  We characterize the existence of minimal idempotent ultrafilters (on N) in
the style of reverse mathematics and higher-order reverse mathematics using the
Auslander-Ellis theorem and variant thereof.
  We obtain that the existence of minimal idempotent ultrafilters restricted to
countable algebras of sets is equivalent to the Auslander-Ellis theorem (AET)
and that the existence of minimal idempotent ultrafilters as higher-order
objects is $\Pi^1_2$-conservative over a refinement of AET.
