  Consider a Shimura variety of Hodge type admitting a smooth integral model S
at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the
Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the
pull-back to S(p^\infty) of the Ekedahl-Oort stratification on the mod p
special fiber of S and the pull back to S(p^\infty) of the fine Deligne-Lusztig
stratification on the mod p special fiber of the flag variety which is the
target of the Hodge-Tate period map. An application to the non-emptiness of
Ekedhal-Oort strata is provided.
