Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures

Abstract

We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on the Cayley tree of order three. We prove the vastness of the periodic $p$-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1(\mathrm{mod}3)$. In fact, for $0<|\theta -1|\_p<|q|\_p^2<1$, there exists a subsystem that 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 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 cases $p=2,3$ and the corresponding Potts--Bethe mapping are also discussed.Furthermore, for $0<|\theta -1|\_p<|q|\_p<1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,p=3$ and $p\equiv 2(\mathrm{mod}3)$ and we could not conclude the vastness 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 $p$.