  Let $k$ be an algebraically closed field of odd characteristic $p$ and $X$ a
proper smooth scheme over the Witt ring $W(k)$. To an object
$(M,Fil^{\cdot},\nabla,\Phi)$ in the Faltings category
$\mathcal{MF}^{\nabla}_{[0,n]}(X), n\leq p-2$, one associates an \'{e}tale
local system $\V$ over the generic fiber of $X$ and a Higgs bundle $(E,\theta)$
over $X$. Our motivation is to find the analogue of the classical Simpson
correspondence for the categories of subobjects of $\V$ and $(E,\theta)$. Our
main discovery in this paper is the notion of periodic Higgs subbundles, both
in positive characteristic and in mixed characteristic. In char $p$, it relies
on the inverse Cartier transform constructed by Ogus and Vologodsky in their
work on the char $p$ nonabelian Hodge theory. A lifting of the inverse Cartier
transform to mixed characteristic is constructed, which is used for the notion
of periodicity in mixed characteristic. We show a one to one correspondence
between the set of periodic Higgs subbundles of $(E,\theta)$ and the set of
\'{e}tale sub local systems of $\V\otimes_{\Z_{p}}\Z_{p^r}$, where $r$ is a
natural number. The notion turns out to be useful in applications. We have
proven, among other results, that the reduction $(E,\theta)_0$ of $(E,\theta)$
modulo $p$ is Higgs stable, if and only if, the corresponding representation
$\V$ is absolutely irreducible over $k$.
