Multivalued entropy of supersymmetric black holes

The supersymmetric ow equations describing the ow of moduli from infinity t o the black hole horizon, and vice versa, are derived in the five-dimension al theories where the moduli space of the very special geometry has disjoin t branches. The multiple solutions are derived from the 'off the horizon' a ttractor equation. Within each branch, the black hole entropy, as usual, de pends only on the near horizon attractor values of moduli, i.e. the entropy depends on the charges and on coefficients of the cubic polynomial. It doe s not depend on the values of the moduli fields at infinity. However, the e ntropy, as well as the near horizon values of the moduli fields, are shown to depend on the choice of the branch specified by the choice of the set of moduli at infinity. We present examples of BPS black hole solutions with t he same Q(I) and C-IJK, whose entropies differ significantly.