Patterns generation and transition matrices in multi-dimensional lattice models

Patterns Generation and Spatial Entropy in Two-dimensional Lattice Models

“…This study extends our previous work [4]. For simplicity, two symbols on 2 × 2 lattice Z 2×2 are considered.…”

“…In the field of lattice dynamical systems (LDS) and cellular neural networks (CNN), the complexity of the set of all global patterns recently 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].…”

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

“…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.…”

Zeta Functions for Higher-Dimensional Shifts of Finite Type

“…Various works relate to this study, including zeta functions and related topics [Artin & Mazur, 1965;Ban et al, 2008a;Bowen & Lanford, 1970;Fried, 1987;Hardy & Ramanujan, 1918;Hardy & Wright, 1988;Kitchens & Schmidt, 1989;Ledrappier, 1978;Lind, 1996;Lind & Marcus, 1995;Lind et al, 1990;Manning, 1971;Markley & Paul, 1979Pollicott & Schmidt, 1996;Ruelle, 1978;Schmidt, 1995], patterns generation problems and lattice dynamical systems [Ban et al, 2001a[Ban et al, , 2002Ban & Lin, 2005;Ban et al, 2007Ban et al, , 2008bBan et al, , 2001bChow & Mallet-Paret, 1995;Chow et al, 1996aChow et al, , 1996bChua et al, 1995;Chua & Roska, 1993;Chua & Yang, 1988a, 1988bChua, 1998;Chua & Itoh, 2003Lin & Yang, 2000, and phase-transitions in statistical physics [Baxter, 1971[Baxter, , 1982Lieb, 1967aLieb, , 1967bOnsager, 1994].…”