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

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