Periodic Points of a p-Adic Operator and their p-Adic Gibbs Measures

In this paper we investigate generalized Gibbs measure (GGM) for p-adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order k >= 2. This model is defined by p-adic parameters lambda(i), i is an element of N. We analyze p-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by p-adic version of Kolmogorov's theorem. We define p-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our p-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters p, k and lambda i we find the number of translation invariant and two-periodic GGMs for the p-adic HC model on the Cayley tree of order two.