turkmath.org

In this talk, I would like to discuss the pattern generation problems which arise from multiplicative integer systems. We investigate the systems by using a method that was developed for studying pattern generation problems in symbolic dynamical systems. The entropy 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 one-dimensional 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. The intrinsic difficulty in the study of the twodimensional coupled system will be demonstrated.