  The spin-glass $q$-state Potts model on $d$-dimensional diamond hierarchical
lattices is investigated by an exact real space renormalization group scheme.
Above a critical dimension $d_l(q)$ for $q > 2$, the coupling constants
probability distribution flows to a low-temperature \emph{strange attractor} or
to the high-temperature paramagnetic fixed point, according to the temperature
is below or above the critical temperature $T_c(q,d)$. The strange attractor
was investigated considering four initial different distributions for $q = 3$
and $d = 5$ presenting strong robustness in shape and temperature interval
suggesting a condensed phase with algebraic decay.
