2βn−1+ 2α1βn−2+ 2α2βn−3+ . . . + 2αn−2β1+ βn−2+ α1βn−3+ . . . + αn−3β1 = 2βn−1+ 3βn−2+ α3βn−3+ . . . + αn−1β1, and qn+1= βn+ βn−1+ 2βn−2+ α3βn−3+ . . . + αn−1β1. Hence, qn+1− 2qn− qn−1= (βn− βn−1− βn−2) as asserted.
Proposition 4.4. Let λn be the quantity as given
in (27b), then λn= L(v(n)) sn = (pn+1+ pn) + (qn+1+ qn) sn .
Proof. It follows directly from Theorem (17b) and
(34).
To evaluate lim supn→∞(pn+1+pn)+(qn+1+qn)
nsn ,
we need the following proposition:
Proposition 4.5. Let λ = 1 +√2. Then the
follow-ing holds.
(i) λ, −1λ and −1 are the characteristic roots of γn.
Here γn+1= γn+3γn−1+γn−2 with γ2 = γ1= 1
and γ2 = 0.
(ii) There are constants cs, ds, cα, dα, cγ, dγ, eγ
for which sn= csλn+ ds −1λ n , αn= cαλn+ dα −λ1 n , and γn= cγλn+ dγ −λ1 n + eγ(−1)n.
Here sn and αn are defined in (28) and (33),
respectively.
Proposition 4.6. The following limit exists.
lim n→∞ qn λn = cα q4 λ3 + q3 λ4 + ∞ X i=4 ki λi ! =: q∗ where ki= βi− βi−1− βi−2, and cα is defined as in
Proposition 4.5 (ii).
Proof. Let A = 2 1
1 0
. Using (35c), we get that qn+1 qn = A qn qn−1 + kn 0
with initial conditions q4 q3 . Note that An= α n+1 αn αn αn−1 ,
where we assume α0 = 0. Using the variation of
constant formula, we then obtain that q n qn−1 = An−4 q 4 q3 + n−1 X i=4 An−1−i k i 0 . Hence, qn= (αn−3q4+ αn−4q3) + n−1 X i=4 αn−iki.
Applying the ratio tests, we conclude that
∞
X
i=4
ki
λi
converges. The proof of the proposition is thus complete.
Remark 4.1. We note that ki > 0 for all i ≥ 4.
Hence the partial sum qn∗ := cα
q4 λ3 + q3 λ4 + Pn i=4λkii converges upward to q∗. Corollary 4.1
(i) Given ε > 0, there exists N ∈ N such that (q∗− ε)λn< qn<(q∗+ ε)λn (37)
whenever n ≥ N.
(ii) limn→∞nsqnn = 0 and, hence,
lim n→∞ L(v(n)) nsn = lim n→∞ pn+1+ pn nsn . Proposition 4.7. lim n→∞ pn nλn = q∗cγ(2λ−1+ λ−2) ,
where cλ is defined as in Proposition (4.5) (ii).
Proof. Let gn= 2qn−1+ qn−2 and B =
1 3 1 1 0 0 0 1 0
! .
We see via the induction that Bn = γn+1 · ·
γn · · γn−1 · · ! . Then pn+1 pn pn−1 = B pn pn−1 pn−2 + gn 0 0 .
Now we let ε > 0 be fixed and N = N (ε) > 0 be given as in Corollary , then for n > N ,
pn pn−1 pn−2 = Bn−N pN pN−1 pN−2 + n−1 X i=N γn+1 · · γn · · γn−1 · · gi, (38) where we set γ−1= 0. Using (38), we obtain that
pn= O(λn) + cγ n−1
X
i=N
λn−igi.
It follows from (37) and (38) that, we have cγ n−1 X i=N λn−i(q∗− ε)(2λ−1+ λ−2)λi+ O(λn) ≤ pn≤ cγ n−1 X i=N λn−i(q∗+ ε)(2λ−1+ λ−2)λi + O(λn) . Hence, (q∗− ε)cγ(2λ−1+ λ−2) ≤ limn →∞ pn nλn ≤ (q∗+ ε)cγ(2λ−1+ λ−2)
Since ε is arbitrary, the assertion of the proposition holds.
