Many systems have been studied as models for spatial pattern formation in biology, chemistry, engineering and physics. Lattices play important roles in modeling underlying spatial structures. Notable examples include models arising from biology[7, 8, 21, 22, 23, 33, 34, 35], chemical reaction and phase transitions [4, 5, 11, 12, 13, 14, 24, 41, 43], image processing and pattern recognition [11, 12, 15, 16, 17, 18, 19, 25, 40], as well as materials science[9, 20, 26]. Stationary patterns play a critical role in investigating of the long time behavior of related dynamical systems. In general, multiple stationary patterns may induce complicated phenomena of such systems.
In Lattice Dynamical Systems(LDS), especially Cellular Neural Networks (CNN), the set of global stationary solutions (global patterns) has received considerable attention in recent years (e.g.[1, 2, 6, 10, 27, 28, 29, 30, 31, 32, 36, 37]). When the mutual interaction between states of a system is local, the state at each lattice point is influenced only by its finitely many neighborhood states. The admissible (or allowable ) local patterns are introduced and defined on a certain finite lattice. The admissible global patterns on the entire lattice space are then glued together from those admissible local patterns.
More precisely, let S be a finite set of p elements (symbols, colors or letters of an alphabet). Where Zd denotes the integer lattice on Rd, and d ≥ 1 is a positive integer representing the lattice dimension. Then, function U : Zd→ S is called a global pattern. For each α ∈ Zd, we write U(α) as uα. The set of all patterns U : Zd→ S is denoted by
Σdp ≡ SZd,
i.e., Σdp is the set of all patterns with p different colors in d-dimensional lattice. As for local patterns, i.e., functions defined on (finite) sublattices, for a given d-tuple N = (N1, N2,· · · , Nd) of positive integers, let
ZN = {(α1, α2,· · · , αd) : 1 ≤ αk ≤ Nk,1 ≤ k ≤ d}
1
2 Pattern Generation Problems be an N1 × N2 × · · · Nd finite rectangular lattice. Denoted by eN ≥ N if Nfk ≥ Nk for all 1 ≤ k ≤ d. The set of all local patterns defined on ZN is denoted by
ΣN ≡ ΣN,p≡ {U|ZN : U ∈ Σdp}.
Under many circumstances, only a(proper) subset B of ΣN is admissible (allowable or feasible). In this case, local patterns in B are called basic patterns and B is called the basic set. In a one dimensional case, S consists of letters of an alphabet, and B is also called a set of allowable words of length N.
Consider a fixed finite lattice ZN and a given basic set B ⊂ ΣN. For larger finite lattice ZNe ⊃ ZN, the set of all local patterns on ZNe which can be generated by B is denoted as ΣNe(B). Indeed, ΣNe(B) can be characterized by
ΣNe(B) = { U ∈ ΣNe : Uα+N = VN f or any α ∈ Zd withZα+N ⊂ ZNe
and some VN ∈ B}, where
α+ N = {(α1+ β1,· · · , αd+ βd) : (β1,· · · , βd) ∈ N}, and
Uα+N = VN means uα+β = vβ f or each β ∈ ZN.
Similarly, the set of all global patterns which can be generated by B is denoted by
Σ(B) = {U ∈ Σdp : Uα+N = VN f or any α∈ Zd with some VN ∈ B}.
The following questions arise :
(1) Can we find a systematic means of constructing ΣNe(B) from B for ZNe ⊃ ZN?
(2) What is the complexity (or spatial entropy) of {P
Ne(B)}Ne≥N ? The spatial entropy h(B) of Σ(B) is defined as follows :
Let
(1.1.1) ΓNe(B) = card(ΣNe(B)),
the number of distinct patterns in ΣNe(B). The spatial entropy h(B) is defined as
(1.1.2) h(B) = lim
Ne→∞
1 Nf1· · · fNd
log ΓNe(B),
1.1. INTRODUCTION 3 where eN = (fN1, fN2, ..., fNd) be a d-tuple positive integers, which is well-defined and exists (e.g. [13]). The spatial entropy, which is an analogue to topological entropy in dynamical system, has been used to measure a kind of complexity in LDS (e.g. [13], [42] ).
