We adapt the analysis of HOD of L[x,G] as a strategy mouse to various admissible structures. Let us fix a natural number n assume that there is a Woodin cardinal. For a real x let a(x) be the least ordinal \beta such that L_\beta[x] is \Sigma_n-admissible and has an inaccessible cardinal and its successor exists. For a cone of reals x, we identify a mouse M and define a class H as a natural analogue of HOD of L[x,G], and show that H=N[Σ], where N is an iterate of M and Σ a fragment of its iteration strategy.
In the analysis of HOD of L[x,G] it is convenient to work with indescernibles of L[x,G]. We find an appropriate replacement for our context. This involves an analysis of a tree T searching for an illfounded structure whose wellfounded part is L_{a(x)} (R), where R is an appropriate subset of the reals. The analysis involves a variant of Ville's Truncation Lemma and a pruning of T which is related to the fine structure of L_{a(x)} (R).
This is joint work with Farmer Schlutzenberg.