2017/08/07 by Ahmad, Mohd Ali Khameini, Liao, Lingmin, Saburov, Mansoor · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.02152
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 ℚ_p for p≡ 1 (\rmmod 3). In fact, for 0 < |θ-1|_p < |q|_p2 < 1, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for 0 < |q|_p2 ≤ |θ-1|_p < |q|_p < 1, there exists a subsystem that 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 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≡ 2 (\rmmod 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 ≤ |θ-1|_p < 1 for all p.