  This paper discusses discrete-time maps of the form $x(k + 1) = F(x(k))$,
focussing on equilibrium points of such maps. Under some circumstances,
Lefschetz fixed-point theory can be used to establish the existence of a single
locally attractive equilibrium (which is sometimes globally attractive) when a
general property of local attractivity is known for any equilibrium. Problems
in social networks often involve such discrete-time systems, and we make an
application to one such problem.