We are ready to state the main result of this paper. Theorem 4.1. Let H = V = 1 1 1 0 , then h(ΣH,V) ≥ q∗cγ(2λ−1+ λ−2) 1 cs (λ + 1) ≥ q∗ncγ(2λ−1+ λ−2) 1 cs (λ + 1) =: hn.
The second inequality holds for all n ≥ 4.
We remark that the known lower and upper bounds of h(ΣH,V), where H = V = 1 1 1 0 , are log 1+2√5(≈ 0.481212) and 12log 2(≈ 0.346574), re-spectively. Our estimate in (39) gives
h(ΣH,V) ≥ h5000 ≈ 0.404089 .
References
Baxter, R. J. [1982] Exactly Solved Models in Statistical Mechanics (Academic Press, NY–London).
Bowen, R. [1971] “Entropy of group endomorphism and homogeneous spaces,” Trans. Amer. Math. Soc. 14, 23–42.
Bowen, R. [1988] “On axion a diffeomorphism (confer-ence board math. sci(confer-ence),” Trans. Amer. Math. Soc. Circuits Syst. 35, 1273–1290.
Chow, S. N., Mallet-Paret, J. & Van Vleck, E. S. [1996] “Pattern formation and spatial chaos in spatially discrete evolution equations,” Rand. Comp. Dyn. 4, 109–178.
Chua, L. O. & Yang, L. [1988a] “Cellular neural net-works: Applications,” IEEE Trans. Circuits Syst. 35, 1272–1273.
Chua, L. O. & Yang, L. [1988b] “Cellular neural networks: Theory,” IEEE Trans. Circuits Syst. 35, 1257–1290.
Juang, J. & Lin, S. S. [2000] “Cellular neural networks: Mosaic patterns and spatial chaos,” SIAM J. Appl. Math. 60, 891–915.
Juang, J., Lin, S. S. & Shieh, S. F. [2000] “The spa-tial entropy of two dimensional Markov shifts of finite type,” Int. J. Bifurcation and Chaos 10, 2129–2140. Juang, J., Shieh, S. F. & Turyn. L. [2002] “Cellular
neural networks: Space-dependent template, mosaic patterns and spatial chaos,” Int. J. Bifurcation and Chaos 12, 1717–1730.
Kasteleyn, P. W. [1961] “The statistics of dimers on lattice,” I. Physica 27, 1209–1225.
Lieb, E. H. [1967] “Residual entropy of square ice,” Phys. Rev. 162, 162–172.
Mane, R. Ergodic Theory and Differential Dynamics (Spring-Verlag).
Markely, N. G. & Paul, M. E. [1981a] “Matrix subshifts
for Zν
symbolic dynamics,” Proc. London Math. Soc. 43, 251–272.
Markely, N. G. & Paul, M. E. [1981b] “Maximal
mea-sures and entropy for Zν
subshifts of finite type,” Collection: Classical Mechanics and Dynamical
Sys-tems, Medford, MA, 1979 (Marcel Dekker, NY), pp. 136–157.
Misiurewicz, M. [1976] “A short proof of the
varia-tional principle for a ZN
+ action on a compact space,”
Ast´erisque 40, 147–187.
Parry, J. [1964] “Intrinsic Markov chains,” Trans. Amer. Math. Soc. 112, 55–66.
Robinson, C. [1993] Dynamical Systems, Stability, Symbolic Dynamics, and Chaos (CRC Press, London). Schmidt, K. [1990] Algebraic Ideas in Ergodic Theory, Conference Board Math. Science. Trans. Amer. Math. Soc. Providence, RI.
Temperley, H. N. V. & Lieb, E. H. [1971] “Relation be-tween the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems assoicated with reg-ular planner lattices: Some exact results for the ‘per-colation’ problem,” Proc. Roy. Soc. London Series A322, 251–280.
