  For certain families of complex maps, we give a formula for the Hausdorff
dimension of the equilibrium measure. In particular, given an endomorphism $f$
of $\mathbb C\mathbb P^k$ of algebraic degree $d \ge2$, and given the
equilibrium measure $\mu$ with Lyapunov exponents $\chi_1\geq \ldots\geq
\chi_k$, we show $\dim_\mathrm{H}(\mu) = \log d\sum_{i\leq k}\frac{1}{\chi_i}$
where $\dim_\mathrm{H}(\mu)$ is the Hausdorff dimension of the measure $\mu$.
This gives an answer to the question of Forn{\ae}ss and Sibony, and proves the
Binder-DeMarco Conjecture.
