We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on Cayley tree of order three. We prove the vastness of the set of the periodic $p$-adic Gibbs measures for such models by showing the chaotic behavior of the corresponding Potts–Bethe mapping over $\mathbb{Q}_p$ for prime numbers $p ≡ 1 (mod\ 3)$. In fact, for $0 < |\theta − 1|p <|q|^2_p < 1$ where $\theta = exp_p (J)$ and $J$ is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for $0 < |q|^2_p ≤|\theta − 1|_p < |q|_p < 1$, 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.
