Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and Z 2ℓ×2ℓ , ℓ ≥ 1, be a fixed finite square sublattice of Z 2 . Function U : Z 2ℓ×2ℓ → S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A 2 which is a q 2 × q 2 matrix, q = p ℓ 2 , can be defined. The recursive formulae of higher transition matrix An on Z 2ℓ×nℓ have already been derived [4]. Now A m n , 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 m n to A m n+1 recursively for n ≥ 2. Furthermore, the lower bound of spatial entropy h(A 2 ) 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 Tm of Cm can also be introduced. In the case of symmetric A 2 , T 2m gives a good estimate of the upper bound on spatial entropy. Combining Cm with Tm helps to understand the patterns generation problems more systematically.

The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width n are introduced for all n ≥ 2. To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps.

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

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Abstract. This study investigates a multiplicative integer system by using a method that was developed for studying pattern generation problems. The entropy and the Minkowski dimensions of general multiplicative systems can thus be computed. A multi-dimensional decoupled system is investigated in three main steps. (I) Identify the admissible lattices of the system; (II) compute the density of copies of admissible lattices of the same length, and (III) compute the number of admissible patterns on the admissible lattices.A coupled system can be decoupled by removing the multiplicative relation set and then performing procedures similar to those applied to a decoupled system. The admissible lattices are chosen to be the maximum graphs of different degrees which are mutually independent. The entropy can be obtained after the remaining error term is shown to approach zero as the degree of the admissible lattice tends to infinity.