Periodic p-adic Gibbs Measures of q-State Potts Model on Cayley Trees I: The Chaos Implies the Vastness of the Set of p-Adic Gibbs Measures

Abstract

We study the set of p-adic Gibbs measures of the q-state Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic p-adic Gibbs measures for such model by showing the chaotic behavior of the corresponding Potts–Bethe mapping over $$\mathbb {Q}_p$$ for the prime numbers $$p\equiv 1 (\mathrm {mod} 3)$$ . In fact, for $$0< |\theta -1|_p< |q|_p^2 < 1$$ where $$\theta =\exp _p(J)$$ and J is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for $$0< |q|_p^2 \le |\theta -1|_p< |q|_p < 1$$ , there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where $$r \ge 4$$ . However, these subshifts on r symbols are all topologically conjugate to the full shift on three symbols. The p-adic Gibbs measures of the same model for the prime numbers $$p=2,3$$ and the corresponding Potts–Bethe mapping are also discussed. On the other hand, for $$0< |\theta -1|_p< |q|_p < 1,$$ we remark that the Potts–Bethe mapping is not chaotic when $$p=3$$ and $$p\equiv 2 (\mathrm {mod} 3)$$ and we could not conclude the vastness of the set of the periodic p-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $$0< |q|_p \le |\theta -1|_p < 1$$ for all prime numbers p.