BOUNDARY INFLUENCE ON THE ENTROPY OF A PROBLEM IN CELLULAR NEURAL NETWORKS
Yu‐Chuan Chang, Jong Juang · 2004
The purpose of this thesis is to shed some light on the open prob- lem raised by Afraimovich and Hsu (2003). Specifically, under some mild con- ditions, we show that for any '1 and n 2 N, except possibly a few pieces of T n '1, T n '1 is contained in an N-shaped tunnel for which its boundary point is an !-limit point of '1 for T. Moreover, we show under a stronger condition, see (3.3), that the entropy h'1,'2(T), see Definition 1.1, of T with respect to '1 and '2 is independent of the choice of '1. It is also shown that h(T) = hD(T) = hN(T) = ln3, where hD(T) and hN(T) are the entropy of T with respect to Dirichlet and Neuman boundary conditions, respectively, see Remark 1.1-(2), and that h'1,'2 (T)(= h'2 (T)) takes on two distinct values ln3 and 0. The necessary and sucient conditions on '2 for which h'2 (T) = ln3 are also obtained.