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

• Main Authors: Ahmad, Mohd Ali Khameini, Liao, Lingmin, Saburov, Mansoor
• Format: Article
• Language: English; English; English
• Published: Springer New York LLC 2018
• Subjects: Q Science (General)
• Online Access: http://irep.iium.edu.my/65048/2/65048_Periodic%20p-adic%20Gibbs%20Measures_SCOPUS.pdf; http://irep.iium.edu.my/65048/3/65048_Periodic%20p-adic%20Gibbs%20Measures_WoS.pdf; http://irep.iium.edu.my/65048/19/65048_Periodic%20p-adic%20Gibbs%20measures%20of%20q-State_MYRA.pdf; http://irep.iium.edu.my/65048/; https://link.springer.com/article/10.1007/s10955-018-2053-6
• Institution: Universiti Islam Antarabangsa Malaysia

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 Qp for the prime numbers p≡1 (mod 3) . In fact, for 0<|θ−1|p<|q|2p<1 where θ=expp(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|2p≤|θ−1|p<|q|p<1 , there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where r≥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<|θ−1|p<|q|p<1, we remark that the Potts–Bethe mapping is not chaotic when p=3 and p≡2 (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≤|θ−1|p<1 for all prime numbers p.