  We employ the recently introduced conformal iterative construction of
Diffusion Limited Aggregates (DLA) to study the multifractal properties of the
harmonic measure. The support of the harmonic measure is obtained from a
dynamical process which is complementary to the iterative cluster growth. We
use this method to establish the existence of a series of random scaling
functions that yield, via the thermodynamic formalism of multifractals, the
generalized dimensions D(q) of DLA for q >= 1. The scaling function is
determined just by the last stages of the iterative growth process which are
relevant to the complementary dynamics. Using the scaling relation D(3) =
D(0)/2 we estimate the fractal dimension of DLA to be D(0) = 1.69 +- 0.03.
