In this present paper, we consider the p-adic Ising Vannimenus model on the Cayley tree of order two. A new measure-theoretical approach (in the p-adic sense) to investigate such a model is proposed. The main result of this paper is to establish the existence of the phase transition for the model. By the phase transition we mean the existence of at least two non-trivial p-adic quasi Gibbs measures, such that one is bounded and the second one is unbounded (note that in the p-adic probability, unlike a real setting, the probability measures could even be unbounded). To prove the main result, we investigate a nonlinear recurrence equation via the methods of p-adic analysis. Note that the methods used in the paper are not valid in a real setting.

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Abstract. In the present paper, by conducting research on the dynamics of the p-adic generalized Ising mapping corresponding to renormalization group associated with the p-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.Mathematics Subject Classification: 46S10, 82B26, 12J12, 39A70, 47H10, 60K35.

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We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.