Malaysian Journal of Mathematical Sciences, May 2020, Vol. 14, No. 2
Dynamic Behaviors of \(p\)-adic Ising-Vannimenus Model on the Cayley Tree of Order Three
Dogan, M.
Corresponding Email: mutlay.dogan@ub.edu.bs
Received date: 12 March 2019
Accepted date: 21 March 2020
Abstract:
Recently, Ising-Vannimenus model on the Cayley tree of order \(k=3\) has been studied by Akin (2017) in a real case. In this study, we continue to investigate Ising-Vannimenus model on the Cayley tree of order \(k=3\) in \(p\)-adic sense. We investigate the dynamic aspects of \(p\)-adic Ising-Vannimenus model on the Cayley tree of order \(k=3\). We show that the recurrent equation 26, is associated to the model, has four non-trivial fixed points. And one of the fixed points lies in \(\varepsilon p\) and the rest of fixed points lie in \(\mathbb{Z}_{p}^{*}\). As a main result of the paper, we show that the fixed point \(u_0\) is attractive and the other fixed points \(u_1\), \(u_2\), \(u_3\) are repellent when \(u_{i}=p-1\), and neutral when \(u_{i}\neq p-1\).
Keywords: \(p\)-adic Gibbs measures, \(p\)-adic dynamical systems, Ising- Vannimenus model, Cayley tree
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