  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 $\mathbb{Q}\_p$ for $p\equiv 1 \
(\rm{mod} \ 3)$. In fact, for $0 < |\theta-1|\_p < |q|\_p^2 < 1$, there exists
a subsystem that isometrically conjugate to the full shift on three symbols.
Meanwhile, for $0 < |q|\_p^2 \leq |\theta-1|\_p < |q|\_p < 1$, there exists a
subsystem that isometrically conjugate to a subshift of finite type on $r$
symbols where $r \geq 4$. 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 $0 < |\theta-1|\_p <
|q|\_p < 1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,\
p=3$ and $p\equiv 2 \ (\rm{mod} \ 3)$ 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 $0 < |q|\_p \leq |\theta-1|\_p < 1$ for all $p$.
