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 Cayley tree order $$k\geq 2$$ . This is defined by parameters $$\lambda_i$$ , $$i\in \mathbb N$$ We analyze functional equation which provides the consistency condition finite-dimensional distributions. Each solutions defines GGM version Kolmogorov’s theorem. define distributions as limit consistent family and show that, our HC tree, such distribution does not exist. Under some conditions $$k$$ find number translation-invariant two-periodic GGMs two.