  While the $\mu$-calculus notoriously subsumes Alternating-time Temporal Logic
(ATL), we show that the epistemic $\mu$-calculus does not subsume ATL with
imperfect information (ATL$_i$) for the synchronous perfect-recall semantics.
To prove this we first establish that jumping parity tree automata (JTA), a
recently introduced extension of alternating parity tree automata, are
expressively equivalent to the epistemic $\mu$-calculus, and this for any
knowledge semantics. Using this result we also show that, for bounded-memory
semantics, the epistemic $\mu$-calculus is not more expressive than the
standard $\mu$-calculus, and that its satisfiability problem is
EXPTIME-complete.
