Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. Let $C_{p^n}$ the cycle subgroup of $S^{1}$ with $p^{n}$ elements.

Question 1 when the map from homotopy fixed points to fixed points
$$ THH^{C_{p}}(X)\rightarrow THH^{hC_{p}}(X)$$ is an equivalence (after $p$-compeltion)?

Question 2 what is the interpretation in algebraic geometry of the the groups $$ \pi_{i}THH(X)^{hS^{1}}$$ up to $p$-completion?