Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

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. If time allowa, an application to the non-emptiness of Ekedhal-Oort strata is provided.