  In this contribution, we deal with the longtime behavior of the solutions to
the fractional variant of the Cahn-Hilliard system, with possibly singular
potentials, that we have recently investigated in the paper `Well-posedness and
regularity for a generalized fractional Cahn-Hilliard system' (see
arXiv:1804.11290). More precisely, we study the omega-limit of the phase
parameter and characterize it completely. Our characterization depends on the
first eigenvalue of one of the operators involved: if it is positive, then the
chemical potential vanishes at infinity and every element of the omega-limit is
a stationary solution to the phase equation; if, instead, the first eigenvalue
is 0, then every element of the omega-limit satisfies a problem containing a
real function related to the chemical potential. Such a function is nonunique
and time dependent, in general, as we show by an example. However, we give
sufficient conditions in order that this function be uniquely determined and
constant.