In a one dimensional case, the above two questions can be answered by using transition matrix. Indeed, for a given basic set B, we can associate the transition matrix T(B) to B. Then the spatial entropy h(B) = log λ, where λ is the largest eigenvalue of T(B) (e.g. [29, 41]). On the other hand, for higher dimensional cases, constructing ΣNe(B) systematically and computing ΓNe(B) effectively for a large eN are extremely difficult.
In the two dimensional case, Chow et al. [13] estimated lower bounds of the spatial entropy for some problems in LDS. Later, using a ”building block” technique, Juang and Lin [29] studied the patterns generation and obtained lower bounds of the spatial entropy for CNN with square-cross or diagonal-cross templates. For CNN with general templates, Hsu et al [27] in-vestigated the generation of admissible local patterns and obtained the basic set for any parameter, i.e., the first step in studying the patterns generation problem. Meanwhile, given a set of symbols S and a pair consisting of a hori-zontal transition matrix H and a vertical transition matrix V, Juang et al [30]
defined m-th order transition matrices TH,V(m) and ¯TH,V(m) for each m ≥ 1 and, in doing so, obtained the recursion formulas for both TH,V(m) and ¯TH,V(m). Further-more, they proved that TH,V(m) and ¯TH,V(m) have the same maximum eigenvalue λm and spatial entropy h(H, V ) = lim
m→∞
log λm
m . For a certain class of H,V, the recursion formulas for TH,V(m) and ¯TH,V(m) yield recursion formulas for λm explic-itly and the exact entropy. On the other hand, for the patterns generation problem Lin and Yang [37] worked on the 3-cell L-shaped lattice, i.e., N=
. They developed an algorithm to investigate how patterns are generated on larger lattices from smaller one. Their algorithm treated all patterns in ΣNe(B) as entries and arranged them in a ”counting matrix” MNe(B). A good arrangement of MNe(B) implies an easier extension to MNee(B) for a larger lattice eeN ⊃ eN and effective counting of the number of elements in ΣNe(B).
Upper and lower bounds of spatial entropy were also obtained. Next, there are some relations with matrix shift [13], that details will appear in section 1.3.4.
Motivated by the counting matrix MN(B) of [37] and the recursion formu-las for transition matrices in [30], this work introduces the ”ordering matrix”
X2 for Σ2ℓ×2ℓ to study the patterns generation and obtain recursion formu-las for Xn for Σ2ℓ×nℓ where ℓ ≥ 1 is a fixed positive integer and n ≥ 2.
The recursion formulas for Xnimply the recursion formula for the associated
4 Pattern Generation Problems transition matrices Tn(B) of Σ2ℓ×nℓ(B), i.e., a generalization of the recursion formulas in [30]. Notably, a different ordering matrix eX2 for Σ2ℓ×2ℓ induces different recursion formulas of eXn for Σ2ℓ×nℓ and fTn(B). Among them, X2 defined in (1.2.9) yields a simple recursion formula (1.3.16) and rewriting rule (1.3.14), which enabling us to compute the maximum eigenvalue of Tn
effectively. The computations or estimates of λn are interesting problems in linear algebra and numerical linear algebra. Owing to the similarity prop-erty of (1.3.16) or (1.3.14) of transition matrices {Tn}∞n=2, we show that for a certain class of B, λn satisfies certain recursion relations and h(B) can be computed explicitly.
In d ≥ 3, the structure of ordering matrix and transition matrices has been explored, and it can be found in [3].
The rest of this paper is organized as follows. Section 1.2 describes a two dimensional case by thoroughly investigating Σ2×2 and introducing the ordering matrix X2 of patterns in Σ2×2. The ordering matrix Xn on Σ2×n is then constructed from X2 recursively. Finally, section 1.3 derives higher order transition matrices Tnfrom T2and computes λnexplicitly for a certain type of T2.