Cayley树上PDMATHAAB-态Potts模型的周期$p$ -adic Gibbs测度：混沌暗示了$p$ -adic Gibbs测度的巨大性
Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures
摘要：
本文研究了三阶Cayley树上$q$ -态Potts模型的$p$-adic Gibbs测度集.通过证明$p\equiv 1 \ (\rm{mod} \ 3)$在$\mathbb{Q}\_p$上对应的Potts-Bethe映射的混沌行为，证明了该模型的周期$p$-adic Gibbs测度的广阔性.事实上，对于$0 < |θ-1|\_p < |q|\_p^2 < 1$，存在等距共轭到三个符号上的全移位的子系统。同时，对于$0 < |q|\_p^2 \leq |θ-1|\_p < |q|\_p < 1$，存在一个与$r$符号上的有限型子移位等距共轭的子系统，其中$r \geq 4$。然而，$r$符号上的这些子移位都与三个符号上的全移位拓扑共轭。讨论了$p$ -adic Gibbs测度对$p=2,3$和相应的Potts-Bethe映射的影响，并对$0 < |θ-1|\_p < |q|\_p < 1,$指出当$p=2,\ p=3$和$p\equiv 2 \ (\rm{mod} \ 3)$时Potts-Bethe映射不是混沌的，不能得出周期$p$ -adic Gibbs测度的广泛性.在即将发表的同名论文中，我们将讨论所有$p$的情况$0 < |q|\_p \leq |θ-1|\_p < 1$。
英文摘要：
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 $p\equiv 1 \ (\rm{mod} \ 3)$. In fact, for $0 < |θ-1|\_p < |q|\_p^2 < 1$, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for $0 < |q|\_p^2 \leq |θ-1|\_p < |q|\_p < 1$, there exists a subsystem that isometrically conjugate to a subshift of finite type on $r$ symbols where $r \geq 4$. 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 $0 < |θ-1|\_p < |q|\_p < 1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,\ p=3$ and $p\equiv 2 \ (\rm{mod} \ 3)$ 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 $0 < |q|\_p \leq |θ-1|\_p < 1$ for all $p$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
