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.