  We look at the maximal entropy (MME) measure of the boundaries of connected
components of the Fatou set of a rational map of degree greater than or equal
to 2. We show that if there are infinitely many Fatou components, and if either
the Julia set is disconnected or the map is hyperbolic, then there can be at
most one Fatou component whose boundary has positive MME measure. We also
replace hyperbolicity by the more general hypothesis of geometric finiteness.
