  We construct logistic maps whose restriction to the omega-limit set of its
critical point is a minimal Cantor system having a prescribed number of
distinct ergodic and invariant probability measures. In fact, we show that
every metrizable Choquet simplex whose set of extreme points is compact and
totally disconnected can be realized as the set of invariant probability
measures of a minimal Cantor system corresponding to the restriction of a
logistic map to the omega-limit set of its critical point. Furthermore, we show
that such a logistic map $f$ can be taken so that each such invariant measure
has zero Lyapunov exponent and is an equilibrium state of $f$ for the potential
$-\ln |f'|$.
