We show that for each positive integer n there is a real quadratic polynomial f whose postritical set is minimal and supports precisely n ergodic and invariant probability measures. We also show that there is a quadratic polynomial whose postcritical set is minimal and supports infinitely many ergodic and invariant probability measures. Furthermore we show that each of these measures is a equilibrum state of f with potential -ln|f'|, and thus that there are real quadratic polynomials with a prescribed number of such equilibrium states.
This is a joint work with Maria Isabel Cortez.