Ahmad, Mohd Ali Khameini and Liao, Lingmin and Saburov, Mansoor
(2018)
Periodic padic Gibbs measures of qState potts model on Cayley Trees I: The chaos implies the vastness of the set of pAdic Gibbs measures.
Journal of Statistical Physics, 171 (6).
pp. 10001034.
ISSN 00224715
(In Press)
Abstract
We study the set of padic Gibbs measures of the qstate Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic padic 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−1p<q2p<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<q2p≤θ−1p<qp<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 padic 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−1p<qp<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 padic Gibbs measures. In a forthcoming paper with the same title, we will treat the case 0<qp≤θ−1p<1 for all prime numbers p.
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