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 $$k\geq 2$$ . This model is defined by $$p$$ -adic parameters $$\lambda_i$$ , $$i\in \mathbb N$$ . 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.

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