FIXED POINT THEOREM, PERIODICITY THEOREMAND BINOMINALAND TRINOMINAL SEQUENCES FOR ITERATION DYNAMICAL SYSTEMS OF DISCRETE LAPLACIANS ON THE PLANE LATTICE

We treat mathematical problems for iteration dynamical systems of discrete Laplacians on the plane lattice. At first we prove a fixed point theorem for even neighborhoods and a periodicity theorem for odd neighborhoods of the dynamical systems, respectively. Then we are concemed with binominal and trinomial sequences associated the dynamical systems and comment some observations. 1. ITERATION DYNAMICAL SYSTEM OF DISCRETE LAPLACIAN We choose the plane lattice which is generated by two families of lines which are orthogonal to each other. We identify a cell of the lattice with one point of the cell, e.g., the midpoint. We call a set of cells which are attached with the reference cell a neighborhood. We call neighborhood even (or odd) if the number of the cells is even (respectively odd). We give several examples. Some of them are well known ([1])